1

THEHANDOUT

3,5,20 !

tan!!53= tan!! 1 + tan!!

14

Paul Turner and Ed Staples

CMA conference, 17 August 2019

ThisarticlesummarisesthefindingsthatwereexploredinaseriesofthreearticlespublishedinAAMT’sAustralianMathsEducationJournalin2019.

https://www.mathematicalwhetstones.com/blog

Ourwholelifeissolvingpuzzles-ErnoRubik

5

320 559.036°45°14.036°

MATHEMATICAL PUZZLES IN THE CLASSROOM 45°

2

MathematicalPuzzlesintheClassroom

The‘threesquarepuzzle’wasmadefamousbytheAmericanpopular

MathematicsandSciencewriterthelateMartinGardner(1914-2010).It

appearedinhisinfluentialbooktitledMathematicalCircusfirstpublishedin

1979.

Thesolutiontothepuzzleiscompellingandourcuriositywassufficiently

arousedtoinvestigateitfurther.Inparticularwebegantowonderwhether

addingmoresquarestothepuzzlewouldleadtootherinstanceswhereangles

formedbyanytwoobliquelines(notnecessarilyadjacent)wouldaddtogether

toproduceananglemadebyathirdobliquelinedrawntoanothersquare.

Wefoundmanyresults,suchastheoneshownabove.Forbrevity,eachresult

becameknownasaGardnertripledenoted 𝑎, 𝑏, 𝑐 where𝑎, 𝑏,and𝑐arethe

‘square’numbersinvolvedwhencountedfromtheright.Forexample,the

exampleabovewouldbedenoted 3,5,8 meaningthattheangleγatsquareS!is

equaltothesumoftheanglesεandηatS!andS!.Mathematically,wecould

writetan!! !!= tan!! !

!+ tan!! !

!whichcanbeeasilyverifiedusingacalculator.

𝛾 = 𝜀 + 𝜂

3

Thesearchforacompletedescriptionofallcasesprovedaformidableone.

Gardnertriplesoftheform 𝐹!!,𝐹!!!!,𝐹!!!! ,where𝐹!isthenthFibonacci

number,werenoticedandtheirexistenceforallnwasprovedassuchusingan

inductionargument.Buttherewereothergroupsofresultsfoundincluding

triplesthatinvolvedLucasnumbersandFibonaccinumbersincombination.Still

moretripleswereuncoveredusingacomputerspreadsheet.

Themysterybegantoresolveitselfwhenitwaseventuallyrealisedthatthere

wasanunderlyingmathematicalstructureexhibitedbyalltriples,definableby

thedifferencedbetweenthefirstandsecondelementofeachtriple.Specifically,

wewereabletodemonstrateafteratimethat 𝑎, 𝑏, 𝑐 wasaGardnertripleifand

onlyif𝑐 = !"!!!where𝑑 = 𝑏 − 𝑎wasdivisibleby𝑎𝑏 + 1.

Itwasamuchhardertasktoshowthat𝑑wasrestrictedtocertainintegers.

Specifically,𝑑couldeitherbe1,aprimeoftheform4𝑛 + 1,acombinationof

primesoftheform4𝑛 + 1,ortheirdoubles.Soforexamplethelistofpossible

valuesofdbegin1,𝟐, 5,𝟏𝟎, 13, 17, 25,𝟐𝟔, 29,𝟑𝟒, 37,41,𝟓𝟎, 53, 57,𝟓𝟖,…where

thenumbersinboldaredoublesofcombinationsof4𝑛 + 1primes.

Since𝑑 = 𝑏 − 𝑎wehave𝑏 = 𝑎 + 𝑑andso= !"!!!

= ! !!! !!!

= 𝑎 + !!!!!.In

otherwords,aGardnertripleoccurswhen𝑑divides𝑎! + 1.Soforexamplefor

𝑎 = 3wehave𝑎! + 1 = 10andthen𝑑canbechosenas1, 2, 5or10.

Ifwechose𝑑 = 2then𝑐 = 𝑎 + !!!!!

= 3+ 5 = 8and𝑏 = 𝑎 + 𝑑 = 3+ 2 = 5.The

Gardnertripleemergesas 3, 5, 8 ,whichhappenstobeatripleoftheFibonacci

group.Thevalues𝑑 = 1, 5and10wouldlikewiseproduceotherGardnertriples.

ItisinterestingtonotethateverythreeconsecutiveFibonaccinumbers

𝐹!!,𝐹!!!!,𝐹!!!!exhibitthepropertythat𝑑 = 𝐹!!!! − 𝐹!!divides𝐹!!! + 1.

4

FurtherinvestigationeventuallyledtotheideaofageneralisedGardnertriple

denoted 𝑎, 𝑏, 𝑐 !wheretheobliquelinesweredrawnfroma‘height’𝑡where𝑡

wasalsoapositiveinteger(includingtheoriginal𝑡 = 1).

Thesymmetricconditionuncoveredwasthat,given𝑑! = 𝑏 − 𝑎,if𝑐 = 𝑎 + !!!!!

!!

isaninteger,thenbysetting𝑑! =!!!!!

!!ageneralisedGardnertriple a, b, c !has

theelements𝑎,𝑏 = 𝑎 + 𝑑!and𝑐 = 𝑎 + 𝑑!.Inthislightwebegantorealisethat

thedivisorsofthesumofsquaresquantity𝑆 = 𝑎! + 𝑡!wasthekeyto

understandingtheentireinvestigation.Wecouldselectanytworelativelyprime

numbers𝑎and𝑡,determine𝑆,andthenlookfordivisorsof𝑆tocreatetriples.

Forexample,selecting𝑎 = 3and𝑡 = 5yields𝑆 = 34.Adivisorof𝑆is2soby

setting𝑑! = 2and𝑑! = 17,thegeneralisedGardnertriplebecomes a, b, c ! =

3, 3+ 2, 3+ 17 ! = 3, 5, 20 !.Thustheanglerelationship,whenstated

mathematically,becomestan!! !!= tan!! !

!+ tan!! !

!"andthiscanbeeasily

verifiedbyacalculator.

Therearequiteanumberofotherresultsthatwerefoundalongtheway,and

thesearediscussedinthethreearticlesthathavebeenpublishedinthenew

AAMTAustralianMathematicsEducationJournal.

Forustheimportantobservationoverthecourseoftheprojectwastodowith

themotivationprovidedbythesimplepuzzleandtheincreasingcomplexityand

sophisticationofourresearch.Wethinkpuzzlescouldhaveasimilareffecton

students. P&E.

𝑎𝑏𝑐

𝑡