PUZZLE CORNER

ALLAN J. GOTTLIEB, '67

Major League Eternity

This being the first issue of a calen

dar year, we again offer a "yearly

problem" in which you are to

express small integers in terms of the

digits of the new year (1,9,9, and 2) and

the arithmetic operators. The problem is

formally stated in the "Problems" sec

tion, and the solution to the 1991 yearly

problem is in the "Solutions" section.

Problems

Y199Z. Form as many as possible of the

integers from 1 to 100 using the digits 1,

9, 9, and 2 exactly once each and the

operators +,-,x(multiplication),/ (divi

sion), and exponentiation. We desire

solutions containing the minimum

number of operators; and, among solu

tions having a given number of opera

tors, those using the digits in the order

1, 9, 9, and 2 are preferred. Parenthesis

may be used for grouping; they do not

count as operators. A leading minus

sign does count as an operator.

JAN 1. Our "first" problem is a comput

er offering from Bob High. Write the

first n numbers in alphabetical (dictio

nary) orders as they are spelled out (i.e.,

one, two, three,...one million,...). To

avoid ambiguity, use no "ands" or

hyphens, so 837,301 would be written

"eight hundred thirty seven thousand

three hundred one". 1,897 is "one thou

sand eight hundred ninety seven," not

"eighteen hundred ninety seven."

Define two functions, F(m,n) and G(m,n)

as follows: F(m,n) is m th number in the

alphabetical list of the first n numbers;G(m,w) is the position of the number m

in this list. (For given n, F and G areinverses.) we ask:

(1) What is F(l,000,1,000,000)? What is

G(1,OOO, 1,000,000)?

(2) What is F(l,000,000, 1,000,000)?

What is G(1,000,000,1,000,000)?

(3) For what numbers n is F(n,n) =G(n,n)=n? List the first dozen.

JAN 2. Robert Bart offers the following

extensions to an old problem from Nob.

Yoshigahara. What is the smallest posi-

SEND PROBLEMS, SOLUTIONS.

AND COMMENTS TO ALLAN J. GOT

TLIEB, '67, THE COURANT INSTI

TUTE, NEWYORK UNIVERSITY, 251

MERCER ST., NEW YORK, N.Y. 10012,

OR TO: COTTUEB@NYU.EDU

tive integer whose square root has a

decimal expansion beginning with ten

distinct digits? Now consider cube roots

instead of square roots. Finally consider

ith roots for i = 4,5,... ,10.

Speed Department

Mark Astolfi wants you to use Major

League rules (with no rainouts) and

name five ways a baseball game can go

on forever.

Solutions

Y1991. The following solution is from David

Brahm, who writes that most of it was worked out

by computer, which ran through the 6 possible

number orderings, the 5 possible orders of operation, and the 7 possible operators (_, +, -, *, /, A, and

A-> in each of the 3 positions.

1 = 1A99! 20=19+lA9 80 = (l+9)*(9-l)

2 = 1A99 + 1 27 = 19 + 9-1 81=91-9-1

3 = 1+9/9 + 1 28 = 19 + 9M 82 = 91-9*1

7 = 9 + 9-11 29 = 19 + 9 + 1 83 = 1-9 + 91

8 = 9-lA19 36 = (l+l)*(9+9) 88 = 99-11

9 = 99/11 38=19 + 19 89 = (l+9)*9-l

10 = 19-9*1 63 = 9*(9-l-l) 90 = 11*9-9

11 = 19-9+1 64 = (9-l)'(9-l) 91 = 1^9*91

12 = 11+9/9 70 = 9*9-11 92 = 1A9 + 91

16 = 9 + 9-1-1 71=9*(9-1)-1 97 = 99-1-1

17 = 1*9 + 9-1 72 = 91-19 98 = 1*99-1

18 = 19-1A9 73 = l+9*(9-l) 99 = 1+99-1

19 = 19*1A9 79 = 9*9-1-1 100 = 1+99*1

A/S 1. We begin with a bridge problem from Don

Boynton who needs to make 7 Hearts against any

defense with an opening lead of the Queen ofClubs.

North

* 2

v 32

« AK2

* AK76543

West East

* K108 * 76543

» S v Q1087

* QJ109 ♦ 876

* QJ1098 * 2

South

* AQJ9

v AKJ964

* 543

Bart Bramley notes that "this is the famous

Vaniva problem, composed in 1928 by the great

player Sidney Lenz for a contest sponsored byVaniva shaving cream.'' David Gross felt this was a

"nice problem with lots of blind alleys." His solu

tion is to win the CA (discard D3) and play the CK.

East has a choice of 1) spade discard, 2) diamond

discard, 3) ruff. Each of these leads to a differentending.

1) After a spade discard play for a trump coup. Play

a low TRUMP on the CK, play the SA and the SQ

(and a 3rd spade if West does not cover) ruffing out

the SK. Then play the heart double finesse, run the

spades and the DA, take a club ruff with East dis

carding a diamond (best), and play the DK. Finally

use a club for the coup and all the tricks.

2) After a diamond discard there is a repeating

triple squeeze against West. Discard the S9 on the

CK, take a heart finesse, the DA, a heart finesse, and

run all the hearts. This will be the position with 1

heart to play:

North

* 2

K2

76

West

* K10

* QJ* j 10

East

* 7654

¥ -

« 8

* -

South

* AQJ

« 4

* 5

* -

On the last heart West cannot discard a spade, so

s/he discards a minor suit stopper while dummy

discards C6. Now South cashes the DK and

whichever minor suit card West has abandoned,

squeezing West in spades and the other minor.

3) After a ruff South can set up the spades via a

ruffing finesse and draw trumps. The play is: over-

ruff, SA, SQ (ruffing this or the 3rd round of spades

whenever West plays SK), heart finesse, drawtrumps and claim.

A/S 2. Thomas Weiss wants you to find a crossword

puzzle using as few squares as possible but satisfy

ing:

(1) All 26 letters of the English alphabet are used atleast once each.

(2) No proper nouns, abbreviations, contractions,

acronyms, or foreign words are used.

(3) All letters are used to form words both horizon

tally and vertically.

(4) Radial symmetry about the center is achieved,

as is common in American crossword puzzles.

I guess the winner is the following 7x7 solution

from an anonymous reader who stayed at theClarion Hotel in Cincinnati.

One might object to the use of a British dictionary

as representing a foreign language (or using two

scrabble dictionaries as representing two foreign

Continued on Page MIT 42

TEOK0L0CT REVIEW JUT 55

. il>l W'<X T l^PTY^f§^$;^$F*}

Fti'H's

"Cokinel^Powers,; '26;Gt, movedi1to■\Gaiifdrnia.andtsooh became...

■ inypIvecl-iirUie'motidnnpicturej ;iHl:

■1p'ictiireieciui'pmerit:!;in1'1940; he'wasrcalleajntT.);t.ne:seiyice as

capfeih in' th'e'llrrriy. Coips of.Engineers to te'ach;miiitaryengjmemig atvQregonState %

J^lll|fc;!^fed

''■;• i -;'■■.'" '.' liianflei;he'established the 3rd.Air-?Fprce Camouflage School'

^.-and served,as'campuflage offi-■'',. ce'rvfprahe. Army-Air Forces and

' "■■■..; -,as ..ehgineerlinspectm--generar■i'*"1 ■■ To'r'tiieiWestern-Regipn. He

■^ "i' -',' .'.i'etii;ed;from theVArmyafter ■:;i.:■*■}. ' '■■ .;,-&.: 'service-in itorea. '' !■■' ■'.'^^r':r'.O\: ^s:'..-,.■ v-o•;;■.. .v ■ :.----;-.- . . -..-

Wfeii■hjs'tS}ifesporisjaictep.ted a'position as professor of

■a? , Englisn«at Arizoh'a'State'University,■■Colonel Powerss;Xenrolled|m:a:cp,urse^n beginning Russian. .He also stud-

s.v ii;«iWlndlana;yhiversi(.y arfdfVvent pii tb'earn the firstArizona SlafeijUniversijy. ;e^er awarded. ..:';

i.i.e wwju&DfoHonmtyfrjfRussian;tfrcgu-. '■>.;'1 .'^'lav.yvHvffri'iris and;'with'hisiwife, an English translation1-$>/;''of jfcpppular Russiaiv novel-, and has translated Russian.

;j t.eclmi'eal)articles!,Chpl6nei Powers has,been a-member ofijil^,Sp^i.ety|p|MotipnjRiGture Engineersjanjd^of the lius-

i":V^jSn^ifp1riprary^'sJocietyipbbf.pslovc),'an(l'is^ /'htf:itlv(>r'A'ri5riria..(S!tori|ei!"of;'the-'AnieHcan Association of

; in v-uJoiado and',

.■■;'; ■»' ; .'."»;■,■

b pursue a' ',

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.ALUMNEWS,

Continuedfrom pageMTT7

the black astronaut who lost his life in the

Gtallenger explosion.

•Stanley Rose, an MIT telethon volun

teer in New York City for more than 15

years, has been remarkably effective in

enlisting project support from his em

ployer. Bankers Trust, notably in provid

ing a site and catering the callers' dinner.

Presidential Citations honor unusual

service to the Institute and the Alumni/ae

body by its member organizations. This

has been a blockbuster year, with citations

going to six alumni/ae groups:

•The Class of 1950 Student Aid Fund,

established at the class's 25th Reunion,

now totals more than $2 million and is

MIT's largest class-supported financial

aid fund. The fund has helped hundreds

of students and their families by provid

ing scholarships and subsidizing loan in

terest.

•The MIT Club of Cape Cod, founded

in 1976, has demonstrated the best that

clubs can offer their members, through an

annual directory, quarterly newsletter,

broad range of speakers, and notable ef

forts to reach out to all members of the

MITcommunity in the region.

• In the three short years that it has been

in operation, the Young Alumni Steering

Committee of Boston has done an exem

plary job of organizing events tailored to

the interests of graduates from the past 10

years.

•Since Greg Turner '74 led a revival of

the MIT Club of South Texas seven years

ago, the club has grown steadily and has

established a positive MIT presence in

Houston.

•The MIT Club of Boston was honored

for its black-tie gala at the Boston Muse

um of Fine Arts to welcome President

Charles Vest and Becky Vest. The event

attracted 450 guests.

•The Productivity Commission "Road

Show," a collaboration of the Alumni/ae

Association staff, MIT clubs in eight cities,

and the MIT Industrial Liaison Program,

was honored for its success in bringing

the Made in America report of the MIT

Commission on Industrial Productivity to

more than 1,600 attendees coast to coast.

The Road Show attracted national media

attention and demonstrated MIT's con

cern for the national agenda to a wide au

dience.—Judith Norkin. □

PUZZLE CORNER.

Continued from Page MIT55

languages. Eugene Sard has a 9x9 solution butneeded an unabridged dictionary for threewords. The following 9x9 from the proposer

used only Webster's Students Dictionary andRandom House College Dictionary.

A

R

K

S

■F

A

R

E

Q

U

1

T

IO

V

E

N

U

N

0

O

1B

O

N

0

A/S 3. Our last regular problem is from Nob.

Yoshigahara. Choose two digits excluding 0

and 1 and consider the set of numbers that contain each of the two digits at least once. For

example, 4 and 8 gives 8848, 4884, 84 and

infinitely many others. Now consider the small

est member of this set that is a multiple of the

two original digits. Call this the LYM (least

Yoshigahara multiple). In our example theLYM is 48; the LYM of 3 and 5 is 3555. Among

the 28 pairs of digits, 4 lead to sets that do notcontain a multiple of the digits and, for these

pairs, the LYM is not defined. For example) all

multiples of 2 and 5 end is 0 so are not in the

set constructed from 2 and 5. The LYM of 2 and

4 is 24, which is the smallest of the LYMs. Whatis the largest?

Apparently considerable searching, guided

by some heuristics such as "casting out nines,"

was needed for this problem. Farrel Powsner

found this question to be a good exercise for

teaching problem solving to high school stu

dents. Powsner's results were as follows.

23-2232

24-24 (smallest)

25 - none

26-2226

27-2772

28-2888

29-;

34-3444

35-3555

36-36

37-37737

38-3888

39-3339

45 - none

Other Respondere

46-4464

47-44744

48-48

49-4,444,444,944

56 •none

57-5775

58-none

59-5,555,555,595

67-76776

68-6888

69-6696

78-7,888,888

79 - 77777,779,779 (largest)

89-8,888,889,888

Responses have also been received from R.

Bart, D. Boynton, S. Bragg, B. Bromley, D.

Church N. Cooke J. Cronin, C. Dale, L Daley, J.

Drumheller, S. Feldman, M. Fountain, J.

Grossman, J. Harmse, W. Hartford, R. High, K.

Kicsel, D. McMahon, A. Ornstein, D. Plass F.

Powsner, G. Ropes, K. Rosato, J. Rudy, D.

Savage, A. Tracht, and H. Zaremba,

Proposer's Solution to Speed Problem

One can have a never-ending tie, constant scor

ing (without making 3 outs), an infinite series

of foul balls, an infinite series of unsuccessfulpickoff attempts, or an eternal rundown.

MIT 42 JANUARY 1992

entitled "Life as Technology: Representing,

Intervening, and Molecularizing" to the History

of Science Group at the University of California at

Berkeley on October 7 Professor Carl Kaysen

was one of three American participants in a semi

nar on "New Elites in Eastern Europe: Who Will

Lead the New Market Economy?" held at Central

European University Center in Prague, in August

1991. Kaysen is a coauthor with DACS Colleague

George Rathjens and Robert McNamara, former

U.S. Secretary of Defense, of "Nuclear Weapons

After the Cold War," published in the current

issue of Foreign Affairs. Kaysen has spent the fall

term at the Institute for Scientific Interchanges in

Turin, Italy.... Professor Kenneth Keniston has

become a member of the NASA Workshop on

Cultural Aspects of SET! (Search for Extra-

Terrestrial Intelligence). The workshop explores

the "effects on humanity of detection of a signal

from an extraterrestrial civilization." NASA's

large-scale investigation will begin on Columbus

Day, 1992.

Professor Leo Marx presented a talk entitled

"Environmental Degradation and the Ambiguous

Social Role of Science and Technology" at the con

ference on Social, Cultural, and Political

Dimensions of the Environmental Crisis in the

U.S. and U.S.S.R. in Moscow Professor

Theodore Postol, '67, SM '72, PhD '76, attended

the International Summer School on Science and

World Affairs in Moscow at the end of June. In

August, he spoke on strategic nuclear weapons to

the National Security Fellows at Harvard

University. He was also a discussant at the Naval

War College First Annual Force Planning

Conference on "Strategy and Forces for a

Changing Security Environment" in Newport, R.I.

Postol's op ed piece, "Whoops! Patriot Missile

Sputters" appeared in the San Diego Tribune on

July 18 Institute Professor Walter Rosenblith

participated in a symposium in Montreal on the

early history of the International Brain Research

Organization (IBRO). He had been an officer of

IBRO in the early 1960s, when IBRO's individual

membership amounted to several hundred. Now

IBRO plays the role of a world federation of near

ly 30,000 neuroscicntists and of the societies in

which they are grouped. In June Rosenblith went

to Beijing as chair of the advisory panel to the

Chinese University Development Project to deliv

er an evaluation report to Li Tieying, head of the

State Education Ministry.... Professor Sherry

Turkle announced the arrival of a daughter,

Rebecca Ellen Willard, in May.

Carcn Addis, an STS graduate student, gave

birth to a son, Gabriel, on August 15. On hand for

the event was her husband and fellow student,

Antonio Botelho Patricia Bentley, gave birth

in May to a daughter, Kathryn Eric

Kupferberg presented a paper, "The Bergey's

Manual and the Solidification of Bacterial

Taxonomy," at the International Conference for

the History, Philosophy, and Social Study of

Biology, July 10-14 at Northwestern University.

He also attended a two-week course on "The

History of Modern Evolutionary Thought" held at

the Marine Biological Laboratory in Woods Hole

and sponsored in part by the Dibncr Institute for

the History of Science Steven Rcber, '81, is

the proud father of baby Charlotte, who arrived

last spring Jessica Wang married MIT

Economics student Brian Sliker on August 23.

Jessica and Brian met as office-mates in the STS

students' office.... Cheng-lin Zhang has been

awarded the Sir RunRun Shaw Fellowship. This

two-year award is funded by the Shaw

Foundation in Hong Kong, and is earmarked for

support of students from the People's Republic of

China who have "a bent towards science and

management courses."—Phyllis Klein, STS

Program, MIT, Rm. E51-128, Cambridge, MA

02139.

DeceasedThe following deaths have been reported to the

Alumni/ae Association since the Review last went

to press: '

Stanford A. Guthric, '15

John Homan, 'IS; August 9,1991; Indian Rocks

Beach, Fla.

John H. Holton, '17; September 1,1991;

Skanealeles, N.Y.

George F. Halfacre, '18; August 9,1991; Palmerton,

Penn.

Edward A. Merrill, '22; August 15,1991; Tulsa,

Okla.

Charles A. Nicholson, II, '22; July 21,1991; Laguna

Niguel, Calif.

Robert Prescott, '22; August 1,1991; Wayne, N.J.

Thomas F. Richardson, '23; September 5,1991

Robert C. Sprague, '23; September 27.1991;

Williamstown, Mass.

Edwin D. Wilson, '23; May 21,1991; Everett, Wash.

Clarence Edwin Miller, '24; August 7,1991;

Hampton, Vir.

Paul E. Pihl, '24; July 18,1991; Charleston, S.C.

Maurice Grushky, '25; December 24,1988;

Charleston, S.C.

John E McMaster, 76; September?, 1991;

Warcham, Mass.

William Crighton Sessions, '26; August 29,1991;

Grosse Pointc Woods, Mich.

Jack B. Peters, '27; September 10,1991; Lenox,

Mass.

Charles St. George Pope, '27; July 7,1991;

Richmond, Vir.

George Clahane, '28; August 29,1991; Acton, Mass.

Harry E. Shoemaker, '29; February 16,1991;

Temple City, Calif.

Philetus H. Holt, '30; August 30,1991; Summit N.J.

Willard A. Morairt, '30; July 23,1991; Mt. Vernon,

Ohio

Carlton E. Vanderwarker, '30; August 21,1991;

Boca Grande, Fla.

Clement H. Hamblet, '31; August 13,1991;

Newark, Dela.

Murray J. Lalone, '31; Closter, N.J.

Alfred Ziegler, '31; September 10,1991; Palmer,

Mass.

Gerard M. Kincade, '33; June 20,1991; Pittsburgh.

Penn.

Waldron S. MacDonald, '33; August 30,1991;

Concord, Mass.

