MathsJam 2015 Donald Bell Curious and Interesting...
Transcript of MathsJam 2015 Donald Bell Curious and Interesting...
MathsJam2015
DonaldBell"CuriousandInterestingTriangles"
orWhatmakesa"nice"puzzle?
.
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!