Chiralitydef is_left_handed(pips):Eventhoughthishasnoeffecton fairness,pips fromonetosixarenotpaintedondiceanywhich way,butso that thepipson theopposite facesalwaysaddup toseven.(Thisconventionmakesit easiertotellwhensomeonetriestousegaffeddicethatleaveoutsomespotvalues.)Ineachcorner ofthecube,onevaluefromeachpairofopposites1-6,2-5and3-4meetstwovaluesfromtheother twopairsofopposites.Themathworksoutcorrectlyinforthe23=8cornersofthecube.
Thisstillleavestwopossibilitieshowtopaintthesepips.Lookatthecornersharedbythefaces1,2, and3,andreadthesenumbersclockwisearoundthatcornerstartingfrom1.Ifthesethreenumbers readoutas1-2-3, thatdieisleft-handed,andif theyreadoutas1-3-2, thatdieisright-handed. Analogoustoapairofshoesmadeseparatelyfortheleftandrightfoot,left-andright-handeddice arein one senseidentical, yet nomatter how you twist and turn, you can't change either one to becomeindistinguishable fromtheother.(Well,atleastnotwithouttakingthatthree-dimensional pancake“ThroughtheLooking-Glass”byflippingitaroundinthefourthdimension!)
Thethreenumbersreadaroundanyothercornerstampthethreenumbersintheunseenopposite sides,andthereforedeterminethehandednessofthatentirediejustasfirmly.Giventhethree-tuple ofpipsaroundacornerreadinclockwiseorder,determinewhetherthatdieisleft-handed.There are only 23*3! = 8*6 = 48 possible pip combinations to test for, but you shouldfind and cheerily exploitasmanyoftheamplesymmetriesinthisproblemasyoucan.
After solving that, imagine that our physical space had k dimensions, instead of merely just the familiar three.Howwould dice even be cast (in both senses of thisword)ink dimensions?How wouldyougeneralizeyourfunctiontofindthechiralityofanarbitraryk-dimensionaldie?