James H. Burke, '34; September 19,1991; St. Simons

Island, Ga.

Stunner B. Sweetser, '34; August 11,1991;

Farmington, Conn.

Florence J. Driscoll, '35; S. Easton, Mass.

John W. Aldrich, '37; August 25.1991; Woodbury,

Conn.

Charles Frederick Healey, '37; August 1,1991;

Jamesburg, N.J.

Haskell R. Gordon, '38; September 17,1991;

Worcester, Mass.

George E. Shea, '38; August 26,1991; Bedford,

Mass.

Harold H. Butler, '39; May 2,1991; Nutley, N.J.

Mortimer A. Schultz, '39; June 29,1991; Mashpec,

Mass.

Yuen N. Lee, '39; June 21,1991; San Francisco, Calif.

Earle D. Benson, '40; August 26,1991; Reading,

Mass.

Augustus Peter Norton, '40; February 22,1991;

Stafford, Vir.

George W. Clark, '41; August 9,1991; Topsfield,

Mass.

William H. Yocotn, '42; August 10,1991;

Bethlehem, Penn.

Warren A. Hurley, '47; August 15,1990;

Hendcrsonville, N.C.

William Atkinson Schmidt, '48; August 19,1991;

Mansfield, Mass.

John Elmer Dacey, '49; May 15,1991; Sarasota, Fla.

Frank G. Lane, '49; September 3,1991; Middlebury,

VI.

Kenneth R. Grcider, '50; July, 1990; Davis, Calif.

Douglas H. Martin, '50; August 1,1991; Allentown,

Penn.

Dellieware R. Nelson, '51; August 25,1991;

Cambridge, Mass.

Eugene Yu-Cheng Hsi, '54; July 28,1991; Baltimore,

Md.

Bryant Gunsenhouser, '56; August 3,1991;

Anderson, Ind.

Richard Lee Unruh, '56; July 26,1991; Long Beach,

Calif.

John William Holland, '59; July 9,1991; La Jolla,

Calif.

Margaret LA. MacVicar, '65; September 30,1991;

Cambridge, Mass.

H. Kent Bowden, '68; September 4,1991; Houston,

Tex.

Percy G. Holton, '69; May 7,1991; Devon, England

Derek G. Ryan, '69; January 3.1990; Elbirlle, West

Germany

Robert B. Foster, '77; September 12,1991; Chicago,

III.

John Dennett Urmston, 78; August 8,1991;

Bolinas, Calif.

Winston Henry Francis, '86; December 31,1989;

Sao Paulo, Brazil

Thomas Rolf Petersen, '90; August 1,1991;

Munich, Germany

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TKCHNOLOGY REVIEW MIT 41

PUZZLE CORNER

ALLAN J. GOTTLIEB, '67

Objet Dart

This June is my class's 25th reunionand I was asked to write a page of

"Puzzle Corner" for the book they

are producing. In preparing for the col

umn I reread the introductions from the

first seven years of "Puzzle Corner" and

I must confess that it was fun to escape

back to those earlier, more carefreedays. I found some of my old words

touching, some boring, and occasionally

some were quite surprising. I guess we

really do change. If any of you have

little vignettes of your past life storedaway, I recommend that, when no one

else is looking, you take some privatetime with your former self.

Problems

F/M 1. Dave Wachsman sent us a hand

he played (as South) with his wife thatwas reported in Truscott's column in

the New York Times.

North

82

A72

Q75

KQ942

West

♦ 973

¥ J865

♦ KJ1032

♦ J

East

♦ 65

¥10 4

♦ 984

♦ A108 653

South

♦ AKQJ104

¥ KQ93

♦ A6

♦ 7

Both sides were vulnerable. The bidding:

South West North East1♦ Pass 2 * Pass

3* Pass 3N.T. Pass

4 ¥ Pass 4 N.T. Pass

6 * Pass Pass Pass

How does Mr. Wachsman bring homethe slam after West leads the dub jack?

SEND PROBLEMS, SOLUTIONS,

AND COMMENTS TO ALLAN]. GOT

TLIEB, '67, THE COURANTINSTI

TUTE, NEWYORK UNIVERSITY, 251

MERCER ST., NEWYORK N.Y. 10012,

OR TO: COTTLIEB@NYU.EDU

F/M 2. John Prussing believes that thefollowing puzzle, which was actually on

the 1989 Putnam exam, seems aboutright for "Puzzle Corner."

A dart hits a square dartboard. If any

two points on the dartboard have the

same probability of being hit, what is

the probability that the dart will landnearer to the center of the board than it

does to an edge?

F/M 3. Our last problem is from my

NYU colleague, Dennis Shasha, and can

be found in his book, The PuzzlingAdventures ofDr. Ecco.

You are given 20 coins. Some are fake

and some are real. If a coin is real, it

weighs between 11 and 11.1 grams. If it

is fake, it weighs between 10.6 and 10.7

grams. You are allowed 15 weighings

on a scale (not a balance). You are to

determine which coins are real andwhich are fake.

Speed Department

Speedy Jim Landau wants to know why

the Kindom of Metrica chose to use along, flat piece of wood as their RoyalStandard of Length.

Solutions

OCT1. We start with a chess problem (which may

well be a computer problem) from Victor Barocas.

It is well known that a knight can tour the chess

board, touching each square once and only once,

and beginning and ending on the same square.

Consider now the generalized knight K(m,n), m <

n, which moves m spaces along one axis of the

board and n spaces along the other (the normal

knight is K(l,2); also see diagram). For what val

ues ofm and n can the knight tour the board?

•

•

•

•

K

•

•

•

•

Moves available to K<2,3) at position e4.

The following solution is from Ken Kiesel:

Each time the knight moves one space in eitherdirection, it,moves to a square of the oppositecolor. Therefore, m + n must be odd, or the knight

can reach only squares of one color.

A knight on one of the four center squares can

move a maximum distance of 4,4. Thus, the possi

ble solutions are 0,1 0,3 1,2 1,4 2,3 and 3,4.

K(0,l) is the only solution besides K(1,2), unless

there is a bug in my program. K(0,3) obviouslydoesn't work.

Each corner square is accessible to only 2

squares. Therefore, one of these squares must be

used to enter the corner, and the other to exit it.

For K(3,4), diagonally opposite corners reach the

same 2 squares; therefore it cannot tour the board.

This leaves only K(l,4) and K (2,3).

The program I wrote found a solution to K(l,2)

in about 7 seconds on my XT clone. It eliminated

K(2,3) in less than that. In fact, K(2,3) can be

shown not to work quite easily. Starting at a8, the

possible first moves are to c5 or d6. Since the two

are equivalent, choose d6. If the second move is

not to g8, g8 will be left with only one accesible

square, f5. It will be impossible to enter and exit it

later. Therefore, the second move must be to g8.

The only possible third move is to f5. If the fourthmove is not to h7, then it will have only one entry

square, f4. But if it is to h7, then b7 will have only

one entry square, c4.

With two rapid results from the program, plus a

test of the trivial K(0,l) and K(0,3) to further veri

fy program operation, I launched it on K(l,4). It

finished after about 80 hours, having found nosolution.

OCT 2. Gordon Rice wonders how many Pythag

orean triangles you can find in which one of the

three sides is 1991.

The following is from Jerry Grossman, who has

secret plans involving this problem:

There are five Pythagorean triangles one of whose

sides is 1991: (10860,1991,11041), (1980,209,1991),

(180180,1991,180191), (1982040,1991,1982041), and

(16320,1991,16441). The "easiest" way to learn thisis by asking a computer algebra package to solve

(in integers) x2+y2=19912 and u2+19912=v2. Iasked Maple and it gave me these answers. Hereis how to do it more or less by hand:

Since 1991=11M81, we can look for three kindsof solutions: primitive solutions (i.e., no common

factor to the three sides) in which one side is 1991,

solutions in which one side is 11 (and then multi

ply all sides by 181 to obtain the desired triangle),and solutions in which one side is 161 (then multiply by 11).

The fundamental fact we need to use is that all

primitive triangles are of the form (2mn, m2-n2,

nA-n2), where m and n are relatively prime positive integers.

Let's start with the solutions in which one side

is 11. Clearly 2mn cannot equal 11, nor can m2+n2.

So the only possibility here is that m=6 and n=5,

giving us the triangle (60,11,61), and hence the

solution to the original problem (60,ll,61)*181=(10860,1991,11041).

Next we look for solutions in which one side is

181. Again we cannot have 181=2mn, but we can

have 181=m2+n2, with m=10 and n=9 (this isthe only way). This gives a triangle (180,19,181),yielding the second solution (1980,209,1991).We can also have 181=m2-n2, only by lettingm=91 and n=90. This gives us the triangle

(16380,181,16381), and hence our third solution(180180,1991,180191).

Finally we look for primitive solutions with one

side being 1991. Again, 1991 is not 2mn. A com

puter search shows that 1991 cannot equal m2+n2.

Continued on Page MET 38

TECHNOLOGY REVIEW MIT 39

Symposium on Kids and Computers" in late

September. She delivered the keynote address for

the Conference on Cultural Issues in

Psychoanalytic Theory in New York in early

October.... Jessica Wang and Bruce Bimber, STS

graduate students, shared the first Siegel Prize for

their essay on the relationship of science and tech

nology to political events in the U.S. The prize

was established in memory of Benjamin M. Siegel,

'38, PhD '40, a pioneer in electron microscopy and

until his death in 1990, professor emeritus of

applied engineering and physics at Cornell.

Wang's essay recounted the security problems of

the scientist E.U. Condon during the loyalty

investigations of the McCarthy period. Bimber's

essay dealt with the development, philosophy,

and politics of the Office of Technology

Assessment Barbara Masl, '86 (II), SM '88.

Wade Roush, and Ken White presented papers

on "Boundaries and Paradigms in Artificial

Intelligence Research" at the 4S meeting

November 15 at the Cambridge Center

Marriott.—Phyllis Klein, STS Program, MIT, Rm.

E51-128, Cambridge, MA 02139.

Deceased

The following deaths have been reported to the

Alumni/ae Association since the Review last went

to press:

G. Hobart Stebbins, '17; September 26,1991;

Bellevue, Wash.

Henry R. Lacey, '18; March 13,1991; Melbourne,

Fla.

Webster W. Frymoyer, '21; October 5,1991;

Arlington Heights, 111.

Eastman Smith, 72; September 18,1991;

Mountain Home, Ark.

Richard H. Frazier, '23; October 24,1991;

Winchester, Mass.

George H. Fuller, '25; May 30,1991; Colonia, NJ.

Hyman Katz, '25; January 6,1987; North Quincy,

Mass.

Seward S. Merrell, '25; October 5,1991; Saint

Petersburg, Fla.

W. Alan Williamson, '26; September 19,1991

Marion E. Knowles, '27; September 18,1991;

Akron, Ohio

Ralph W. Stober, '27; October 1,1991; Newton

Highlands, Mass.

Howard S. Root, '28; September 24,1991;

Harrington Sound, Bermuda

George A.Roman &

Associates Inc.

Architecture, Planning,

Interior Doslgn

Institutions!

Commercial

Industrial

Residential

Site Evaluation

Land Uso Planning

Master Planning

Programming

Interior Space

Planning

Colleges

Hospitals

Medical Buildings

OHIca Buildings.

Apartments

Condominiums

Oeorrjo A. Roman,

A.I.A. -SS

Donald W. Mills, '84

One Gateway Center

Newton, MA 021SS

(617) 332-5427

Henry D. Addison, '30; October 10,1991;

Northampton, Mass.

Robert A. Lytle, '30; June 14,1991; Grosse Point,

Mich.

Watson E. Slabaugh, '30; August 31,1991;

Mansfield, Ohio

D. Malcolm Fleming, '33; August 12,1991;

Rockvillc Centre, N.Y.

Charles N. Debes, '35; August 31,1991; Rockford,

111.

Joseph K. Racs, '35; August 3,1991; Las Vegas,

Nev.

Albert J. Del Favero, '36; September 18,1991;

Vista, Calif.

O. William Muckenhim, '37; September 2,1991;

Toledo, Ohio

Robert D. Williams, '37; September 17,1991; Lake

Helen, Fla.

Robert R. Chase, '39; January 12,1991; Austin,

Tex.

Joseph W. Harrison, '39; October 14,1991; New

London, N.H.

Charles A. Lawrence, '39; July 12,1991; Seattle,

Wash.

Edward A. Ruckner, '41; September 12,1991;

Annapolis, Md.

David G. Edwards, '42; December 29,1990;

Pacific Grove, Calif.

Anthony P. Batbato, '44; July 17,1991; Kettering,

Ohio

Warren H. Howard, '44; September 29,1991;

Sunapce, N.H.

Louis H. Roddls, Jr., '44; September 15,1991;

Charleston, S.C.

Henry F. Lloyd, '46; June 12,1991; St. Augustine,

Ha.

Wilton M. Fraser, '47; February 14,1991; Naples,

Fla.

C. Gregory Bassett, Jr., '48; September 25,1991;

Hilton Head Island, S.C.

Aubert L. Mowry, '48; March 16,1991;

Northridge, Calif.

Warren W. Houghton, '49; October 20,1991;

Manchester, Mass.

Philip A. Lynn, '49; October 29,1991; Reading,

Mass.

William B. Martz, '50; May 2,1991; Winchester,

Mass.

Douglas Porter, '50; June 10,1991; Watertown,

Mass.

Morgan L Foster, '51; January 11,1990;

Meadville, Pa.

Peler Bishop, '54; September 29,1991; Falmouth,

Maine

Francisco Torras, '54; October 3,1991; Fairfield,

Conn.

Joseph A. Kissinger, Jr., '55; June 1,1990; La

Habra, Va.

J. William A. Tyler, '55; February 1,1990;

Monroe, Ohio

John A. Welsh, '55; September 26,1991;

Richardson, Tex.

Herbert Curt Burrowes, Jr., '56; October 1,1991;

Concord, Mass.

Stanley L Lopata, '56; October 9,1991; Natick,

Mass.

Charles V. Brown, '58; July 27,1991; Bridge of

Allan, Stirling, Scotland

Marvin H. Cantor, '59; May 16,1991; Rockville,

Md.

Irving Levinson, '61; February 26,1991; Silver

Spring, Md.

Frederik O. Jeppesen. '64; April 23,1991;

Denmark

Elsie P. Arnold, '65, February 25,1991; Richmond,

Ontario

George T. Onega, 70; March 23,1990; Slidell, La.

Ronald G. Jackson, 71; June 1,1991; Pytnble,

New South Wales, Australia

Daniel R. Siegel, '83; October 20,1991; Wilmette,

m.

Peter L. Armstrong, '64; August 27,1991; Waban,

Mass.

PUZZLE CORNER.

Continued from Page MIT 39If 1991=m2-n2=(m+n)(m-n), then cither wehave m+n=1991, with m-n=l; or we have

m+n=181 with m-n=ll. This gives m=996,

n=995 in the first case; m=96, n=85 in the sec

ond. The resulting triangles arc our fourth

and fifth solutions: (1982040,1991,1982041)

and (16320,1991,16441).

OCT 3. That famous riverboat gambler. Bob

High, was inspired by 1989 JUL 5 to ask a

two-part question about shuffling cards. First,

in a shuffled deck, what is the average

(expected) number of cards occupying their

original position? (This is to ask, for n = 52,

what is the average number of fixed points of

a permutation of n things.) Second, which is

more likely in a random shuffle (permutation)

of n things: exactly one fixed point, or exactly

none?

Gordon Rice has a fine analytic proof, a

copy of which can be obtained from Faith

Hruby at Technology Review. Curiously, Rice

was in the process of formulating a similar

problem when he read OCT 3. The following

shorter solution is from John Chandler, who

believes he might be a 25-year veteran as a

reader of "Puzzle Corner."

Consider any specific card. After a random

izing shuffle, its chance of being in its original

position is simply 1/52. Obviously, the a pos

teriori probability of a second card being in

'its* original position will depend on the actu

al position of the first card. Still, there is no

preferred treatment of any of the cards, so the

overall expectation of cards remaining in

position after the shuffle must be just 52 x

1/52 = 1. For n « 52, this is easily proven by

enumeration of all the permutations of n

things. Moreover, it is simple to write down

recursion formulas for the count of permuta

tions with a given number of fixed points in

terms of the counts for smaller n. For exam

ple, N(n,l) = N(n-l,0) » n, N(n,2) = N(n-2,0) •

n * (n-1) / 2, and so on. The table begins:

n= 012345 6 7

f=0 1 0 1 2 9 44 265 1854

1 1 0 3 8 45 264 1855

2 1 0 6 20 135 924

3 1 0 10 40 315

4 1 0 15 70

5 1 0 21

6 1 0

7 1

This suggests a further formula: N(n,0) =>

N(n,l) + (-l)n, and that can, in fact, be proved

by induction. The answer to the second ques

tion is, thus, that exactly one fixed point is

(slightly) more probable if n is odd and exact

ly none if n is even.

Better Late Than Never

OCT SD. Dan Drucker notes a typo: S should

bc(D-1)/2not(D+l)/Z

Other Responders

Responses have also been received from

Matthew Fountain, Coe Wadelin, Mayer

Wantman, Frank Carbin, Winslow Hartford,

John Woolston, Eric Lund, Jim Landau, Ken

Rosato, Steven Feldman, Rolph Person,

William Waite, Ronnie Haige, Eugene Sard,

Max Hailperin, Warren Jasper, Scott

Berhenblit, Avi Omstein, Thomas Lewis, AlanFriot.

Proposer's Solution to Speed Problem

Because a plank's constant.

MIT 58 FEBRUARY/MARCH 1992

chips. Sieve and Lesli Oaklander have set their

wedding date for May 24,1992 at the Boca Raton

Hotel and resort in Florida Edward Wong, SM

'91, has a new position as staff engineer at theSouthern California Gas Co.—Fay Wallslrom,

Management of Technology Program, MIT Rm.

E56-304, Cambridge, MA 02139.

XVIAERONAUTICS AND

ASTRONAUTICS

From Oslo, Norway, Tore Christiansen, SM '82,

reports: "After nine years of working for Det

Norske Veritas ship classification society, I have

returned to student life at Stanford University

(oops!) to pursue a PhD in civil engineering (oops

again!). My studies will be in the area of construc

tion management, with thesis work on application

of communication theory and information engi

neering to the organization and management of

large multi-functional design teams."... Byron K.

Lichtcnberg, SM 75, ScD '79, writes: "I'm in the

final stages of training for my second space flight

scheduled for March 1992. Our crew of seven will

be conducting experiments investigating the

upper atmosphere, the impact of solar energy on

the Earth, and natural and artificial auroras."...

