Mark Saul

Program Director

Elementary, Secondary, and Informal Education

National Science Foundation

…Or Fabric?

Embroidery…

City University of New York Research Foundation

New York City Interscholastic Mathematics League

American Regions Mathematics League

Research Science Institute, MIT

Bronxville High School (ret.)

Bronx High School of Science

Bronxville High School (ret.)

Bronx High School of Science (ret.)

1974

2004

There are 22= 4 subsets of the set {a,b}. Arrange these in a sequence so that each subset differs from the previous one in having exactly one new element added or exactly one old element deleted.

  Ø Ø {a} {b}

{a} {b} {a,b} {ab}

{a,b} {a,b} {b} {a}

{b} {a} Ø Ø

Examples (reading down):

Ø a

b ab

There are 23=8 subsets of the set {a,b,c}. Arrange these in a sequence so that each subset differs from the previous one in having exactly one new element added or exactly one old element deleted.

  Ø Ø Ø

a b c 

ab cb ab 

b c a 

bc ac ab 

c a b 

ac ab bc 

abc abc abc 

Ø a

bab

bc

acc

abc

a b c 

0 0 0 (0,0,0){a} 1 0 0 (1,0,0){b} 0 1 0 (0,1,0){c} 0 0 1 (0,0,1){a,b} 1 1 0 (1,1,0){b,c} 0 1 1 (0,1,1){a,c} 1 0 1 (1,0,1){a,b,c} 1 1 1 (1,1,1)

Ø

“Characteristic Functions”

(0,0,0)(1,0,0)

(0,0,1)

(1,1,1)

(0,1,1)

(0,1,0)

X

YZ

(1,0,1)

(1,1,0)

Binary Counting 0 0001 0012 0103 0114 1005 1016 1107 111

Binary Gray Code

X

YZ

0 0001 1002 1103 0104 0115 0016 1017 111

Hamiltonian Circuits

Tower of Hanoi

Chinese Rings

Space-Filling Curves

Campanology (Bell-ringing)

Subsets

Counting

Geometry

Dimension

Characteristic function

Coordinates

Graph theory

Hamiltonian Circuits

Symmetry

Induction

Recursion

What else?

IS THIS EMBROIDERY ON THE FABRIC OF

MATHEMATICS?

OR IS THIS THE FABRIC OF

MATHEMATICS?

• NSF 04-600• EMSW21 (Enlarging the Mathematical

Science Workforce for the 21st Century)• MCTP (Mentoring Through Critical

Transition Points)

Funding? Check out:

Or talk to:John Conway, DMS: jconway@nsf.gov

Liz Teles, DUE: ejteles@nsf.gov