MathsJam2015

DonaldBell"CuriousandInterestingTriangles"

orWhatmakesa"nice"puzzle?

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TheCharacteristicsofa"nice"Puzzle

•  Challengingbutnotimpossible(solutiontime10-20minutes)

•  Nottoohard(orexpensive)tomake•  Wellpresented(looksgoodonthecoffeetable)

•  Containsasurprise(an"Aha!"moment)

Let'slookatthe"Hexasperation"and"3-4-5Symmetry"puzzles

Easytomake?-yes,lotsofsimilarpiecesLooksgoodonacoffeetable?-yes,inawellXittingframeChallenging?-well,no!notevenforathree-yearold(butit'sastart)

Versionzero:Equilateraltrianglesinaregularhexagonalframe

Developingthe"Hexasperation"puzzle

Developingthe"Hexasperation"puzzle

Itispossibletohavesixdifferentscalenetrianglesmakingupthehexagon.Thatmeansupto12differentlengthsand18differentangles.Butitistooeasytoputthematchingedgestogether.AnditismuchtodifXiculttomake!

Developingthe"Hexasperation"puzzle

Ifthelengthsofthe"spokes"(a,b,c,d,eandf)aremadethesameastheedges,thenthepuzzlebeginstobeabitmorechallenging.

Thethreecolouredtriangleseachhaveoneedgeoflengtha.Sowedon’timmediatelyknowwhichtrianglesshouldbetouching.

Developingthe"Hexasperation"puzzle

Ifthenumberofdifferentlengthsoftheedgescanbereducedtofour,thenthepuzzleisalotmorechallenging.

Thethreecolouredtriangleshaveoneedgeoflengthd.Andfourofthetriangleshaveatleastoneedgeoflengthb.

Developingthe"Hexasperation"puzzle

Butunlessthelengthsofthetrianglesarechosenwithsomecare,thecentralangleswillnotaddupto360degrees

Thecosineformulaforcalculatinganangleofatriangleis:cos C = (a2 + b2 - c2) / 2ab An Excel spreadsheet is needed to get the six triangle shapes right

The"Hexasperation"puzzlespreadsheetWithsome"trialanderror"(andaspreadsheettodothetrigonometry),thisistheXinalversionof"Hexasperation".

Thesidesareintegers(5,6,7and8)andtheanglesatthecentreaddupto360.26degrees(whichisamuchsmallererrorthantheerrorsinwoodworking)

The"Hexasperation"puzzle

Thesidesareintegers(5,6,7and8)andtheanglesatthecentreaddupto360.26

Evaluatingthecriteriaforagoodpuzzle:•  Challenging?–certainly•  Easytomake?–well,threeofthesixtrianglesareisosceles•  Wellpresented?–no,acloseXittingframewouldmakethepuzzletooeasy.

•  Asurpriseor"AHa!"moment?–alas,no.

"thisslideisdeliberatelyblank"

Itiscoveringupthenextstepsintheargument,becausethatwouldtellyoutoomuchabouttheXinalpuzzle

The"3-4-5Symmetry"Puzzle

Surprisingly,theXinalpuzzlehasXivetriangles,notsix,anditmight,ormightnot,haveanythingtodowithhexagons.Andallthetrianglesareidentical.ButitDOESsatisfythefourconditionsofa"nice"puzzle!

The"3-4-5Symmetry"Puzzle

CanyoutakeFIVEofthesetrianglesandmakeasymmetricalXigure.Thatis,onewherethelefthalfisamirrorimageoftherighthalf.Andtherearenoholesinit.ThetrianglesmustlieXlatonthetable.Theymayberotatedorturnedover,butmaynotoverlap.

(thisis,obviously,nottherightanswer)

The3-4-5triangleisright-angled(because32+42=52)

The"3-4-5Symmetry"Puzzle

Challenging?–yes,ittakesquiteawhile

Easytomake?–yes,allthepiecesarethesame

Nicelypresented?–itcansitinaholder,likethis

Asurpriseor"Aha!"moment?–oh,yes

(copiesavailablefromme,eithercardorhardboard)Ifyousolveit,pleasekeepithidden.

DoesitXitthefourcriteria?YES!