Marc P. Genain, SM 75, is working on the opera

tional and ergonomic design of the Channel

Tunnel control centers. The operations project is

slated to be completed in August 1993 From

Concord, Mass., Edward B. Bokhour, SM '88,

reports: "My spaceflight experiment flew success

fully aboard STS-48 Discovery this past September.

I was hardware development manager at Payload

Systems for the MODE experiment, under con

tract to M1T-SERC. MODE was structural and

fluid dynamics."... From Big Bear Lake, Calif.,

Creighton Cook, SM '54, sends word that he con

tinues as an aerospace consultant in the Southern

California area Charles Vehlow, SM '77,

writes: "I have been selected as the Longbow

Apache Program Director at McDonnell Douglas

Helicopter Co. located in Mesa, Ariz."... In the

November/December 1991 issue we incorrectly

reported that John Kenneth HavUand, PhD '61,

retired in 1969. He retired from the University of

Virginia's Department of Mechanical and

Aerospace Engineering as professor emeritus on

June 1,1991.

George T. Onega, SM 70, of Slidell, La., died on

March 23,1990. Upon graduation from MIT he

joined the Bell Aerospace Co. as an engineer

where he spent his entire professional career

working on engineering applications for the

aerospace and marine fields, rising to the position

of director of advanced design. His work at

Textron Marine Systems (formerly Bell

Aerospace) involved surface effect ships for the

U.S. government The Alumni/ae Association

has been notifed that Captain Henry F. Lloyd, SM

'46, USN (ret.), of St. Augustine, Fla., died of a

heart attack on June 12,1991. There was no fur

ther information provided.

XVII POLITICAL SCIENCE

Franklin D. Margiotta, PhD 79, writes that as

president and publisher of Brassey's (U.S.), Inc.,

he has "just published President Bush's National

Security Strategy of the U.S. 1991-1992. As a press

that supports the competition of ideas, Brassey's

will soon publish a book developed by another

MIT graduate. Representative Los Aspin, PhD 79

(XIV), chair of the House Armed Services

Committee, entitled Defense for a New Era: Lessons

of the Persian Gulf War."... Vann H. Van Diepen,

SM '83,,reports from Arlington. Va.: "I have been

named director of the Office of Weapons

Proliferation Policy in the Bureau of Politico-

Military Affairs in the U.S. Department of State.".

.. Margaret O. Meredith, Stephen D. Nelson, and

Albert H. Teich, '64 (VIII), PhD '69, are the edi

tors of AAAS Science and Technology Policy

Yearbook 1991 (AAAS, 1991). According to the

book jacket, "The editors have assembled a

unique collection of informed discussions of the

major science and technology policy issues of

1990-91. Topics range from the financial health of

academic science to the state of the art in measur

ing payoffs from research to fraud and miscon

duct in science."

XVIII MATHEMATICS

From Urbana, III., Douglas B. West, PhD 78,

writes: "I have been promoted to a full professor

ship in the Mathematics Department at the

University of Illinois, effective August 1991."...

Bob Leonard, SM '57, sends us a clip from the San

Diego Business Journal. The article details the

takeover of Ticketron by Ticketmaster, the Los

Angeles-based ticketing firm run by Leonard. The

company's 1991 revenues are projected at more

than $1 billion, up from $650 million in 1990.

Leonard said Ticketmaster was able to take the

lead in the ticketing business with superior com

puter systems, aggressive marketing, and its

introduction of telephone sales. Ticketron fol

lowed a few years later, but Leonard said he

believes it was too late to stage a comeback

The Mathematical Association of America held

elections last September and has two MIT alumni

on board. Donald L. Kreider, PhD '59, professor

of mathematics and computer science at

Dartmouth College will be the Association's 45th

president. Kreider will serve a one-year term as

president elect followed by a two-year term as

president, ending with a one-year term as past-

president. Sharon Cutler Ross, '65, has been elect

ed the Association's second VP. Ross teaches

mathematics at DeKalb College in Atlanta, Ga.

Kreider has been an MAA member for 31 years

and Ross has been a member for 21.

XXAPPLIED BIOLOGICAL

SCIENCES

Jorge L. Sintes, PhD 78, writes: "I am currently

associate director of Advanced Technology-Oral,

at Colgate Palmolive Co.'s Technology Center in

Piscataway, in New Jersey. I am manager of a

department in charge of development of new

products and technologies for professional use."..

. Aaron L. Brody, '51, PhD '57, is now the manag

ing director and partner of Rubbright Brody, Inc.

According to a compa

ny news release, "The

countless honors

[Brody] has received

from his peers in the

Institute of Food

Technologists and

Institute of Packaging

Professionals attest to

his accomplishments,

ability, and integrity.

Brody brings to us over

20 years of hands-on

industry managerial

experience and more

A.L. Brody than 10 years of consulting." The company will conduct market and

business research, strategic planning, venture and

acquisition analysis, and opportunity identifica

tion, emphasizing food and food packaging.

Brody was technologist on the development of

microwave food heating and is the inventor of

controlled atmosphere food preservation as well

as one of the aseptic packaging systems.

From 1981 to the present, Susan Kathleen Alfano

Van Der Vynckt, SM '73, has been woking with

UNESCO in Paris. She is a program specialist

responsible for the design and execution of

UNESCO's programs in nutrition, health, and

home economics education, including the development of training and instructional materials for

formal and non-formal education programs. She

is also responsible for the supervision and man

agement of country field projects in nutrition,

health, and home economics. She is responsible

for interagency cooperation in nutrition and

health and is the representative to the U.N.

Administration Committee on Coordination and

Subcommittee on Nutrition. She is editor of the

UNESCO Nutrition Education Series The

Alumni/ae Association has been notified that

Elsie P. Arnold, SM '65, of Richmond, Ontario,

died on February 25,1991. She was employed in

Canada's Health and Welfare Toxicology

Evaluation Division. There was no further infor

mation provided.

XXI HUMANITIES

Thomas S. Kuhn, professor emeritus of philoso

phy at MIT, has been awarded an honorary

degree conferred at a special convocation at the

University of Chicago last September in celebra

tion of the university's centennial. Kuhn was rec

ognized for his writings and for "his ground

breaking contributions to understanding the ways

in which various scientific views are supported

and discounted over time."... Jeanne Bamberger,

professor of music at MIT, has recently completed

The Mind Behind the Musical Ear: How Children

Develop Musical Intelligence (Harvard University

Press, 1991). According to the book's jacket. "By

empasizing the idea that each 'hearing' of musical

composition is a 'performance,' one among many

possible hearings, Bamberger suggests that there

are different ways of constructing meaning and

that the processes of perception and conceptual

ization are reciprocal.

XXIINUCLEAR

ENGINEERING

Martin Zimmennann, SM '89, sends word: "I am

currently working as a research associate at the

Institute of Robotics at the ETH in Zurich,

Switzerland, where I am in charge of the design

for the control system of a combined wheeled and

legged robot with insect-like intelligence. Besides,

I am heavily involved in teaching students and

working on a new course on 'Designing Smart

Mechatronic Products.' Most important, my wife,

Petra, and I are the proud parents of our first

daughter, Pia Andrea."... From San Jose, Calif.,

W. H. D'Ardenne, PhD '64, writes: "I have been

elected chair of the Nuclear Power Plant

Standards Committee of the American Nuclear

Society."... Joseph W. Synan, SM '69, NUE '69, is

founder and president of Leadingwell Associates,

a leadership and team excellence consulting prac

tice in Houston H. Keith Roe, SM 74, NUE

74, has added the title of chair to those of presi

dent, CEO, of Burns & Roe, Inc., in Oradcll, N.J.

STSPROGRAM IN SCIENCE,

TECHNOLOGY & SOCIETY

Professor Leo Marx gave the Lowe Lecture at

Macalester College in Saint Paul, Minn, in mid-

October. The talk was entitled "On the Grace of

Nature: The Religious Aspect of Literary

Pastoralism in America."... Professor Eugene

Skolntkoff, '49 (VI), SM '50 (VI), PhD '65 (XVII),

reported to the October MIT Faculty meeting on

the study he chaired last year on the subject of

MITs international relationships.... Professor

Sherry Turkle was a featured speaker at the "MIT

TECHNOLOGY REVIEW MIT 37

PUZZLE CORNER

ALLAN J. GOTTLIEB, '67

Fiddling with the Roof

Mary Lindenberg reports that her

husband Martin, when inter

viewing high school students

applying to MIT, always encourages

them to take advantage of sailing the

Tech dinghies. Mary usually shows

them "Puzzle Corner" and points out an

interesting problem. Martin includes

comments in his report if they solve the

problem.

It has been at least a year since I speci

fied the size of the backlogs for the vari

ous kinds of problems that are printed.

Let me do so now. When the size of the

column was reduced and the number of

regular and speed problems per issue

halved, the backlog had an instanta

neous doubling (measured in months,

not letters). I now have nearly 2 years

worth of speed problems and even more

for regular problems. However, I have

very few of the special (chess, bridge,

computer, go) problems. So if you have

any of these special problems, send

them in. If not, I may soon just merge

the special with the regular problems

and print three from the combined list

each issue instead of the current 1 spe

cial, 2 regular policy.

Problems

APR 1. A wild one from Jorgen Harmse.

White is to move and mate in 12.

p

R

P

B

P

P

K

R

P

K

P

P

P

B

APR 2. Jerry Grossman needs help

training his robots. Ten workers in a

certain high-tech office have purchased

a fleet of voice-activated robots to help

them with various tasks, such as getting

coffee, delivering mail, and carrying

SEND PROBLEMS, SOLUTIONS,

AND COMMENTS TO ALLAN]. COT-

TUEB, '67, THE COURANT INSTI

TUTE, NEW YORK UNIVERSITY, 251

MERCER ST.. NEW YORK. N.Y. 10012.

OR TO: COTTUEB@NYU.EDU

furniture. The robots are expensive, so

they bought only seven of them, assum

ing that no more then seven workers

would be needing robots simultaneous

ly. Each robot can serve any number of

masters (though only one at a time), but

in order to respond to voice commands,

a time-consuming and expensive train

ing session is needed between each

robot and each master it will serve. To

avoid waste, the workers want to con

duct as few training sessions as possi

ble. Determine what training sessions

should be conducted to achieve these

aims.

APR 3. Liet Van Phan wants you to find

angle DAC without using either

Pythagoras's theorem or trigonometry.

Measured in degrees, angles B and C

are each 90 and angle D is 75 and CB is

half as long as DC.

B

IT

Speed Department

Geoff Landis notes that Allen, Beth, and

Chandra built houses with identical

planforms, but decided on different roof

designs as shown below. The pitch of

the roofs are identical. Which has the

greatest roof surface area and which the

least?

sible contract of 6NT. However, West leads the

spade 4, which East wins with the ace and returns

the spade queen. How can South now make his

contract assuming best defense from this point

onward?

North

West

A

¥

*

*

* 108743

¥ 7

« Q10 9

* Q832

*

¥

»

*

5

AK8

A83

AJ10754

A

¥

*

*

South

KJ62

Q1093

K74

K9

East

AQ9

J6542

J652

6

Both

Allen

Solutions

N/D 1. We begin with a bridge problem from

Winslow Hartford in which South is in an impos-

Thc following solution is from Jonathan Hardis.

William Tripp remarks that the problem is an

example of a guard squeeze.

After the ace of spades is played and East leads

the queen, it's easy to count 9 tricks:

*-K,J

¥-A,K,Q

♦ -A,K

*-A, K

The problem is to develop three more. South cov

ers the queen of spades with his king, and dis

cards the diamond 8 (not a club, not the 3!) in

dummy. South next leads the club 9 from his

hand. If West plays low (best defense). South

overtakes with the 10, which becomes the first

extra trick. If West plays the queen. South over

takes with the ace and dummy's clubs provide all

three extra tricks.

South next leads the heart 8 from the board. If

East plays the jack. South overtakes with the

queen and the 10 of hearts becomes the second

extra trick. If East plays low. South plays the 3

and the 8 becomes the second extra trick. At this

point. West's hand looks like this:

4-10 xx

♦ -Qxx

*-Qxx

South cashes the ace and king of hearts,

returns to his hand with the king of

clubs, and cashes the queen or 10 of

hearts. West must find three discards for

the hearts. (1) If he discards a club, the

club jack becomes the third extra trick.

(2) If he discards two spades, South's

fourth spade becomes the third extra

trick. (3) He discards at least two dia

monds. (East protects the diamond jack.)

South plays accordingly: (1) On the fourth heart,

discard the diamond 3, cross to dummy with the

diamond ace, and cash the ace and jack of clubs,

Continued on Page MIT 45

lUX'STRATIONS: DAVID HRIBV TECHNOLOGY REY1EV MIT 47

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PUZZLE CORNER

Continued from Page MIT 47

pitching the diamond and spade losers. (2) On

the fourth heart, discard a low club,

cash the spade jack and 6, discarding two low

clubs, cash the diamond king, cross to dummy

with the diamond ace, and take the ace of clubs,

pitching the diamond loser. (3) On the fourth

heart, discard a low club, play the diamond 4 to

dummy's diamond ace, pitch the spade 6 on

dummy's ace of clubs, then lead back the (well

preserved) three of diamonds, finessing East'sjack.

N/D 2. Matthew Fountain wants to know how

large is the volume that lies within two inches

of all the corners of a two-inch cube. All the vol

ume must lie within two inches of all the cor

ners.

John Salmon notes that this problem was

solved in his thesis, which I happen to know

involves a very sophisticated solution technique

for hierarchical N-body problems. Indeed, therequired formula is actually a numbered equa

tion (6.8). The solution below is from EuceneSard:

It is convenient to use a rectangular coordinatesystem with x-, y-, and z-axes parallel to the

cube faces, and whose origin is at the center ofthe cube. The 8 cube comers are then at the 8arrangements of (±1, ±1, ±1). The distance of the

origin from each corner is (12+12+12)1/2=31/2 aluj

the volume in question is a small symmetrical 8-

surface "nut" around the origin. Each of the 8

surfaces subtends a solid angle of it/2 between

x-y, y-z, and x-z planes.

Consider the surface where x, y, and z arc all

positive. Each point on this surface is 2 inchesfrom the corner at (-1,-1,-1), or ((x+l)2+(y+»2

+(z+l)2)'/J=2. Recasting for a fixed z gives a

boundary equation of a circle in an x-y plane,(x+l)*+(y+l)2=4-(z+l)*, with center P(-1, -1) andradius PA+(4-(z+l)2)"/'.

The complete shaded area of interest is alsobounded by the lines x=0 and y=0. Other keydimensions are PB=OB=1, 0P=2"', and 0A=AB

-0B=(3-(z+l)')>/M. From the law of cosines,angleO= arccos (O+(3-(z+l)*)"/2)/(2(4.

<z+l)2))i/3).Thus 28 varies from .33984 to 0 rad.as z varies from 0 to 2"* -1. The shaded area is

the difference between the full sector area and

twice the area of triangle OAP, or A=8(4-<z+l)2)-

sinei2(4-(z+l)*))i/2. Finally the "nut" volume,

V=8jn A dz. Numerical integration givesV=.12im4 cubic inches to 6 significant figures.

N/D 3. Robert Sackheim notes that all readers

know that if a man leaves home, walks a milesouth, then walks a mile west, shoots a bear,

then walks a mile north and finds himself back

home, that the bear is white because the man'shome is at the North Pole and the bear is a polarbear. Sackheim wonders if there is any otherplace on earth where a person can go a mile

south, then a mile west, then a mile north and

be back at the starting point?

The answer is clearly yes, there are an infinite

number of solutions near the South Pole. Severalreaders pointed out that no bears will be found.

The following solution is from Jack Bross:

There are an infinite number of latitudes fromwhich one may go 1 mile south, 1 mile west, 1mile north, and find oneself back where one

started: alt of them are slightly more than a mile

away from the South Pole (accumulating on the

circle 1 mile from the pole). The idea is that if

going one mile south brings you near the pole,

west will circle around the pole several times.

For any n, we can find a distance so that one

mile wraps exactly n times around the pole.

Then, north will take us back to where we start

ed. The actual values depend on one's assumption about the curvature of the earth near the

pole, but of course the earth is fairly flat at a dis

tance of one mile, so they are well approximatedby 1+1/2idi miles.

Other Responders

Responses have also been received from E Biek,W. DeHart, M. Deskey, S. Feldman, M.Fountain, D. Fraser, N. Gevirtz, C. Keavney, K.

Kiescl, M. Lindenberg, A. Ornstcin, W. Pulver,

A. Reed, K. Rosato, P. Sanchez, R. Schweiker, A.

Silva, D. Smith, J. Uretsky, D. Wachsman, C.

Whittle, J. Wilson, J. Woolston, H. Zaremba.

Proposer's Solution to Speed Problem

All three are the same.

the Nature Conservancy and the objective is toimprove the management of natural resources

throughout Central America while promoting sus

tainable development projects. PACA is working

in Belize, Guatemala, Honduras, El Salvador,

Nicaragua, and Costa Rica. The project consists of

four technical components: wildland manage

ment, environmental education, conservation

information, and strategic planning Gregg F.

Martin, SM '88, has mentioned that the College of

Naval Command and Staff is outstanding. There

are only 200 students—100 Naval officers, 34

Army, 30 Air Force, 25 Marine, and the remainder

Coast Guard and civilians. The orientation is on

joint military operations, strategy, and national

policy. So far they have analyzed historical military case studies ranging from antiquity to mod

ern times Marina Skumanich, SM '88, has

joined Battelle's Environmental Policy and Social

Research Center as a research associate Simon

Stokes, SM '88, is working for a patent and corpo

rate law firm in London. He expects to qualify as

a solicitor in 1992. After Christmas he hopes to

take a part-time course in European law at King's

College in London We have heard that Terry

Tumipseed, SM '88, has married a redheadednurse.

Jennifer Croissant, SM '89, taught "Women in

Science and Engineering" at Rensselaer High

School Summer Program. She also completed her

qualifying and dissertation exams in the Scienceand Technology Studies doctoral program at

Rensselaer.... Jeff Dieffenbach, SM '89, is cur

rently a project manager at IBIS Associates, an

engineering consulting firm in Wellesley, Mass.

He is also working on issues for the Paul Tsongas

for President Campaign Aaron Todd"

Curtis, SM '90, is on staff at the Boeing

Commercial Airplane Group in Seattle, Wash., as

a safety systems engineer Mark Roberts, SM

'90, has been in Washington, D.C., for a year now,working at the Congressional Office of

Technology Assessment. He has found a great

number of TPP alumni/ae at OTA. He is currently

working on a study of defense conversion beating

missiles into microwave ovens James "Jamie"Winebrake, SM '90, is presently working on a

PhD at the University of Pennsylvania's Centerfor Energy and the Environment Paul Chan,SM '91, has joined the staff of Koencman CapitalManagement, Pte, Ltd., which is an international

investment firm based in Singapore. The firm wasstarted by two MIT alumni. Paul is currently

working on a data visualization project which hefinds both interesting and challenging.—ReneSmith for Richard de Neufville, Technology and

Policy Program, MIT, Rm. E40-252, Cambridge,

MA 02139.

