Wood Stack – Where is the Limit?

Discovering the Harmonic Series

Aron Brunner, Johann Rubatscher Intention The question on how far equal wooden cuboids (e.g.dominoes, CD‐covers) with the dimensions l, b, and h in astack can bemoved without the stack collapsing has beenasked for centuries. Themaximal overhang s of the cuboidontopshouldbediscoveredhere.Newdevelopmentscanbefoundinthebibliography.

https://de.wikipedia.org/wiki/Harmonische_Reihe

Background of Subject Matter Astackofcuboidsdoesnotcollapseifthefootofthecommoncentreofmassofthecuboidsdoesnotleavethecuboidbelow.Themaximalshiftdistancescanbediscoveredbyfindingthecentreofmasswhichisjustbarelylocatedonthesurfaceofthecuboidbelow.Throughaddingnewcuboids,scanbedescribedthroughtheseries:

/2 /4 /6 /8 /10 /12 /14 ⋯,

divisionbyl/2leadsto:

/2 ⋅ 112

13

14

15

16

17

⋯

Thebracketscontaintheharmonicseries,whichdoesnotconverge.Thus,addingmorecuboidsleadstoalargershiftdistance. Working out the Results Thefirstcuboidontopcanbeshiftedby /2asthenthecentreofmassrestsovertheedgeofthecuboidbelow.Twocuboidstogethercanbeshiftedby /4asduetosymmetry,thecentreofmassisinthemiddleofthetwoindependentcentresofmass.Theshiftdistanceforthreecuboidsisbestsolvedwithanequation.Thecommoncentreofmassislocateddirectlyabovetheedgeofthefourth cuboid below. Thus, the mass‐share of the three cuboids and therefore of the individualcuboidsleftofthecentreofmassequalthoseontheright.Bexthelocationofthecentreofmassinrelationtotheleftedgeofthethird.Asaresult,xequals .Asthecuboidsareshiftedlongitudinally,thelengthsreplacethemasses:

 

Totalofindividualmassesleft Totalofindividualmassesright Totalofindividuallengthsleft Totalofindividuallengthsright

/4 /4 /2 /4 /4 /2 Solvingtheequationleadsto: /6Fourcuboidsbeget :

/6 /6 /4 /6 /4 /2 /6 /6 /4 /6 /4 /2

Solvingtheequationleadsto: /8Continuationfor , , ,…generates:

⋯ /2 /4 /6 /8 /10 /12 ⋯AnAlternativeDerivation:Thecentreofmassoftwocuboidsislocatedinthemiddleofthecentresofmassoftwocuboidsontheirown(symmetry).Thecentreofmassofthreecuboids,however,isnotlocatedinthemiddleofthecentreofmassof the twocuboidsand the thirdbut is shifted in thedirectionof thecentreofmassofthetwocuboidsasthetwopossessmoremass.Duetothis,thecentreofmassisshiftedintheratio2:1,whichisequivalenttoathirdofl/2.Itisvalidthat:

1/3 ⋅ /2 /6Thecentreofmassoffourcuboidsisshiftedaccordingtotheratio3:1,whichleadsto:

1/4 ⋅ /2 /8Thecentreofmassoffivecuboidsisshiftedaccordingtotheratio4:1,whichleadsto:

1/5 ⋅ /2 /10

Thecontinuationoftheharmonicseriesshouldbepointedouttothepupils.Aspreadsheetprogramcancalculatetheharmonicseriesuptoabout 200.000.Bibliography Pöppe, C. (2010):TürmeausBauklötzen –MathematischeUnterhaltungen, SpektrumderWissen‐

schaft,September2010,S.64Paterson, M., Peres, Y., Thorup, M., Winkler, P., Zwick, U. (2008): Maximum Overhang, arXiv.org,

CornellUniversityLibrary,http://arxiv.org/pdf/0707.0093v1.pdf

 

Pupils‘ Results

 

 

Wood Stack – Where is the Limit?

https://de.wikipedia.org/wiki/Harmonische_Reihe

1 Description of the Task and First Experiments Taketwoequalwoodencuboids(dominoesorCD‐covers)oflengthl,widthb,andheighth(withhalot smaller than l and b), stack them on top of each other andmove the cuboid on top so far inlongitudinaldirection that it sticksoutas faraspossible.Puta thirdequalcuboidunderneath thetwofirstonesandmovethetwouppercuboidsagainasfaraspossible.Continuethisprocess.2 Calculation Considerwithpen,paper,pocketcalculator,etc.,howfaryoucanmovethecuboidssuccessively ifthetotalnumberofblocksis 2, 3, 4, 5, …?Howfaristhetotalshiftdistancesineachcase? 3 How far? Which is thenumber ofwoodenblocks,withwhicha stackof thismannercanbebuilt thatdoesnotcollapse?4 With Help from the Computer Calculate withaspreadsheetprogram. 5 Further Considerations Howbigiss,ifeachcuboidismovedbyl/3orl/4?Whatwouldhappenwithwoodendisks?Whatwouldchangeifyouwereallowedtousecounterweightsoranyotheradditions? Readthefollowingarticles:Pöppe, C. (2010):TürmeausBauklötzen–MathematischeUnterhaltungen, SpektrumderWissen‐

schaft,September2010,S.64Paterson, M., Peres, Y., Thorup, M., Winkler, P., Zwick, U. (2008): Maximum Overhang, arXiv.org,

CornellUniversityLibrary,http://arxiv.org/pdf/0707.0093v1.pdf