STSPROGRAM IN SCIENCE,

TECHNOLOGY & SOCIETY

Professor Jill Conway gave a talk about

Australian history entitled "Fatal Shore or Luck

Country" last December at the Smithsonian

Institution Professor Deborah Fitzgerald is

editing a new series. Revisiting Rural America, for

The Johns Hopkins University Press ProfessorLoren Graham delivered the keynote address at

the joint meeting of the Society for the History of

Technology and the History of Science

Association in Madison, Wise, last November. He

and Professor Eugene Skolnikoff, '49, SM '50

(VI), PhD '65 (XVH), traveled in early December to

Moscow to speak to leaders of the Soviet and

Russian Academies of Science about the restruc

turing of Soviet and Russian science Professor

Lily Kay presented a paper on the technologicalroots of early molecular biology at the 1991 4S

meeting. She is spending the spring term at the

Max Planck Institute for Physical Chemistry in

Gottingen Professor emeritus Leo Marx gave

a talk in Sharon, N.H., entitled "Henry Thoreau

and the Humanistic Perspective on EnvironmentalDegradation."... Professor Leon Trilling present

ed a paper on engineering education and interna

tional perspectives at the seventh IEEE Careers

Conference in Denver last October.

The following are STS graduate student notations. Bruce Bimber delivered a paper, "The

Politics of Expertise and the Separation of

Powers" at the annual American Political ScienceAssociation meeting in Washington last

September.... Dan Grossman, '82 (VIII), SM '86

(XVII), and Scth Shulman were the coauthors of

"Over There: The US. Military's Toxic Reach," in

the November 28,1992, issue of Rolling Stone

David Mindell went to the Galapagos Islands in

November and December as part of the Jason

Project, an educational program aimed at getting

secondary school students interested in science

and technology. Mindell is control system engi

neer and navigator for this year's project, having

developed the undersea robot "Jason Junior" in

his work at the Woods Hole Oceanographic

Institute.—Phyllis Klein, STS Program, MIT, ES1-128, Cambridge, MA 02139.

Deceased

The following deaths have been reported to the

Alumni/ae Association since the Review last wentto press:

L.G. Lee Thomas, '20; November 22,1991;Naples, Fla.

TECHNOLOGY REVIEW MIT 45

StorchEngineers

Engineers

Architects

Surveyor*

Planners

Goologl*t»

Soil Scientists

Mundpal Specialists

Landscape Architects

Environmental

Consultants

Florham Pork, HJ

201-822-2600

Robblnsvllle, NJ

609-25941640

Manchester, NH

603-623-5544

Rocky Hill, CT

203-529-7727

Now York, NY

212-371-4675

Jericho, NY

516-338-4500

Boston, MA

617-783-0404

Provldenco, Rl

401-751-2235

Washington, DC

202-785-8433

DesignContinuum, Inc.

New Product Planning, Design and Development

J. Zlndler, '55

H. Koselko, '88

R. Miller, '87

R. von Turkovlch

A. Zleglor, '87

646 Beacon Street

Boston, MA 02215

Telephone: (617) 267-5115

Teletax: (617) 267-3923

FeldTechnologies,

Inc.

Custom Software Development

Network Installation and Support

Information Technology Consulting

Bradley A. Feld, '87, SM '88

David J. Jllk, '85

Peggy B. Ng, '89

Bonnie C. Steele, '91

155 Federal Street

Boston, MA 02110

(617) 451-0055

Warren K. Brimblecom. '21; December 18,1991;

South Yarmouth, Mass.

Thomas B. Davis, '21; September 20,1991;

Memphis, Tex.

Irving D. Jakobsen, '21; October 21,1991;

Bayville, NY.

Henry C. Tainlor, "21; August 25,1991; Mission

Viejo, Calif.

Richard B. Oakes, '22; November 4,1991;

Auburndale, Mass.

William H. Blandy, '23; June 29,1991; Delray

Beach, Fla.

Laurence Edmund Noble, '23; August 9.1991;

Enid, Okla.

Clarence j. Odell, '23; November 27,1991; Lee,

Mass.

Dunbar L. Shanklin, '23; December 21,1991;

Delray Beach, Fla.

Frederick B. Stevens, '23; December 17,1991;

Akron, Ohio

G. Baer Connard, '25; October 26,1991; Bath,

Maine

Robert B. Hatton, '25; November 26,1991;

Andover, Mass.

Jesse L. Maury, '25; April 4,1991; Arlington, Va.

William R. Wheeler, '25; November 2,1991;

Southbury, Conn.

Reginald Frank Terrill, '26; November 14,1991;

Concord, N.H.

Samuel S. Auchincloss, '27; November 5,1991;

Englcwood, Fla.

Edward R. Coop, '27; December 15,1991;

Kumford, R.I.

David R. Knox, '27; November 25,1991; Lantana,

Fla.

Thomas F. Russell, '27; July 6,1991; Naples, Fla.

Warren Crier Armstrong, '28; October 26,1991;

Newark, Del.

Robert F. Schuler, '28; December 16,1991;

Newton Centre, Mass.

Milton H. Clapp, '29; December 1,1991;

Edgecomb, Maine

Ruth C. Dean, '29; November 13,1991; Wellesley,

Mass.

John J. Jarosh, '30; December 1,1991; Wilmington,

Calif.

Stanley G. Russell, '30; October 16,1991;

Annisquam, Mass.

Willard W. Selden, '30; November 29,1991;

Ludlow, Mass.

David Nicoll, '31; November 16,1991; La Jolla,

Calif.

Max Kate, '32; January 7,1992; Topsfield, Mass.

Richard A. Lobban, '32; December 10,1991;

Haverford, Pa.

Halsted R. Warrick, '32; August 30,1991;

Hendersonvillc, N.C.

John C. Herbert, '33; October 29,1991; Erie, Pa.

Eugene Rohman, '33; November 25,1991; West

Hartford, Conn.

Frank R. Milliken, '34; December 4,1991; Tucson,

Ariz.

John M. Thompson, '34; March 2,1991; Rye, NY.

W. Olmstead Wright, '34; August 26,1991;

Lombard, III.

Herbert C Zitzewitz, '34; October 19,1991; Port

Republic, Md.

Arthur M. King, '35; August 13,1991; Louisville,

Ky.

John Ayer, Jr., '36; November 13,1991; Denver,

Colo.

D. Elliot Cullaty, '36; June 9,1988; Weston, Mass.

William A. Healy, '36; November 26,1991;

Concord, N.H.

Merwin Miller, '36; October 4,1991; Berkeley,

Calif.

Samuel Norton Miner, '36; December 8,1991;

Lakeville, Conn.

Carl M. Peterson, '36; October 3,1991; Pittsburgh,

Pa.

Edward M. Fischer, '37; October 22,1991; Olathe,

Kan.

Robert Nedbor, '37; June 18,1991; Plantation, Fla.

William C. Wold, '37; October 19,1991; Dover,

N.H.

George F. Wollinger, Jr., '37; November 19,1991;

Laguna Hills, Calif.

Murray H. Hayward, '38; November 13,1991;

Troy, Mich.

Louis B. Tura, '40; December 28,1991; Chatham,

Mass.

Marion Loren Wood, '40; October 3,1991;

Briarcliff, N.Y.

Lawrence E. Beckley, '42; December 30,1991;

Winchester, Mass.

Donald Stein, '42; October 13,1991; Falls Church,

Va.

Marvin N. Stein, '42; November 1,1991; Brcwster,

Mass.

David J. Crawford, '43; April 15,1991;

Poughkeepsie, N.Y.

Jonathan H. Sprague Jr., '43; October 2,1991;

Houston, Tex.

Katherine A. Kulmala, '44; January 5,1992;

Carlisle, Mass.

Keith W. Cramblel, '45; March 20,1990; Santa

Ana, Calif.

Richard S. Bettes, Jr., '44; August 17,1990;

Mountain Lakes, N.J.

George K. Landon, Jr., '45; November 18,1991

Edmund L. Czapek, '46; October 31,1991; Quaker

Hill, Conn.

John M. McMlllin, Jr., '46; July 20,1991;

Cincinnati, Ohio

John M. De Bell, Jr., '47; December 14,1991;

Toms River, N.J.

R. Ellsworth Annig, Jr., '48; August 17,1991; West

Yarmouth, Mass.

Roy Oringer, '48; November 16,1991; Quincy,

Mass.

Robert H. Ruth, '49; October 4,1991; Lodi, N.Y.

Paul E. Weamer, '49; December 5,1991;

Chesterfield, Mo.

Maurice CM. Grandpierre, '50; September 5,

1991; Nancy, France

Frederic D. Grant, '50; November 11,1991;

Wellesley Hills, Mass.

Yaichi Ayukawa, '52; November 30,1991; Tokyo,

Japan

Albert L. McManus, Jr., '52; November 3,1991;

Framingham, Mass.

Paul C. Watson, '52; December 25,1991; Davis,

Calif.

KJell Loevaas, '53; July 14,1991; Oslo, Norway

Gerald P. Richards, '53; November 18,1991;

Framingham, Mass.

Roswell I_ Derby, '54; November 7,1991;

Longmeadow, Mass.

Domenic B. Vassallo, '56; November 26,1991;

Bethesda, Md.

George W. Walsh, III, '57; October 10,1991;

Durham, N.C.

Larry G. Glassmaker, "58; September 29,1991;

Palo Verdes Pen, Calif.

Franklin W. Mohney, '61; May 2,1991; New

York, N.Y.

Ralph J. Bahnsen, '65; June 16,1991; Wappingers

Falls, N.Y.

Preston G. Pollock, Jr., '67; November 27,1991;

Cruz Bay, St. John, U.S.V.I.

Merrill L Andrews, '68; August 23,1991;

Beavercreek, Ohio

John E. Litt, 70; November 1,1991; Mountain

View, Calif.

John Chaiken, 71; December 4,1991

William K. Kietrich, 71; September 4,1991;

Houston, Tex.

Charles F. Scott, Jr., 73; November 4,1991;

Boston, Mass.

Carlos Thompson, 74; July 3,1990; Greenville,

N.C.

Kenneth R. Gobeille, '81; August 31,1991;

Montreal, Quebec, Canada

Gregory O. Tomlinson, '81; November 8,1991;

Boston, Mass.

Douglas K. Tannatt, "87; October 19,1991;

Philadelphia, Pa.

MIT 46 APRIL 1992

PUZZLE CORNER

ALLAN J. GOTTLIEB, '67

Coriolis Aloft

By the time you read this article I

will be packing for Australia! The

International Symposium on Com

puter Architecture is being held there

this year. I chaired the program commit

tee and am anxious to see if my commit

tee did a good job in selecting papers.

But of course I am even more interested

in seeing the southern cross! As you

can probably tell, I have never crossed

the equator before. I was planning to

empty a bathtub on the plane to watch

the circling water get confused but, alas,

the airlines would have none of it (bath

tubs fit neither under a seat nor in an

overhead rack).

Problems

M/J 1. We begin with a Bridge problem

that Winslow Hartford sent us from the

London Sunday Observer. In the hand

shown, West missed the killing dia

mond opener against 7H and instead

lead the spade jack. How can South now

make the grand slam?

\)1

'"

*

*

♦

West

J10 8 7

4 2

10965

964

North

Q95

10653

AQ

J872

South

* A

♦ akq;

♦ 32

* AKQ

A

V

♦

|87

10

East

K6432

9

KJ974

53

M/J 2. Gordon Rice is thinking of four

positive integers

0<A<B<C<D

that have a curious property. When

numbers are written in base D

AB = A (modulo C)

and BA = B (modulo C).

SEND PROBLEMS, SOLUTIONS,

AND COMMENTS TOAUAN /. GOT

TLIEB, '67, THE COURANT INSTI

TUTE, NEWYORK UNIVERSITY. 257

MERCER ST., NEW YORK, N.Y. 10012,

OR TO: COTTUEB@NYU.EDU

For what values of D do solutions exits?

Are they unique? Note that AB does

not represent AxB. Instead it signifies

juxtaposition, e.g., if A=24 and B=345,

AB is 24345.

M/J 3. Daniel Morgan wants to know

the expected point count for a randomly

dealt Bridge hand of 13 cards? High

cards are valued as Ace=4, King=3,

Queen=2, and Jack=l. In addition a void

(no cards in a suit) contributes 3 points,

a singleton contributes 2, and a double-

ton contributes 1.

Speed Department

Speedy Jim Landau wants to know the

smallest possible number of pitches in a

complete baseball game and how many

calls does the plate umpire make during

this game?

Solutions

JAN 1. Our "first" problem is a computer offering

from Bob High. Write the first n numbers in

alphabetical (dictionary) orders as they are

spelled out (i.e., one, two, three,...one million,...).

To avoid ambiguity, use no "ands" or hyphens, so

837,301 would be written "eight hundred thirty-

seven thousand three hundred one." 1,897 is "one

thousand eight hundred ninety-seven," not "eigh

teen hundred ninety-seven." Define two func

tions, F(m,n) and G(m,n) as follows: F(m,n) is m

th number in the alphabetical list of the first n

numbers; G(m,n) is the position of the number m

in this list. (For given n, F and G are inverses.) we

ask: (1) What is F( 1,000, 1,000,000)? What is

G(l,000, 1,000,000)? (2) What is F(1,000,000,

1,000,000)? What is G(1,000,000, 1,000,000)? (3)

For what numbers n is F(n,n) = G(n,n) = n? List

the first dozen.

Speedy Jim Landau sent us a detailed solution

to this problem and an extension of it. Interested

readers should write to Faith Hruby at TR for a

copy. A summary of Landau's solution follows:

Consider the numbers beginning "eight." There

are, in alphabetical order

number

eight

eighteen

eighteen thousand xxx

eight hundred xxx and

eight hundred thousand xxx

eight thousand xxx

eighty

eighty x

eighty thousand xxx

eighty two

quantity

1

1

1,000

100

100,000

1.000

1

8

10,000

1

112,112

The case of leading "one" is different because

"one million" must be included and the two-digit

and five-digit numbers beginning with the digit

one fall alphabetically under the second digit

(e.g., "eighteen").

Now we are ready to start answering questions.

(1) What is FUOOO, 1M)?

F0002,1M) is "eighteen thousand two," which is

the highest number alphabetically in the "eigh

teen thousand" series. Working backwards,

FdOOO, 1M) is "eighteen thousand twenty two"

and FdOOO, 1M) is "eighteen thousand twenty."

ILandau's full situation also tabulates numbers

with other leading digits. He then proceeds:)

What is G(1000, 1M)? The "one thousand" series

falls at the end of the "one" series, which means

Gdxxx, 1M) runs from 549,552 to 550.551. Since

1000 falls at the beginning of the "one thousand"

series, GdOOO, 1M) is 549,552.

(2)WhatisF(lM,lM)?

"Two" sorts highest alphabetically, and can only

be followed by "hundred" and "thousand." The

last number alphabetically is F(1M, 1M) = two

thousand two.

What is GUM, 1M)?

The "one" series goes

one

one hundred xxx

one hundred thousand xxx

one hundred twenty, twenty two, and two

one million

one thousand xxx

Using the answer to d), we find G(1M, 1M) =

GdOOO, 1M)-1=549,551.

(3) For what n does F(n,n) = G(rt,n) = n?

There are exactly 64 such n. The first 4 such n are:

one

two

two hundred '

two hundred two

Notice the pattern 2, 200, 202. It will repeat itself

below. The next 4 n are:

two thousand

two thousand two

two thousand two hundred

two thousand two hundred two (which is

F(IM.IM) above)

There will be no new n until we find a suffix for

"two" which sorts alphabetically higher than

"thousand." The first such is "trillion." The next 8

nare:

two trillion

two trillion two

two trillion thousand two hundred two

JAN 2. Robert Bart offers the following extensions

to an old problem from Nob. Yoshigahara. What

is the smallest positive integer whose square root

has a decimal expansion beginning with ten dis

tinct digits. Now consider cube roots instead of

square roots. Finally consider ith roots for i =

4,5 10.

Daniel J. Weidman not only solved this problem

but, as with Landau above, extended the problem

and solved the extension. Once again interested

readers should contact Ms. Hruby for a copy.

Weidman's solution to the original problem fol

lows. Note that we are interpreting the "decimal

expansion" to begin after the decimal point.

Continued on Page MIT 54

TECHNOLOGY REVIEW MIT 55

25 years, he and his associates) discovered!

nent of the problem; he turned his attentio^to Los Angeles smogvHe chairedJan. . X'--^-industry-wide contmift'ee that organized a *concerted natioi^^ack on airjpollution>|Its effpiits res

gasolfn,(wcomj

Laboratories, where he remained until, he

GIFT OF CAPITAL: Charitable Remainder ■

Unitrust to establish theJohn M. Campbell

(1925) Undergraduate Student Loan Fund.

QUOTE: A charitable remainder gift to MIT

is like having your cake and eating.it, too.

My wife Marie and I receive a lifetime

income that is almost double what we for

merly earned from, the same amount of

capital and, of course, there is no capital .

gains tax. It is also ah excellent way to.return some of the profit that comes from

an MIT education and to assist the next

generation of technologists.

For more information about gifts of capital,

write or call Frank H. McGrory or

D. HugH Darden'at MIT,

77 Massachusetts Avenue, Room. 4-234;

Cambridge,' Massachusetts 02139-4307;

(017)253-3827. . ■

I'lmtii: Luiry Hire, SliTling Heights, Michigan

'S

^v

——PUZZLE CORNER—

Continued from Page MTT 55The 2 th root of 143 is 11.95826O7431014

The 3 th root of 939 is 9.792386145009786

The 4 th root of 633 is 5.015923768441686

The 5 th root of 8117 Is 6.051723946894983

The 6 th root of 896 is 3.104926578310817

The 7 th root of 551 is 2.463729851098231

The 8 th root of 558 is 2.204597318658172

The 9 th root of 759 is 2.089425371646355'The 10 th root of 667 is 1.916075348263711

Other Responders

Responses have also been received from D.

Church, D. Eckhardt S. Feldman, M. Fountain,

I. Shalom,

Proposer's Solution to Speed Problem

28. One player on the home team hits the first

pitch for a home run. Every one else grounds

out on the first pitch. The game is called on

account of rain after 4 1/2 innings. The home

plate umpire makes 1/4 of a call. The ground

outs arc called by the first base umpire but

the plate umpire participates in the decision

to call the game. D

►: PUZZLE CORNER

ALLAN J. GOTTLIEB, '67

Dot's Entertainment

It has been a year since I reviewed the

criteria used to select solutions for

publication. Let me do so now.

As responses to problems arrive, they

are simply put together in neat piles,

with no regard to their date of arrival or

postmark. When it is time for me to

write the column in which solutions are

to appear, I first weed out erroneous

and illegible responses. For difficult

problems, this may be enough; the most

publishable solution becomes obvious.

Usually, however, many responses still

remain. I next try to select a solution

that supplies an appropriate amount of

detail and that includes a minimal num

ber of characters that are hard to set in

type. A particularly elegant solution is,

of course, preferred as are contributions

from correspondents whose solutions

have not previ ously appeared. I also

favor solutions that are neatly written,

typed, or sent via e-mail, since these

produce fewer typesetting errors.

Problems

JUL 1. We begin with a bridge problem

from Richard Hess, who (I guess)

always seems to get low point count

hands and likes to see how far they can

go. Inspired by the 1991 Jan 1 problem,

Hess asks for the lowest number of high

card points that North and South can

have (combined) and still make 7NT

against best defense.

JUL 2. Matthew Fountain suggests we

tackle the "hold that line" problem

devised by Sid Sackson and appearing

in his book A Garnet of Games.

"Hold That Line" is a game in which

two players alternate drawing straight

lines between dots on a 4 x 4 dot field.

The player to draw the last line loses.

The first diagram shows a game in

which the lines are numbered in the

order they were drawn. Restrictions are

that lines after the first shall only be

drawn from the free end of a previously

drawn line. All lines must be straight

and start and end at a dot. A line may

SEND PROBLEMS, SOLUTIONS,

AND COMMENTS TOALLAN /. GOT

TLIEB. -67, THE COURANT INSTI

TUTE. NEWYORK UNIVERSITY. 251

MERCER ST.. NEWYORK. N.Y. 10012.

OR TO: GOTTUEB@NYU.EDU

connect more than two dots if all are in

a straight line. No line shall be drawn to

a previously connected dot or cross

another line.

The second diagram shows a game in

progress where the first player has

drawn his first line along a long diago

nal of the field. Is this a winning or a

losing move? There can be no ties.

Game lost bv first player Game in progress

JUL 3. Geoffrey Landis has found our

previous cryptarithmetic problems

(where you are given an arithmetic

equation such as XXX + Y = YZZZ and

must find which digits to assign to each

letter, in this trivial case X=9, Y=l, Z=0)

"rather uninteresting." So he offers a

challenge. Find a cryptarithmetic prob

lem with (precisely) two solutions based

on two (completely) different keys, i.e.,

no letter is assigned the same digit in

both solutions. I would not be surprised

to find that Nob. Yoshigahara has a few

dozen of these sitting under his socks in

his bureau.

Speed Department

Pete Chandler wants you to figure out

each of these eight brain teasers.

Solutions

F/M 1. Dave Wachsman sent us a hand he played

(as South) with his wife that was reported in

Truscott's Column in The New York Times.

North

* 82

• A72

♦ Q75

* KQ942

West

* 973

* 1865

* KJ1032

* J

East

* 65

¥ 104

* 984

* A108653

South

* AKQJ104

v KQ93

* A6

* 7

Both sides were vulnerable. The bidding:

South West North East

1 * Pass 2 * Pass

3* Pass 3N.T. Pass

4» Pass 4N.T. Pass

6 * Pass Pass Pass

How docs Mr. Wachsman bring home the slam

after West leads the club jack?

Larry Shiller sent us the following solution. If

East lets club K win, declarer draws trumps and

leads to dummy's diamond Q for 12th trick. Oth

erwise, declarer wins East's return in hand (ruff

ing high if a club), draws trumps, cashes diamond

A, crosses to dummy with the heart A, and leads

the club Q, discarding a diamond from the closed

hand, squeezing West.

F/M 2. John Prussing believes that the following

puzzle, which was actually on the 1989 Putnam

exam, seems about right for Puz zle Comer.

A dart hits a square dartboard. If any two points

on the dartboard have the same probability of

Continued on Page MFT 46

SAND

B

L

<*■E

Cycle

Cycle

Cycle

He's/Himself

Her

ECNALG

GI

ccccccc

cccccc

ccccc

cccc

Death/Life

0

MD

PhD

MA

TECHNOLOGY REVIEW MIT 47

5.J-

in

ITiW!

Alessandro Morclli, '44; November 18,1991;

Cohassct, Mass.

Paoli E.C. Massaglia, '45; January 14,1992; Stuart.

Fla.

Theodore E. Gerber, "46; February 7,1992; Guil-

ford. Conn.

Emerson H. Newton, '46; March 14,1992; Arling

ton, Mass.

Norman f. Gold, '47; March 5,1987; Newton

Highlands, Mass.

Thomas K. Hughes, '47; April 10,1991; Dickin

son, Tex.

William M. Hunt, '47; January 15,1992; Urn-

bertville, N.J.

Robert L. Kamm, '47; January 27,1992; Birming

ham, Mich.

Vance A. Myers, '47; February 9,1992; Fairfax,

Va.

James P. Storm, '47; 1991; Mcnlo Park, Calif.

Donald S. Floyd, '48; May 2,1991; Alexandria,

Va.

William Nicholson, '48; 1992; Easton, Md.

Edward T. Podufaly, '48; January 20,1992; Sher

wood, Md.

Roger L Sisson, '48; January 22,1992; Lafayette

Hill, Pa.

Edward N. Strait, Jr., '48; November 12,1991; St.

Paul, Minn.

William S. Hutchinson, Jr., '49; February 2,1992;

Jacksonville, Fla.

Willcm E. Lower, '49; 1990; Holton, The Nether

lands

Ferdinand G. Mikel, '49; March 3,1992; Silver

Spring, Md.

Chien-Hou Chang, '50; January 14,1991; Tianjin,

China

Theodore S. Huang, Jr., '50; November 2,1991;

Alexandria, Va.

Charles W. Ellis, 3rd, '51; January 25,1992; New

ton Square, Pa.

Matthew Goodwin, '52; July 26,1990; Culver

City, Calif.

Roger E. Ladd, '52; January 9,1992; Manchester,

Mass.

Malcolm C McQuarrie, '52; January 3,1992; Oak

land, Calif.

Gabriel Palmero, '52; 1991; Washingtonville, N.Y.

Francis B. Van Wyk, '52; 1991; Wallingford, Pa.

Morris B. Carter, '53; November 29,1991;

Columbia, Tenn.

William T. Wootton, '53; June 1,1991; Santee,

Calif.

Richard F. Merrill, '56; February 17,1992; Balti

more, Md.

Harold H. Rolhstein, '56; 1991

Lester Y. Sen, '56; 1991

Robert K. Boese, '57; August 6,1991; Glen Cove,

N.Y.

Lovett R. Smith, Jr., '57; April 17,1991; Danbury,

Conn.

Rene E. Unson, '57; February, 1991; Manadaluy-

ong, Philippines

Edwin R. Rose, '58; January 1,1992; Houston,

Tex.

Gabriel T. Kerekes, '60; 1991

Alan M. Edwards, '61; February 11,1992;

Crownsville, Md.

Keihachiro Moriyasu, '62; January 4,1992; Her-

miston. Ore.

Albert O. Riordan, Jr., '62; January 23, 1992; Hor-

nell, N.Y.

Leonard H. Edwards, '64; March 26,1991; Cincin

nati, Ohio

Freeman K. Keyte, '66; February 19.1992;

Nepean, Ontario

Willard J. Basner, Jr., '69; January 24,1992;

Franklin, Mass.

Edward A. Parks, '69; 1991; Big Oats, N.Y.

Richard E. Brackeen, 75; February 5,1992; Great

Falls, Va.

Osvaldo A. Jaeggli, '80; August 20,1990

Michael H. Bulat, '81; 1985

Clinton C Bourdon, '83; February 2,1992;

Ipswich, Mass.

■ PUZZLE CORNER.

Continued from Page MIT 47

being hit, what is the probability that the dart will land nearer to the center of the board than it

does to an edge.

George Blondin chararactcrizcs the solution

(4V2 -5)/3as "a really neat answer." Blondin writes: For a 2 by 2 square with center at (0,1), equating the

squared distance from the center, x^+O-y)^, to the squared distance from the bottom edge y^gives a parabola: y=(x2+l)/2 which intersects the lower right diagonal (y=!-x) at Xi, Yi whereXi=</2 -1, Yi=2W2. The area nearer the edge in this quarter square (whose gross area is 1.0) is thearea under this parabola plus the two triangles with sides = Yi.

Integrating Ay . dx gives (x3+3x)/6. Evaluating between -Xi and +Xi then adding Yi2 for the

triangles gives (8-4\/2)/3 as the chance of hitting nearer the edge. l-(8-4\/2)/3=(4\/2-5)/3 is

the chance of hitting nearer the center.

F/M 3. Our last problem is from my NYU Colleague, Dennis Shasha, and can be found in his

book. The Puzzling Adventures of Dr. Ecco.

You are given 20 coins. Some are fake and some are real. If a coin is real, it weighs between 11

and 11.1 grams. If it is fake, it weighs between 10.6 and 10.7 grams. You are allowed 15 weigh

ings on a scale (not a balance). You are to determine which coins are real and which are fake.

Our last solution is from Edgar Rose:

1.Divide the coins into five groups of four. We must determine, in three weighings or less, the

nature of each coin in a given quartet.

Let's take one group of four and mark the coins A, B, C, and D respectively. Also we will use

"f" and V when referring to fakes and real coins. As the last preparatory step, we set up a table

of weight ranges for the three possible pairs (f,f; f,r; and r,r) and the four possible trios (f,f,f; f,f,r;

f,r,r; and r,r,r); i.e., W(f,0 = 21.2-21.4 grams, W(f,r) = 21.6-21.8 grams, etc. There are no overlaps

between the ranges, therefore each weighing will identify how many fs and r's there are in the

weighed group.

2.Weigh A+B+C

Zl If f,f,f or r,r,r—weigh D for identification.

2.2 If f,f,r—use chart below.

2.3 If f,r,r—use chart below but change all *Vs" in the chart to "Ps," and vice versa.

W(A+B+C+)

W(C+D)

f,fI

f

W(B)

B=f

C,D=f

A=r

Other Responders

r

B=r

C,D=f

A=f

W(B+D)C,D=rA,B=f

f,f

B,D=f

C=r

A=f

f,r

D=r

B,C=f

A=r

B,D=r

C,A=f

Responses have also been received from J. Abbott, J. Bitsky, C. Brooks, F. Carbin, W. Guett, N.

Cook, D. Detlcfs, D. Eckhardt, S. Feldman, E. Field, E. Freudcnthal, E. Friedman, J. Grossman, B.

Huntington, J. Landau, M. Lindenberg, E. Lund, D. McMahon, G. Parks, R. Record, S. Root, K.

Rosato, J. Rudy, E. Sard, L. Saunders, C. Taubman, D. VanPatter, W. Woods, H. Zaremba.

Proposer's Solution to Speed Problem

Sand box, tricycle, backwards glance, life after death, see-through blouse, he's beside himself

over her, GI overseas, 3 degrees below zero.

MrT46 JULY 1992

XXI HUMANITIES

Peler C. Perdue, Course XXI associate professor

and head of the History faculty, has been awarded

the 1992 Levitan Prize in the Humanities. The

$20,000 prize was established through a gift from

James A. Levitan, '45 (X), a member of the MIT

Corporation and a senior partner in the law firm

of Skadden, Arps, Slate, Meagher, and Flom of

New York City. The prize, first awarded in 1990,

supports innovative and creative scholarship in

the humanities by faculty members in the School

of Humanities and Social Science. Perdue, a lead

ing specialist in modern Chinese history, will use

the prize to continue research into the Chinese

conquest of Central Asia from 1680-1760

Irene R. Heim, has been appointed to the Class of

1943 Career Development Chair in linguistics and

philosophy. Heim is recognized as one of the most

prominent and most influential figures in contem

porary semantics, an area that is among the most

technically complex branches of linguistics. As a

teacher, Heim is noted for her ability to introduce

her students to the strands of semantic argument

and then showing them how these are woven into

the broader picture of the field. In her publica

tions and colloquium and conference presenta

tion, she has been a force for reshaping the field of

semantics and is seen as a leading voice affecting

the choice of the central questions to be asked and

their answers as well.

Douglas J. Forsyth, has been appointed to the

Class of 1922 Career Development Chair in histo

ry. Forsyth is noted for his research on the politi

cal economy of 20th century Europe and for his

innovative teaching. He is the author of the forth

coming Monetary and Financial Policy and the Crisis

of Liberal Italy, which examines the political econo

my and the crisis of parliamentary government in

Italy from 1914-22.... Christopher D. Sawyer

Laucanno, lecturer in the Program in Writing and

Humanistic Studies, has been awarded a $10,000

Translators Fellowship Grant from the National

Endowment for the Arts. The grant will support

the translation from Spanish of Sobre Los Angeles

(Concerning the Angels), a book of poems by

Rafael Alberti Catherine V. Chvany, profes

sor of Russian in the Foreign Languages and Lit

erature Section at MIT, has received the Distin

guished Scholarly Career Award from the

American Assocation of Teachers of Slavic and

East European Languages. The award, for out

standing contribution to scholarship, was present

ed at the group's 50th anniversary meeting in San

Francisco. Chvany, an internationally known

expert pn the Russian language and linguistics, is

coordinator for MIT's interdisciplinary program

in Russian studies. She came to MIT in 1971 and

was promoted to full professor in 1985.

The Association of Alumni and Alumnae has

been notified that Osvaldo A. Jaeggli, PhD '80, of

Los Angeles, died on August 20,1990. Jaeggli had

been a professor in the Linguistics Department at

UCLA. There was no further information provided.

and therapeutic purposes, and the understanding

of the health effects of ionizing radiation.

XXIINUCLEAR

ENGINEERING

Robert W. Sawdye, SM '78, ScD '85, writes: "I am

in a new postion as principal consultant at Devon-

rue, Ltd., in Boston. We have new family mem

bers due soon, twin girls to join 3-year-old son,

Alex. Elaine and I are dreaming of a vacation,

someday."... Jacquelyn C. Yanch has been

appointed to the W. M. Keck Career Development

Chair in nuclear engineering/Whitaker College.

Yanch is a Course XXII assistant professor with a

secondary appointment at the Whitaker College of

Health Sciences and Technology. Her research

currently involves investigation of the uses of ion

izing radiation in medicine for both diagnostic

TPPTECHNOLOGY & POLICY,

PROGRAM

Diana B. Dutton, SM '76, is the author of Worst

than the Disease: Pitfalls of Medical Progress (Cam

bridge University Press, 1988) Hans Peter

Brondmo, SM '87, was featured in an article enti

tled "Ideas Galore, but Where Are the Goods" in

the February 10,1992 issue of Business Week.

Brondmo is co-founder of DIVA in Cambridge, a

Macintosh software developer spun off from the

MIT Media Lab.... Christoph Hilz, SM '90, is

currently on the staff in the Commission of the

EC, DGI-PHARE in Belgium. His thesis for TPP,

International Toxic Waste Trade, has been pub

lished by Van Nostrand Reinhold in New York

and is being introduced to the market later this

month.—Rene Smith for Richard de Neufvillc,

Technology and Policy Program, MIT, Room E40-

252, Cambridge, MA 02139

STSPROGRAM IN SCIENCE,

TECHNOLOGY & SOCIETY

Professor Leo Marx was the keynote speaker at a

March conference on Nature and Arts in the

Humanities sponsored by the Maine Alliance for

Arts Education in Portland, Maine. His essay,

"The American Ideology of Space," available as

STS Working Paper #8, has been published by the

Museum of Modern Art Professor Merrill Roe

Smith spoke at the Cornell University STS Pro

gram in March. His topic was "Technology,

Industrialization, and Social Change in 19th Cen

tury America." He also commented on two papers

at a March conference on "The Transformation of

the New England Landscape" at Old Sturbridge

Village Professor Sherry Turkle delivered the

keynote address, "Computation and Psychology,"

for the Educational Telecommunications Confer

ence on New Strategies in Distance Learning. The

conference was held in Lexington, Mass., at the

end of February.... Brian O'Donnell gave the

Hayes Seminar at St. Joseph's University in

Philadelphia in February. His paper was entitled

"Industrialization and Deindustrialization in the

Merrimack Valley."

Deceased

The following deaths have been report

ed to the Alumni/ae Association since

the Review last went to press:druthers A. Coleman, '16; February 5,1992; Lex

ington, Ky.

Elizabeth G. Paltee, 16; March, 1991; Hight-

stown, N.J.

Richard F. Lyon, '20; 1991; Laguna Hills, Calif.

Samuel Schenberg, '20; April 23,1991; Miami,

Fla.

Clifton B. Morse, '22; November 4,1990; Carmel

Valley, Calif.

Atherlon Hastings, '23; February 8,1992; Flo

rence, Ala.

Laurence S. McLane, '23; November, 1990;

Poway, Calif.

George B. McReynoIds, '23; 1991; Montecito,

Calif.

Clifford P. Swaine, '23; February 6,1992; South

Hamilton, Mass.

Edward A. Abdun-Nur, '24; October 3,1991; Den

ver, Colo.

Carleton Shugg, '24; January 23,1992

Hubert D. Barnes, '25; January 2,1992; Milwau

kee, Wis.

Leslie B. Bragg, '25; February 24,1992; Skaneate-

les, NY.

Ray E. Lucey, '25; February 3,1992; Fort Laud-

erdalc, Fla.

Samuel B. Maddock, '25; 1991; Portsmouth, N.H.

Philip S. Mancini, '26; January 29,1992; Beverly

Hills. Fla.

Donald S. Nelson, '26; January 4,1992; Dallas,

Texas

Francis H. Whitaker, '26; February 9,1992;

Wayne, Pa.

Stanley C. Boyle, '27; January 29,1992; Terre

Haute, Ind.

Paul E. Parker. '27; August 5,1991; Fair Oaks,

Calif.

Alden G. Reed, '27; December 29, 1991; Wood-

burn, Ore.

John H. Wever, '27; February 24, 1992

Hector E. Hagerdorn, '28; April 10,1991; Madrid,

Spain

Henry F. Kohlcr, '28; January 21,1992; Alexan

dria, Va.

William H. Phillips, '28; 1990; Albany, N.Y.

Floyd W. Buck, '29; January 15,1992; Hamdcn,

Conn.

Harry F. Dickinson, '29; September 20,1991;

Rotunda West, Fla.

John T. Hallahan, '29; February 25,1992; New

York, N.Y.

Francis S.F. Leong, '30; 1988; Honolulu, Hawaii

Milton Mezoif, '30; February 25,1992; Provi

dence, R.I.

Robert G. Marcus, '31; January 16,1992; Prince

ton, N.J.

Doyle L. Northrup, '31; December 15,1991; Satel

lite Beach, Fla.

Samuel B. Pritchard, '31; 1991; Saint Petersburg,

Fla.

Arthur C. Sugden, '31; 1991; Tuxedo Park, N.Y.

Charles M. Davis,'32; 1991

Robert B. Freeman, '32; February 21,1992; Kent-

field, Calif.

Rene G. Hochreutiner, '32; November 12,1991;

Vaud, Switzerland

Herbert J. Pfingslag, '32; June 11,1983

Philip W.Snyder,'32; 1990

Jack F. Andrews, '33; December 29,1991;

Lawrenceville, N.J.

James M. Dunlap, '33; December 30,1991; St.

Louis, Mo. .

Warren G. Webster, '33; March 3,1992; Lexing

ton, Mass.

Radcliffe G. Edmonds, '34; June 30,1991; Canaan,

N.H.

Edward L. Wemple, '34; February 27,1992;

Darien, Conn.

George A. Revell, '35; 1991; Cornwall, Ontario

Kenneth B. Galr. '37; January 17,1992; Alamo,

Calif.

James W. Pearce, '37; January 18,1992; Cincinnati,

Ohio

Gerald L. Simard, '37; January 30,1992; Winter-

port, Maine

William C. Wulbem, '37; September 17,1990;

Charleston, S.C.

Gordon L. Foole, '38; October 24,1991; Colum

bus, N.C.

Marvin M. Kahn, '38; April 27,1990; Sharon,

Mass.

J. William Blattenberger, '40; January 26,1992;

Lawrcnccville, N.J.

James E. Fifield, '40; January 23,1992; Darien,

Conn.

Kenneth R. Fox, '40; February 22,1992; Rockport,

Mass.

Thomas P. McConville, '40; June, 1990; Santa

Maria, Calif.

Karr Parker, Jr., '41; August 3,1991; Buffalo, N.Y.

William H. Reeder, 3rd, '41; 1991; Warren, N.J.

Giulio Ascoli, '42; February 4,1992; Urbana, 111.

Charles S. Hofmann, '42; January 31,1992; West

Yarmouth, Mass.

Willard D. Nalchajian, '42; February 10,1992;

Chelsea, Mass.

Maynard S. Renner, '42; December 31,1991; Gro-

ton. Mass.

TECHNOLOGY REVIEW MIT 45

PUZZLE CORNER

ALLAN J. GOTTLIEB, '67

Out on a Limerick

Imust be very brief this issue as I am

about to leave for Australia and NewZealand. After the "short" flight

from New York to Los Angeles, Ichange planes for the hop to Sydney,which is your basic 14-hour nonstopendurance test. Wish me luck.

Problems

A/S 1. Unfortunately, APR 1 was misprinted so that white and black pieceswere indistinguishable. As a result weare offering it again (now as A/S 1) but

with the colors indicated as intended.We apologize for the error.

White is to move and mate in 12.

p

R

P

B

P

P

K

m

p

IF

if

IF

i

A/S 2. A real cute one from Jan Daviswho writes: •

The wife of a man who grew barleyWas also the sister of Charlie.Her Neighbour grew hay

And was married to Ray,

And one of these girls was named Carly.

The girl who was married to WayneLived next to the farm that grew grain.She liked to eat celery

That was grown by Valerie,Andsheweighed 80pounds morethanJane

The woman whose husband grew dillWas never married to BillWhen Jane married Benny

And Ray married Jenny,

She went out drinking with Jill.

Norc: Only one couple has rhyming names.

SEND PROBLEMS, SOLUTIONS,

AND COMMENTS TOALLAN /. COT-

TUEB, '67, THE COURANTINSTI

TUTE, NEWYORK UNIVERSITY, 251

MERCER ST., NEW YORK N.Y. 10012,

OR TO: GOTTUEB@NYU.EDU

A/S 3. Our last regular problem is"Golomb's Gambits" edited by SolomonGolomb in the Johns Hopkins magazine.You are to dissect the figure below intofour congruent pieces.

Speed Department

Tom Lydon asks: Two days ago I was38. Next year I start my 42nd year. Howold am I, what is my birthday, and whatis today's date?

Solutions

APR I. As indicated this problem was misprintedin April and the corrected version appears above.

APR 2. Arthur Wascrman sent us the followingsolution. The general problem of minimizing thenumber of training sessions required when thereare n robots serving n+k users is solved as easilyas the n=7, k=3 case. Clearly no robot can betrained to less than k+1 users; if a robot has only kpotential users and precisely those users do notrequire the services of a robot then the remainingn users cannot be satisfied. Thus the minimum is

at least n(k+l). This number is easily seen to beadequate. Number the users from 1 to n+k andtrain robot j to user j and also to the k users n+1 ton+k. If any of the first n users require the servicesof a robot, they use the correspondingly numbered robot; if they do not require the use of arobot their robot is freed up to serve one of the kusers n+1 to n+k.

Robert High notes that we are assuming thatwhen it is time to assign robots to workers, weknow in advance all the workers to whom robotsare to be assigned.

APR 3. Our final solution is from Joel Brainard:

First, draw an equilateral triangle in a circle ofradius "a" and bisect the triangle with anotherradius as shown below. (While it is not necessaryto use the circle, it is a convenient device thathelps show what is going on.)

By construction we know that the length of linesegment BD is a/2 and that angle A is 60 degreesHence angle a1 is 30 degrees.Second, draw chords BE and EC

B

By construction, triangle ABE is isosceles with thelength of sides AE and AB equal to a. This meansthat angles B and E must equal one another andbe 75 degrees each.

By drawing parallel (and perpendicular) linesas shown below we can see that we have constructed the figure presented in APR 3 where AFis a/2 in length and the angle in question is 75degrees.

Other Responders

Responses have also been received from M. Fountain, K. Rosato, S. Theriault, L. Steffens, R. Whitman, A. Cangahuala, W. Hartford, R. Arrison IrE. Sard, G. Stallings, E. SignoreUi, H. Hodara, P.Card, S. Root, N. Wickstrand, B. Gunther.

Proposer's Solution to Speed Problem

39,31 December, 1 January.

TECHKOLOCYREVIEW SIIT47

MIT LIFE INCOME FUNDSALBERT J. O'NEILL

HOME: Milton, Massachusetts

CAREER: The son of a contractor and builder,Mr. O'Neill decided during high school to go into

the construction industry. MIT was offering a new

course in building engineering and construction,and Mr. O'Neill knew a family that sent four sonsthere, so the decision to attend MIT was easy.

Earning his S.B. in 1932, he worked in govern

ment geological surveys for three years, before

becoming a construction engineer for private

contractors. He spent the war years as a super

intendent of construction, specializing in marine

installations, for the U.S. Navy, and then joinedMIT's building engineering department, whichwas later absorbed into civil engineering. Hetaught, conducted research and published inthe field of materials—primarily wood, metals

and masonry. He later became involved withMIT's Technology and Development Program,

a project funded by the U.S. Agency for Inter

national Development to help developing countries strengthen their engineering capabilities.

He retired in 1988.

Mr. O'Neill's hobbies are travel, photography,

gardening and local senior activities. He and hiswife Helen have been married 42 years.

GIFT OF CAPITAL: The Albert J. O'Neill (1932)and Helen J. O'Neill Fund in the MaclaurinPooled Income Fund.

QUOTE: When I considered the effect MIThas had on my life, both professional andpersonal, I had no trouble deciding to makea gift to the Maclaurin Pooled Income Fund.It has been a tax advantage for me and,

at the same time/has benefitted MIT.

For more information about gifts

of capital, write or call Frank H.McGrory or W. Kevin Larkin at

MIT, 77 Massachusetts Avenue,

Room 4-234, Cambridge,Massachusetts

021394307;

(617)253^827.

Photo:

Richard Howard

w- ■

ii

PUZZLE CORNER

ALLAN J. GOTTLIEB, '67

Roving Riverward

Since this is the first issue of a new

academic year, I once more review

the ground rules under which this

department is conducted.

In each issue I present three regular

problems (the first of which is chess,

bridge, go, or computer-related) and

one "speed" problem. Readers are invit

ed to submit solutions to the regular

problems, and three issues later, one

submitted solution is printed for each

problem; I also list other readers who

responded. For example, solutions to

the problems you see below will appear

in the February/March issue and this

issue contains solutions to the problems

posed in May/June. Since I must submit

the February/March column in Novem

ber, you should send your solutions to

me during the next few weeks. Late

solutions, as well as comments on pub

lished solutions, are acknowledged in

subsequent issues in the "Other Respon

dents" section. Major corrections or

additions to published solutions are

sometimes printed in the "Better Late

Than Never" section.

For speed problems the procedure is

quite different. Often whimsical, these

problems should not be taken too seri

ously. If the proposer submits a solution

with the problem, that solution appears

at the end of the same column in which

the problem is published. For example,

the solution to this issue's speed prob

lem is given below. Only rarely are com

ments on speed problems published.

There is also an annual problem, pub

lished in the January issue of each year;

and sometimes I go back into history to

republish problems that remained

unsolved after their first appearance.

Problems

OCT1. We begin with a Bridge problem

from J. Harmse who notes that the high

est possible declarer score is obtained

by playing INT redoubled vulnerable

making all 13 tricks. The problem is to

devise a distribution of the cards in

which the above occurs with "normal"

bidding and play.

SEND PROBLEMS. SOLUTIONS, AND

COMMENTS TO ALLAN). GOTTLIEB,

■67, THE COURANT INSTITUTE,

NEW YORK UNIVERSITY, 251 MER

CER ST., NEWYORK, N.Y. 10012, OR

TO: COTTUEB@NYU.EDU

OCT 2. The following problem is from

Robert Sackheim. A is 73 feet from a

straight river, and B is on the same side

of the river but not so far from it. M and

N are the points on the river nearest to

A and B respectively. The length of AB,

MN and BN are whole numbers of feet.

Joan walks from A to B via the river

(i.e., at one point she is at the river), tak

ing the shortest possible route, and this

is also a whole number of feet.How far

does she walk? What is the direct dis

tance from A to B?

OCT 3. Richard Hess entitles this one

'The missing term" and writes: Given

the series

...,35,45,60,x,120,180,280,450,744,1260,...

the problem is to find a simple continuous

function to generate the series and from it

to determine the surprise answer for x.

Speed Department

There are 13 diamond cards in a card

deck. How many diamonds are on those

13 cards?

Solutions

M/J 1. We begin with a Bridge problem that

Winslow Hartford sent us from the London Sunday

Observer. In the hand shown. West missed the

killing diamond opener against 7H and instead led

the spade jack. How can South now make the grand

slam?

North

* Q95

v 10653

* AQ

* J872

West East

* J1087 * K6432

* 42 v 9

* 10965 ♦ KJ974

* 964 4. 53

South

* A

* AKQJ87

* 32

* AKQ10

The following solution is from Jerry Grossman.

Win the spade ace, cash five hearts, pitching the

diamond from dummy, and play the A, K, Q of

clubs. Now with this 4-card ending, lead the last

club to the Jack:

North

* Q9

West

* 108

» KJ6

* 109

« A

* JEast

* K6

♦ KJ

South

*

v 7

* 32

* 10

On the club. West must discard. If he discards a

spade.then you can lead the Q of spades offdummy

for a ruffing finesse against East's king, pinning

West's ten. The ace of diamonds provides the entry

back to the board. So assume that West discards a

diamond.

Now East is squeezed. If he discards a spade,

then ruff a spade and dummy is good. If he dis

cards a diamond, then cash the ace of diamonds,

dropping the opponents' remaining cards in that

suit, and your hand is good. A neat trump squeeze.

M/J 2. Gordon Rice is thinking of four positive integers

0<A<B<C<D

that have a curious property. When numbers arewritten in base D

ABsAimoduloO

and BA a BOnoduloQ.

For what values of D do solutions exits? Are they

unique? Note that AB does not represent AxB.

Instead it signifies juxtaposition, e.g., if A=24 and

B=345,AB is 24345.

Robert High writes: Gordon Rice's relationship is

far from unique,-1 found 7,268 solutions with D<100

and 98 with D<20. Since0<A<B<C<D, the condi

tions boil down to the simultaneous congruences

AD+B n A mod C

B-D*A b B mod C

(The fact that A and B are "written in base D" is

really irrelevant.)

A little manipulation leads to the conclusion that

these conditions are satisfied if and only if we can

find A < B < C < D such that

(B+/0-D is divisible by C

and (B-AMD-2) is divisible by C.

These conditions are satisfied by many families of

quadruples; a simple three-parameter family is

C=B+A; D=K-C+2, but there are many other solu

tions as well, such as (1,7,12,18) or (13,20,21,98).

As noted by Richard Hess, it is easy to see that no

solutions exist for D<5, unique solutions exist for D=5

and D=6, and many solutions exist for every D>6

M/J 3. Daniel Morgan wants to know the expected

point count for a randomly dealt Bridge hand of 13

cards? High cards are valued as Ace=4, King=3,

Queen=2, and Jack=l. In addition a void (no cards

in a suit) contributes 3 points, a singleton con

tributes 2, and a doubleton contributes 1.

Stephen janowsky sent a solution involving fairly

little calculation (i.e., a computer was not required).

He uses the notation E0 for expected value, P0 for

probability and # for "number of and writes:

The expected high card value of a bridge hand is

easily determined using the additivity of expected

values:

E(ace points) = 4E(# aces) - (4)(4) E(# spade aces)

= (4)<4) PCspadc ace) = (4)(4){l/4) = 4. Thus

E(high card pts) = 4 + 3 + 2 +1 = 10.

E(void pts) = 3E(# voids) = (3X4) E(# spade

voids) = (3)(4) Pfcpade void) = (3)(4) (39 choose

13)/(52 choose 13)

E(singleton pts) = (2X4) P(spade singleton) =

Continued on Page MIT 54

TECHNOLOGY REVIEW MIT 55

MH

B

I

Robert W. Barker, '21; January 21,1992; Narbelh, I'a.

Richard P. Windisch. '21; May 4,1992; Naples, Fla.

H.FeltonMetcalf,'22;Mayl2,1992;Newmarkel,N.H.

Charles W. Ufford, '22; April 29,1992; Newlown. Pa.

Edwin M. Goldsmith, Jr.,'23; May 4,1991; Wyncote,

I'a.

Herbert L Hayden, '23; May 2,1992; Lancaster,

Mass.

Max W. Tetlow, '23; May 14,1992; New London, N.H.

Jacob Lurie, '24; April 9,1992; Denver, Colo.

John E. Black, '25; February 2,1992;Clearwater, Fla.

Homer S. Davis, '25; June 6,1992; Seattle, Wash.

F. Cushing Foss, '25; May 7,1992; Westford, Vt.

Isaac W. Cleason. '26,SM '27; May 21,1992; Middle-

bury, Vt.

Albert S. Goleman, MAR'25; November 23,1991;

Houston, Tex.

NaomiC Turner, '26; May 12,1992; Arlington, Mass.

Winfred F. Dunklee, '27; July 25,1991; Hamden,

Conn.

Arthur Dunlevy, '27; May 1,1992; North Falmouth,

Mass.

Robert C Wallace, '27; April 21,1992; St. Charles, 111.

Homer A. Burnell, '28; June 26,1992; Chicago, III.

Harold L. Geiger, '28; April 18,1992; Minneapolis,

Minn.

Robert Walker Hunn, Jr., '28, MAR '29; January 1,

1992,Santa Monica, Calif.

Willis G. McCown, '28; May 6,1992; Chicopee Falls,

Mass.

James Cooper, '29; May 25,1992.

Alfred H. Hayes, '29, SM '30; March 10,1991; Whit

ing, Ind.

Laurence A. Horan, '29; May 8,1992; North

Chatham, Mass.

Robert EJackson. '29; May 3,1992; Marblehead,

Mass.

Annand M. Morgan, SM '29; April 25,1992; Port

land, Me.

Edward B. Papenfus, SM '29; June 5,1992; Vancou

ver, BC, Canada.

Leonard C Peskin,'29, SM'31, ScD'36; July 12,1991;

Wyncote,Pa.

GAZETTEMIT ALUMNI JOB LISTINGS

A bi-weekly bulletin giving em

ployers the opportunity to reach

experienced MIT graduates,

and MIT graduates the oppor

tunity to scan the market.

If you are an employer looking

for talent, or an MIT graduate

looking for new directions, con

tact Bonny Hafner at the ad

dress below.

For a subscription to the

Gazette clip and mail this ad,

enclosing a check (made out to

MIT) for $12 for six months or

$20 for one year.

NAME

STREET

CITY STATE ZIP

MAIL TO: MIT ALUMNI

CAREER SERVICES, RM 12-170.

CAMBRIDGE, MA 02139

617-253-4733

MarkC Cuibrcath, '30; April 12,1992; Kaymore, Md.

Charles D. May, '30; June 13,1992.

Lloyd E. Montgomery, '30; April 29,1991; Colorado

Springs, Colo.

George D. Love, '31; April 28,1992; Portland, Me.

Carrington Mason, '31; March 12,1992; Houston, Tex.

Robert Sanders, '31; May 31,1992; Annapolis, Md.

Howard L. Richardson, '31, SM '32; May 1,1992;

New Britain, Conn.

Allan L. Dunning, SM '32; April 22,1992; Stoning-

ton. Conn.

j. Richard Rafter, '32; May 18,1992.

Donald C Sanford, '32; April 28,1992; Woodbury,

Conn.

HarlandH. Young, Jr., PhD'32; December 31,1991;

Columbus, Ohio

Gardner Harvey, '33; February 23,1991; Wilming

ton, Del.

Frederick A. Ladd, Jr., '33; November 12,1991; Lees-

burg, Fla.

Hugh W. MacDonald, '33; April 29,1992; Green

Brook, N.J.

Meredith E. Morgan, '33; May 3,1992; Kerhonkson,

N.Y.

Charles E Quick, '33; October 26,1990; Traverse

City, Mich.

Emst W. Spannhake, '33, SM '35; April 30,1992;

Akron, Ohio

Robert C. Wellwood, '33; May 15,1992; Saline, Mich.

Jack Delmonte,SM '34; April 15,1992;GlcndaIe,

Calif.

Albert M. Grass, '34; May 29,1992; Wollaston, Mass.

Louis T. Montant, Jr., '34; 1992.

Arthur O. Williams, Jr., '34; May 16,1992;

Doylestown, Pa.

John L Fuller, PhD'35;June8,1992;Cambridge,

Mass.

Chester H. Brown, Jr., '37; September 22,1989; Pitts

burgh, Pa.

Rolf E. Schneider, '37; January 24,1991; Parkers-

burg, W.V.

F. William Brown, III, '38; May 15,1992; Kensington,

Calif.

Richard H. Koehrmann, '38; May 3,1992; Alton, III.

John J. Perkins, '38; November 15,1991; New Bem,

N.C.

William S. Quigley, Jr., '39; June 6,1992; North Scit-

uate. Mass.

George E.B. Hill/ 40; May 7,1992; Kentficid, Calif.

Charles S. Butt II, '41; April 16,1992; McLean, Va.

Charles Margnettl, '41; June9,1986; West Roxbury,

Mass.

John W. Clarke, '42; April 10,1992; Missouri City,

Tex.

John Hinchman, '42; May 11,1992; West Cornwall,

Conn.

Bernard Brindis, '43; May 29,1992; Boca Raton, Fla.

Tan Chih Lu, SM '44; June 2,1992; Tustin, Calif.

Walterj. Loughlin, '46; May 2,1992.

James E Haggett, '47; May 26,1992; Shrewsbury,

Mass.

Victor H. Pomper, '48, SM '50; May 9,1992; Weare,

N.H.

Roger D. Smith, '48; March 30,1992; New Paltz, N.Y.

James C Buck, '49, SM '65; August 25,1990; Coron-

ado, Calif.

Donald P. Germeraad, '50; May 11,1992; Under

wood, Mass.

Gerard J. Griesmer, '52; May 29,1990; Aurora, Colo.

Howard K. Larson, '52, SM '54; August 5,1991;

Saratoga, Calif.

George L. Tuer, Jr., ScD '55; March 5,1991; Aikcn, S.C.

Dorothy O. Schlag, '56,1992; Redlands, Calif.

John E. Murray, Jr., '57; 1992; Framingham, Mass.

Amarjit Singh, '61; July 11,1991; New Delhi, India.

George Piotrowski, '64, SM '65; May 24,1992; Cain-

seville, Fla.

Dirk Berghager, '62, SM '64; January 29,1984; Cas-

cais, Portugal

William J. Day, '67; May 4,1992.

Charles J. Sheehan, SM '67; May 12.1992; Andover,

Mass.

Kathleen S.Sargent, PhD'71; February 18,1991;

Winchester, Mass.

Mark E Schaefcr, PhD'77; April 13,1992; Atlanta, Ga.

Albert B. Lester, '82; June 7,1992; Westwood, Mass.

Paul J. Ramos. '86; May 24,1992; Seekonk, Mass.

PUZZLE CORNER.

Continued from Page MIT 55

(2)(4) (13X39 choose 12) / (52 choose 13)

E(doubleton pts) = 4 Ptspade doubieton) = (4)

(13 choose 2)(39 choose 11) / (52 choose 13)

Combining the above, E(distribution pts) =

(19)(29)(31)(37)(239) / [ (23)(41)(43)(47)(49) 1

=" 1.61748

So E(pts)=" 11.61748

Robert High assumes that the proposer "DANIEL

MORGAN is your MAIN GOREN LAD," which

just goes to show what happens when you start

hanging out with Nob. Yoshigahara.

Better LateThan Never

M/J 1. Darold Rorabacher and George Blondin

noticed that numbers ending in one were inad

vertently omitted. For example, the fourth n

with F(n,n)=G(n,n)=n is "two hundred one."

There are 95, not 64 solutions as previously

claimed.

SD. I really do not normally include comments

on speed problems but quite a ruckus has

occurred regarding the minimum number of

pitches in a complete baseball game and the num

ber of calls by the first place umpire. I somehow

cannot resist printing the following from Tony

Carpentieri but will try hard to refrain from

speed problem follow-ups in the future

"I disagree on the number of pitches/plate

umpire calls in a complete game. There are

things that a pitcher can do, such as (I believe)

touching his tongue to his pitching hand that

result in one ball being called. So, each half

inning goes like this: Pitcher licks hand (or other

stuff) 12 times in a row. All calls made by field

umpires. He then picks off the three runners,

with calls naturally made by field umpires. Well,

this goes on for a bunch of half innings, let's say

17. Then a pitcher licks his hand 16 times, and

walks a run home in the bottom of any inning

after 8. Pitches: 0, plate umpire calls: 0."

In addition, Joseph Gurland tells me that our

problem was printed in rec.sport.baseball, a

popular electronic newsgroup. Gurland sent

copies of several other no-pitch solutions found

by newsgroups readers.

Other Responders

Responses have also been received from M.

Chellino, S. Feldman.M. Fountain, D. Grant, A.

Halbcrstadt, M. Handel, R. Hedrick M Herbert,

B. Inadomi, R. Loesch, L. Nissim, A. Ornstein,

X. Peng, K. Rosato, J. Rosenthal, G. Schwartz,

and A. Wasserman.

Proposer's Solution to Speed Problem

81. Each of the 13 cards has a diamond in the

upper left corner and one in the lower right

corner, for a total of 26. The Jack.Queen, and

King have no other diamonds on them. The

Ace throughlO have 1+2+3+4+6+7+8+9+10=55

additional diamonds.

MIT 54 OCTOBER 1992

accomplished essayist who has written widely on top

ics in thehumanities and social sciences.

XXIINUCLEAR

ENGINEERING

Robert W. Davis, SM '80, NUE 'BO, writes: "My

employer is TRW Space & Technology Group in

Redondo Beach, Calif. I am the advanced system

manager in LightSat Systems. We are working

on the leading edge of satellite technology and

space systems, with emphasis on the powerful,

miniaturized technologies that enable the vastly

increased capabilities of small "LightSats" of the

300-800 lb. class, at low costs for development,

fabrication, and costs. This new position is

responsible for continued growth (through gov

ernment programs, TRW investments, and

acquisitions) and presence in this new market

segment. Lead for S380M in sales in 1991. Work

ing additional S740M in sales for 1992."

Mark Melvin, a Course XXII graduate student

from 1984-86, died June3,1992 from complications

resulting from AIDS. Melvin received a BS from the

University of Michigan in 1984. He was active in

the American Nuclear Society Student Chapter,

serving as its social chair in 1985. Hecame to MIT

from Bloomfield Hills, Mich., and his research

focused on fusion reactor design studies.

XXIVLINGUISTICS

AND PHILOSOPHY

John J. McCarthy, PhD79,aUniversityofMas

sachusetts professor in the Department of Linguistics,

has received a Guggenheim Fellowship. He is the direc

tor of the Linguistics Department Graduate Program.

He will beusing his grant money tocompleteresearchin theoretical linguisticsand finish a book. The book.

cowritten with Allan Prince, a linguistics professor at

Rutgers University, concerns the processes ofword for

mation in relation tosound structure. McCarthy will

spend this academic ycaron sabbatical.

TPPTECHNOLOGY AND

POLICY PROGRAM

Kent W. Hughes, SM '85, has informed us that his

homeand all its possessions were destroyed in the

October20,1991,firestorm in the Oakland/BeverlyHills area (his father's homewasalso destroyed). They

arc rebuilding and would love to hear from otherTFP

alums.... Steven C. Anderson,SM'87, will be back in

theCambridge area beginning in September. He will be

pursuing a PhDat the JFKSchool ofGovernment atHarvard University.... JessicaStem,SM '88, received

a PhD in public policy from Harvard in last June's com

mencement. ... Dava Newman,SM '89, SM '89 (XVI),

PhD '92 (XVD, has been elected to the MITCorporation

asone of the "representatives from recent classes."Congratulations! Dava is thesecond TPP alum to

receive this honorand recognition in leadership in tech

nology and policy, in the past few years, joining Robin

Wagner,SM '86.... Lee Newman,SM '92, SM '92 (XV),

joined the staffat McKinsey& Co., Inc., in Chicago, III.,

this past September.... Michael "Mick" Rookwood,

SM '92, is an environmental consultant with the Eastern

Research Group, Inc.—Richard de Neufville, Technol

ogy and PolicyProgram, MIT,Room E40-252, Cam

bridge, MA 02139.

STSPROGRAM IN SCIENCE,

TECHNOLOGY &

Professor Deborah Fitzgerald lectured on sustain

able agriculture in the United States in May. Her

talk was part of a conference entitled En/Gender

ing Environmental Thinking, sponsored by the

MIT Women's Studies Program Professors

Loren Graham and Kenneth Keniston have written chapters in The Outlookfor STS, edited by Sheila

Jasonoff and published by the Department of Sci

ence and Technology Studies at Cornell University.

... Professor Emeritus Carl Kaysen has initiated a

project exploring evolving norms that legitimize

international intervention in sovereign states. The

project is under the auspices of the American

Academy of Arts and Sciences Professor Emeritus Leo Man spoke at a conference on American

Studies in Germany in early June at Bad Munsteram Stein, Germany Professor Merrltt Roe

Smith organized and chaired a session on the Euro

pean Roots of the American System at the annual

meeting of the Society for the History ofTechnol

ogy in Uppsala, Sweden, in August Professor

Leon Trilling headed a two-week workshop for 60

middle-school teachers in July. The theme of theworkshop, held at MIT, was "How Does a City

Work?"... Professor Sherry Turkle was keynote

speaker in May at a Tokyo symposium on

advanced robotics. The conference, entitled Cybera-

nimism: Will Robots Control Human Beings or Live

Together in Harmony in the Year 2019?, took place at

Tokie University.—Phyllis Klein, STS Program,

MIT, Room E51-128, Cambridge, MA 02139

Deceased

The following deaths have been reported

to the Alumni/ae Association since the

Review last went to press:

John C Bradley, '07; May 24,1992; Waterbury,

Conn.

Harold C. Wells, '18; January 26,1991.

George W. Cann, '19; December 30,1991; Conestoga,Pa.

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TECHNOLOGY REVIEW MIT 53

PUZZLE CORNER

ALLAN J. GOTTLIEB, '67

Parking Lotto

Ihad an interesting experience yester

day (23 August). My wife Alice had to

talk at the National Psoriasis Founda

tion in Orlando during the weekend so

the family used the cheap summer air

fares for a chance to see Disneyworld.

Our routing home was Orlando-Miami-

NYC (don't ask). Hurricane Andrew

arrived this morning in Miami. Well, yes

terday at Orlando the final word from the

airline was that if you went to Miami, you

were "on your own" and "the Miami air

port is all glass." There was a noticeable

time warp that occurred at the airport

entrance. While we were driving from

Orlando, Andrew was reported as due to

hit Miami in 14 hours. Once we were

inside the airport, the only word we could

hear is that it would be "soon," and ten

sions were getting pretty high. We decid

ed to chance it, in part because there were

no seats available leaving Orlando for

anywhere. We were also encouraged by

the words of a wise pilot, who noted that

they would not fly a plane into Miami

unless they were dam sure that the plane

could get out. Extrapolating (dangerous

ly?) I concluded that the airline would not

allow its four o'clock flights to fly in

unless it expected they would all get out.

Bottom line: we left Miami at 5:45 in calm

weather. But no flights were landing—the

four o'clock batch was the last

Readers may be interested in an

impressive compendium of puzzles

entitled Index to Mathematical Problems

1980-1984, edited by Stanley Rabin-

owitz and published by MathPro Press.

Also, my NYU colleague Dennis Shasha

has just written Codes, Puzzles, and Con

spiracy, a second Dr. Ecco book. Finally,

our frequent contributor, Nob Yoshiga-

hara, has written a book including a few

problems from "Puzzle Corner." Nob's

book is in Japanese; the only word I can

read is "Puzzle" set in big type on the

cover.

Problems

N/D 1. We begin with a computer-relat

ed problem that Max Hailperin heard

SEND PROBLEMS, SOLUTIONS, AND

COMMENTS TO ALLAN]. COTTUEB,

■67. THE COURANTINSTITUTE.

NEW YORK UNIVERSITY. 251 MER

CER ST.. NEW YORK. N.Y. 10012. OR

TO: GOTTUEB@NYU.EDU

from Albert Faessler. A primitive

Pythagorean triple (PPT) is a triple of

positive integers (a,b,c) such that

a2+b2=c2 and a, b, and c have no com

mon factor (this last condition is what

makes the triple "primitive"). The area

of a PPT is ab/Z. Euler found that the

smallest area shared by three PPTs is

13123110. What is the next smallest area

shared by three PPTs?

N/D 2. Gordon Rice wants you to find

(non-equilateral) triangles containing a

60-degree angle. How about a 30-degree

angle?

N/D 3. Tom Harriman wonders for

what values of X does the following

"infinite exponential" converge.

Speed Department

Speedy Jim Landau sent us a speed

problem from Eric Weill. To avoid the 5-

dollar parking fee at Atlantic City casi

nos, it is necessary to place a bet. What

bet should you make to get your free

parking with the least risk?

Solutions

JUL 1. We begin with a bridge problem from

Richard Hess, who (I guess) always seems to get

low point count hands and likes to see how far they

can go. Inspired by the 1991 Jan 1 problem, Hess

asks for the lowest number of high card points that

North and South can have (combined) and still

make 7NT against best defense.

The following solution is from Edward Sheldon:

South is assumed declarer, and West is on lead.

North/South (NS) must have the ace of every suit

in the West hand. Therefore West cannot have 13

cards of one suit. If West has 3 or 4 suits, NS must

have 3 or 4 aces, for 12 or more points. Since at least

one solution of less than 12 exists, there is no mini

mum solution where West holds more than 2 suits.

For all minimum solutions. West holds exactly 2

suits.. Further, all 13 NS tricks must come from the

same 2 suits, to avoid a third NS ace. There are 26

cards in 2 suits; with 13 in the West hand, the other

13 must be played one per trick to win a grand

slam; therefore they must all be in one hand.

Assuming West and South share the hearts and

spades, with South arbitrarily long in spades, and

that East holds all the point cards in clubs and dia

monds, the following 11-point solution is obvious:

West South

» K A.Q.J->2

• K->2 A

This solution is also a minimum. South's aces have

already been proved necessary, and since the West

spade K must be singleton, the spade Q and/or ]

cannot be moved to the West hand, and by the 2-

suit restriction, they cannot be moved to the East

hand, thus they must stay in the South hand. Since

all 4 honors are necessary, no lower solution exists

and the above is a minimum solution.

JUL 2. Matthew Fountain suggests we tackle the

"hold that line" problem devised by Sid Sackson

and appearing in his book A Garnet of Games.

"Hold That Line" is a game in which two players

alternate drawing straight lines between dots on a 4

x 4 dot field. The player to draw the last line loses.

The first diagram shows a game in which the lines

are numbered in the order they were drawn.

Restrictions are that lines after the first shall only be

drawn from the free end of a previously drawn line.

All lines must be straight and start and end at a dot.

A line may connect more than two dots if all are in

a straight line. No line shall be drawn to a previous

ly connected dot or cross another line.

The second diagram shows a game in progress

where the first player has drawn his first line along

a long diagonal of the field. Is this a winning or a

losing move? There can be no tics.

Game lost bv first flayer Came in fkocress

Steve Altchuler advocates a "copy your oppo

nent" strategy. The second diagram shows an easy

win for the first player. S/he has divided the board

in half, and can now mirror whatever move the sec

ond player makes onto the other half of the board.

Thus, the two halves will always remain identical,

and whenever the second player Is able to make a

move, the first player will be able to make the same

move on the other half.

This strategy means there are many other first

moves which guarantee a win for the first player.

Using spreadsheet-type notation on a 4x4 board,

not only is A4-D1 a win, so are: A1-D4, A2-D3, A3-

D2,Bl-C4,andCl-B4.

This being the case, "Hold that line" now has all

the strategic appeal of tic-tac-toe, unless you ban

first moves that bisect the board.

JUL 3. Geoffrey Landis has found our previous

cryptarithmetic problems (where you are given an

arithmetic equation such as XXX + Y = YZZZ and

must find which digits to assign to each letter, in

this trivial case X=9, Y=l, Z=0) "rather uninterest

ing." So he offers a challenge. Find a cryptarith

metic problem with (precisely) two solutions based

on two (completely) different keys, i.e., no letter is

assigned the same digit in both solutions. I would

not be surprised to find that Nob Yoshigahara has a

few dozen of these sitting under his socks in his

bureau.

Bob High sent us some that even "make sense."

ONE

ONE

ONE

ONE

ONE

ONE

ONE

Continued on Page MIT 46

TECHNOLOGY REVIEW MIT 47

■ • TT1. Tfr AiWW ,

HOME:,ConcG^|Massachiisgtls.

gX0; ,

Sc.D. (1051) degrees from MIT^ncivil.engineerr- .;*?'■'

:.ing,.pr. Aiclrich.'served'six^ears.bntheMIT [ '■'■. . .'; .•'faculty, teachirig^ourses^jisoirmechanicsand ;£: .foundation engirieeringsln^^j.ihe eo.-founded :;f.the consulting engineeringfiriTi Haley & Aldrich;' ■' 'Inc. Based in Cambridg^; the1 "firm spocializesjn

^tdtJ ilgg^gvgeoscit'nces,';withr3/)Uemployeesandeiglitpffices, ■locatedprimaiily i"nthe Northeast:.'Dr. Aldrich is :

.now chairman emeritus of t.R&3U;ni. . Ss:^-"-1 .

Thr6ughoiit'hisx^ireer);Pr. Aldrich has been ; *"

active at MIX He^vafpresident otHhe Aluiruii^-Associatioivin 1980r-81.and served on the |Cmipo-■'ration from'1980 tlfrough 1986; forthree of thoseyears,'he was oii.the Executive Committee. He is

vice chairman:of the Boston Campaign Comniit-

tee.of the Camp;tign>'for the future.

FGAEITAIWT

Gharit!alj||lR(

JiiToJsneadMlny life. Eaenk

Lion for-the pla

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significant contribution that'strengthens this' ™

world-class institution, while providing us with ,.;;

For more information about gifts,o££apiwrite or cairHugh Darden, Frank McGroi^

orKevinLarkinatMIT, . '-"S77 Massachusetts Avenue, Room 4-234;s«J;C;imbticJge?Massa|lius^ts02139-43()7^f:;V..

;Ht$|Mfe:*.P:■i. ■

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toward <i second field, studies of the relationship

of gender and science. Her works in this field

include her biography of Barbara McClintock, A

Feeling for the Organism (written several years

before McClintock won her Nobel Prize); articles

on the relationship between gender, the concep

tion of science, and the idea of objectivity; her

volume. Reflections on Gender and Science; and con

tinuing work on the relationship between gender-

linked concepts and work in biology. In the last

five years, Keller has increasingly turned toward

studies of the history of the life sciences, in partic

ular of molecular and evolutionary biology. Keller

expects to teach at both the undergraduate and

graduate level; her courses will focus on gender

and science and on the recent history of evolution

ary biology. Next year, she will oversee the

weekly Monday-afternoon STS Colloquia.

Hugh Gustcrson, an anthropologist who has

studied nuclear weapons production, joined the

faculties of the Anthropology/Archaeology Pro

gram and the STS Program as assistant professor.

Gusterson received a bachelor's degree in modern

history from Cambridge University in 1980. He

subsequently studied cultural anthropology at the

University of Pennsylvania, where he received a

master's degree, and at Stanford, where he

received a PhD in cultural anthropology in 1991.

He is currently a Weatherhead Postdoctoral Fel

low at the School of American Research in Santa

Fe. Gusterson is best known for his ethnographic

studies of nuclear weapons producers and their

opponents at the Livermore Laboratory. He has

also written on orientalism and the discourse of

nuclear deterrence. In STS, his teaching will focus

on the ethnographic and cultural study of science

and technology Victor McElheny and Eugene

Skolnikoff, '50, SM '50 (VI), PhD '65 (XVII),

attended the second meeting of the Working

Group on "Cultural Aspects of SETI" in Santa

Cruz last May Leon Trilling headed a work

shop for 60 middle-school teachers last July at

MIT. The theme of the workshop was "How Does

a City Work?"

GAZETTEMIT ALUMNI JOB LISTINGS

A bi-weekly bulletin giving em

ployers the opportunity to reach

experienced MIT graduates,

and MIT graduates the oppor

tunity to scan the market.

If you are an employer looking

for talent, or an MIT graduate

looking for new directions, con

tact Bonny Hafner at the ad

dress below.

For a subscription to the

Gazette clip and mail this ad,

enclosing a check (made out to

MIT) for $12 for six months or

$20 for one year.

NAME

STREET

CITY STATE ZIP

MAIL TO: MIT ALUMNI

CAREER SERVICES, RM 12-170,

CAMBRIDGE, MA 02139

617-253-4733

The Mellon Fellowship Committee has selected

three Fellows in the History and Social Studies of

Modern Life Science for the academic year

1992-93. The Fellows will participate in STS semi

nars, colloquia, and workshops while pursuing

their research at MIT. According to Charles

Weiner, committee chair, their research interests

are especially relevant to the theme of the 1993

Mellon Workshop on international aspects of

genetic engineering and biotechnology, including

cultural and political significance. The 1992-93

Mellon Fellows are Richard Doyle, Michael A.

Fortun, and Herbert Gottweis. Doyle is an

instructor in the Department of Rhetoric at the

University of California at Berkeley. He is in the

final stages of writing his doctoral dissertation,

"On Beyond Living: Vital and Post Vital Rhetorics

in Molecular Biology," and has taught "The

Rhetoric of Technology" at Berkeley. His next pro

ject will deal with the use and impact of comput

ers and computational metaphors in contempo

rary molecular biology. Fortun is completing his

doctoral dissertation, "Mapping Genes, Science,

and Society: Charting the Human Genome Pro

ject," in the Department of History and Science at

Harvard. He has also served as a teaching fellow

in the history of science. His next project will be an

historical and sociological study of the context of

the genome project in molecular biology and in

the political economy of the biological sciences

and the biotechnology industry. Gottweis is assis

tant professor in the Department of Political Sci

ence at the University of Salzburg. He is editor of

the A ustrian Journal of Political Science; his pub

lished works include studies of the politics of

biotechnology in Europe. At Salzburg he has

taught a course entitled "Biotechnology, Politics,

and Society in Comparative Perspective." At MIT

he will focus on completing his book, Retextualiz-

ing Life: Genetic Engineering and the State in Western

Europe (MIT Press). Gottweis is a member of the

Advisory Committee for the Mellon Work

shops.—Graham Ramsay, STS Program, MIT,

Room E51-128, Cambridge, MA 02139.

Deceased

The following deaths have been reported

to the Alumni/ae Association since the

Review last went to press:

James M. Ralston, '16; June 13,1992; Trenton, NJ.

HarveyM. King, MAR '23; July 6,1992; Ft. Walton

Beach, Fla.

Howard A. Lockhart, '23; June 18,1992; HaverhiU,

Mass.

Arthur R. Stuckcy, '23; July 15,1992; Tucson, Ariz.

Hyman J. Verner, '23; June 9,1992; Pikesville, Md.

Neil L. Olken, '24; December 12,1988; NewtonCen

tre, Mass.

George A. Whinery, SM '25; July 9,1992; Grand

Rapids, Mich.

Avedis M. Kazazian, '27; January 27,1988; La

Canada, Calif.

Robert G. Loomis,'29, SM '30; August 30,1991;

Manchester, Conn.

Michael Anthony, '32; July 13,1992; North Falmouth,Mass.

Robert H. Hansen, '32; May 19,1992; Lakeside, Calif.

Norman M. Johnsen, '34; January, 7,1992; Atlantic

Beach, Fla.

Willard F. Bixby, '35, SM '36; July 12,1992;

Novelty, Ohio

J. Lawrence Tobey, '36; June 24,1992; Wenham,

Mass.

William A. Williams,SM '36; February 4,1990

Herbert P. Gusdane, '37, SM '38; December 1,1991;

Mentor, Ohio

Charles J. Palmer, SM '37; July 26,1992; East Sand

wich, Mass.

William J. Pattison, '37; June 28,1992; Camden, Maine

Norman E. Weeks, '38; July 8,1992

Donald B. Peck, '39; July 1,1992; Bloomfield,Conn.

William W. Pomeroy, SM '40; July 14,1992; Alexan

dria, Va.

Michael B. Bever,SM '42, ScD '44; July 17,1992; Cam

bridge. Mass.

PUZZLE CORNER:

Continued from Page MIT 47

ONE

ONE

ONE

ONE

ONE

ONE

ONE

MANY

Solutions are: ONE = 362, MANY = 5068; ONE

= 438, MANY = 6132.

TWO + ... + TWO = WAYS (29 TWOs).

Solutions are: TWO = 147, WAYS = 4263; TWO

= 271, WAYS = 7859.

TRICKY

TRICKY

EUREKA

Solutions are: TRICKY = 152697, EUREKA =

305394; TRICKY = 436901, EUREKA = 873802.

Other Responders

Responses have also been received from S.

Altchuler, J. Grossman, D. Garcia, W. Hartford,

R. High, D. Kahn, J. Keilin, J. Libby, E. Lund, A.

Omstein, G. Rice, J. Rosenthal, D. Wachsman,

and R. Yassen.

Proposer's Solution to Speed Problem

Go to a 5-dollar crap table and bet 5 dollars on

"Come" and 5 dollars on "Don't come"

simultaneously.

Robert V. Coleman, '44;January 9,1992; McLean, Va.

Frank J. Huddleston, '45; April 8,1992; Bowie, Md.

Keith Knutzen, '47; October 14,1991; Bronxville,N.Y.

Richard C.B. Berry, '48, SM '49; August 8,1992;

Danielson,Conn.

Earl D. Hoyt, '48; June 6,1992; Menlo Park, Calif.

George G.CK. Mah, PhD '48; May 28,1990;

Evanston, Ml.

Manfred G. Wentzel, '48; June 14,1992; Indian Har

bor Beach, Fla.

Nesblt L. Duncan, '49; June 15,1992; Belmont, Mass.

Richard W. Henderson, '50;July 13,1992; Colonia,

N.J.

Bernard Edelman, '53; June 28,1992; La Mesa, Calif.

Howard W. Wong, '53; August 30,1990; Silver

Spring, Md.

Arthur W. Haines, '54; August 6,1992; Sacramento,

Calif.

William N. Tatmers, PhD '54; July 9,1992; Flushing,

N.Y.

John R. Segal, PhD '59; January 31,1990; NewYork,

N.Y.

Immo-Ragnar H. Nordstrom, '60; April 27,1992; Fair

Haven, N.J.

Robert Akullian, '70; July 20,1992; SanJose, Calif.

Sandra A. Wadsworth, MAR '74; September 15,1991;

Brookline,Mass.

Steven K. McClung, '84;July 11,1992; Atkinson,N.H.

Correction: April's Deceased List erroneously

listed the date ofdeath ofLaurence Edmund

Noble, '23, ofEnid, Okk.,as August 9,1992. It

was actually Mrs. Laurence Edmund Nobte,'23,

theformer Gladys Farmer, whodied on that

date;her late husband wasan alumnusofHar

vard. Ourapologiesfor the error.—Erf.

MIT 16 NOVEMBER/DECEMBER 1992

XVIIIMATHEMATICS

Eugene Gath, PhD'89, writes: "1 have been lecturing

in mathematics at the University of Limerick since

1990. We are a new technological university in Ire

land, aspiring to be 'Ireland's MIT! I leach mathe

matics to scientists and engineers as well as continue

my own research. This year 1 won theTeaching

Excellence Award (worth~$l,700) at the University

of Limerick. I hope to visit MIT to meet with the few

friends there who still haven't graduated!" ... Lance

Forlnow, I'hD '89, assistant professor of computer

science at the University of Chicago, has been named

a National Science Foundation Presidential Faculty

Fellow. One of the 30 young scientists and engineers

recognized for excellence in research and teaching,

Fortnow will receive S500.000 to support his work

over the next five years. Fortnow is an expert in com

putational complexity theory—the study of how

much computer time and memory are ne«led to

solvedifferent types ofcomputational problems. In

his recent research, he has studied methods for veri

fying long, complex computer programs and has

investigated the power of "interactive proofs," a

technique that uses a trustworthy small computer to

check the results of a much larger, not necessarily

reliable computer. He hopes to build on recent work

that has shown that the interactive proof technique

places a limit on how well computers can estimate

answers to some extremely difficult problems.

William W. Roberts, Jr., '64, PhD '69, professor

of applied mathematics at the University of Vir

ginia's School of Engineering and Applied Science

in Charlottesville, Va., has been named Common

wealth Professor. As director of the university's

Mathematical-Computational Modeling Labora

tory, Roberts' professional work focuses in part on

developing the mathematical equations that

express events that are observed in nature and in

the experimental laboratory. He has developed

algorithms for computer models that allow

researchers to study the motion, drag, and stability

of fibers in steady and turbulent gas flows, the

dynamics of hypersonic gas flows, and the struc

ture and dynamics of galaxies. His olher research

areas include intelligent-control systems and simu

lations, intelligent control, manageability, feed

back, computational fluid dynamics, shock waves

and nonlinear phenomena, and prototype tech

nologies for fiber processing operations. Roberts

has been at the UVa since 1969.

XXAPPLIED BIOLOGICAL

SCIENCES

Charles]. Bates, PhD

'57, VP of the technical

group at American

Maize Products Co.,

was elected to the board

of directors of the

American Association

of Cereal Chemists

(AACC). Bates has been

with American Maize

Products Co. for 19

years. He previously

was in the Food Prod

ucts Division and

Industrial Detergents

Division of Procter &

Gamble Co. for 15

years. His work at Procter & Gamble included bak

ery mixes and bulk shortenings and some of his

early work led to the Duncan Hines Angel Food

Cake Mix. Other career responsibilities have

included process and product development for

both new and existing products, customer service,

sales and marketing, and most recently, process

development and plant design and construction for

high fructose corn syrup. A member of AACC since

1972, Bates has served the cereal chemistry profes

sion in many capacities including as member of the

C.J. Bates

AACC short course faculty of Introduction to

Cereal Science and Technology, member of the

AACC Cereal Foods World F.ditorial Advisory

Board, president and treasurer of the Institute of

Food Technologists, part of the VP program of the

Calumet Council BSA and chair, vice chair, and

member of the anniversary celebration committee

of the AACC Carbohydrate Division.

The Association of Alumni and Alumnae has

been notified that Kathleen S. Sargent, PhD '71, of

Winchester, Mass.,died February 19,1991,asa

result of an automobile accident. No further infor

mation was provided.

XXIINUCLEAR

ENGINEERING

Patrick M. Hogan,SM '89, writes: "1 am currently a

seniorengineer with ABB Impcll Corp. in Lin

colnshire, HI. Presently, I am the project engineer on a

project that involves performing various engineering

analyses to support an upcoming dual unit outage

and the associated modification work for the Zion

nuclear power plant in Zion, III."... Philip F.

Palmedo,SM '58 (X), PhD '61, has been named presi

dent andCEOof the newly formed Long Island

Research Institute. 1.1 RI was formed when three of

Long Island's major research institutions joined forces

to accelerate the flow of technology to industry and to

contribute to Long Island's economic development.

Concludinga year of discussions, the University of

Stony Brook, Associated Universities (the managing

corporation of Brookhaven National Laboratory), and

Cold Spring Harbor Laboratory agreed to create the

URI. The non-profit corporation will focus on bring

ing new technologies from the academicsphere to the

world ofcommerce. Collectively, the three founding

institutions employ more than 1,500 PhDorMD

researchers and annually conduct more than S400 mil

lion in sponsored research. In its technology transfer

activities, the Institute will have an active program of

technology evaluation and will facilitate the applica

tion of technologies through licensing agreements,

joint ventures, and company start-ups.... Richard R.

Sonstelie, SM '68, has been named president and CEO

of Puget Sound Power & Light Co. in Bellevue, Wash.

Prior to this May 1992 promotion, Sonstelie was presi

dent and COOof the company.

Navy Commander David W. Hearding, SM '77,

ENE '77, recently received the Legion of Merit.

Hearding was cited for "outstanding performance

as commanding officer aboard the submarine USS

Pargo, homeported in New London, Conn."

According to a Navy release, "As an extremely

competent and resourceful leader, he performed

his duties in an exemplary and highly professional

manner. His superb leadership abilities were

instrumental in the exceptional performance of the

Pargo, and the remarkable improvement in its

material condition and operations readiness.

Under Hearding's expert guidance, the Pargo per

formed superbly and set standards of excellence

during numerous and ever-changing at-sea opera

tions." Hearding is currently assigned with Subma

rine Development Squadron 12, Naval Submarine

Base, New London, Conn. He joined the Navy in

1968.... Jacquelyn C. Yanch, the W.M. Keck

Career Development Assistant Professor in

Biomedical Engineering and assistant professor of

nuclear engineering, has received a 525,000 Junior

Faculty Career Award from the School of Engineer

ing under a program supported by the GE Founda

tion. The foundation's Faculty for the Future pro

gram is designed to increase the number of women

and members of underrepresented minority

groups on the engineering, science, and business

faculties in the United States. The foundation has

committed $330,000 over three years to MIT.

Yanch, a member of the Course XXII faculty and

Whitaker College since 1989, is involved in three

major projects: the design of neutron beams for

boron-capture therapy for brain cancer victims;

studies of dose distributions of isotopes injected in

human joints; and development of data-visualiza

tion methods for tomography in medical diagnosis.

TPPTECHNOLOGY AND

POLICY PROGRAM

MIT Professors Joel Clark, SM '75, (XV), ScD '72

(III), David Marks, and Richard de Neuiville. '61.

SM '61, PhD '65 (1), traveled to Norway together in

May, along with Frank Field, '78 (XXII), SM '81

(XXII), SM '81 (TPP), PhD '85 (III), as part of an MIT

team establishing relations with a consortium of

Norwegian companies and the Norwegian Techni

cal Hockschule At the end ofJuly, Professor de

Ncufvillc was the featured speaker at the Mexican

Academy of Engineering, speaking on the develop

ment of TPP worldwide and at MIT James

Durand, SM '78, is now designing simulators for

power plant operations at TRAX Corp. in Virginia.

... Paul M. Hauge, SM '83, has been accepted at

Harvard Law School beginning September 1993—

Having completed a PhD in economics at Sloan,

Mark J. McCabc, SM '86, PhD '92 (XV), is now

employed in the Antitrust Division of the U.S.

Department of Justice. There are so many TPPers in

Washington, D.C. Friends can contact him at 703-

329-1718 After spending a year and a half in

Argentina, and running his own company for a

while, Richard Tomlinson, SM '87, has entered the

British Foreign Office, working in the Environmen

tal, Science, and Energy Department.

Michael Masslmlno, SM '88, SM '88 (II), ME '90

(II), PhD '92 (II), has accepted a position with

McDonnell Douglas in Houston, Tex. He will be

working on research and operations for the space

station.... David Gold, SM '90 (XVI),SM '90, is

currently the Associate Director of the Manufactur

ing Technology Center Program at the National

Institute of Standards and Technology. There are

presently seven centers, each in different states,

which help small and medium-sized manufactur

ers modernize and become more competitive

Ziad Oueslati, SM '90, is working for Citibank in

Tunis.... Michael R. Berube, '89 (I), SM '92 (XV),

SM '92, has started a new job with Chrysler Corp.

as an environmental planning specialist working

on alternative fuels, fuel economy, global warming,

and electric vehicle issues.... Lola Matysiak, '91

(I), SM '92, has relocated to Los Alamos, N. Mcx.,

where she is working at the Los Alamos National

Laboratory.—Richard de Neufville, Technology

and Policy Program, MIT, Room E40-252, Cam

bridge, MA 02139.

STSPROGRAM IN SCIENCE,

TECHNOLOGY & SOCIETY

Professor Merritt Roe Smith became the director

of the Program in Science, Technology, and Soci

ety on July 1,1992. He succeeded Kenneth Kenis-

ton, who has completed a five-year term as direc

tor. Smith joined the MIT faculty shortly after the

founding of STS in 1977. He played a major role in

the development of the doctoral program in the

History and Social Study of Science and Technol

ogy, which he has directed since 1987. Smith has

been Metcalfe Professor of Engineering and Lib

eral Arts, co-director of MIT's Context Initiative,

and immediate past president of the Society for

the History of Technology. He is author or editor

of three books and numerous articles and reviews.

He also edits the Johns Hopkins Press series on

the history of technology and serves on the boards

of trustees of the Hagley Museum and Library, the

Museum of American Textile History, and the

Charles Babbage Institute as well as the history

advisory committees of NASA and the Secretary

of the Air Force Professor Evelyn Fox Keller,

known for her work on gender and science and on

the history of biology, has joined the STS faculty

as professor. Keller was most recently on the fac

ulty of the University of California at Berkeley.

Keller began her scholarly career as a natural sci

entist, with a PhD in physics from Harvard. She

went on to work in mathematical biology, and

became known for her collaborative studies of

chemotaxis and aggregation. She then turned

TECHNOLOGY REVIEW MIT 45