Covering a chessboard with dominoes

Season 02 • Episode 12 • Covering a chessboard with dominoes

Covering a chessboard with dominoes

Season 02

Episode 12 Time frame 2 periods

Objectives :

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Materials :

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1 – Seven problems ^{75 mins}

Students work by pairs. Eah pair of students has to solve seven problems and tell the

answerto the teaher. They get 3points for every problemsolved.

2 – The solutions Remaining time

The solutions are shown with a beamer and then handed out to the students. Pairs of

students an behosen randomlytoomment onthe solutions.

Covering a chessboard with dominoes

Season 02

Episode 12 Document Problems

Problem 1

Is it possible toover awhole hessboard with dominoes?

Problem 2

One orner has been removed from a hessboard. Is it possible to over the remaining

portion of the board with dominoes so that eah domino overs exatly two squares?

What if two opposite ornersare removed?

Problem 3

Two arbitrary but adjaent squares have been removed from a hessboard. Is it possible

to over the remaining portionof the board with dominoes?

Problem 4

Two arbitrary squares of dierent olors have been removed from a hessboard. Is it

possibleto overthe remainingportionof the board with dominoes?

Problem 5

Two arbitrary pairs of squares of dierent olors have been removed from a hessboard.

Is it always possible to over the remainingportion of the board with dominoes?

Problem 6

Three arbitrarypairsof squares ofdierentolors have been removed froma hessboard,

sothatthehessboarddoesnotsplitintotwoormoreseparatepiees.Isitalwayspossible

to over the remaining portionof the board with dominoes?

Problem 7

A dominohas two edges, a long edge and a short edge. Two adjaent dominoes must be

in one of the only three possible ongurations : long edge to long edge, short edge to

short edge and long edge to short edge. In a domino tilingof a hessboard, what is the

minimum number of long-edge tolong-edge pairs?

Problem 8

Prove that in any over of a whole hessboard with dominoes, the number of horizontal

dominoes with a blak left square and the number of horizontal dominoes with a white

Covering a chessboard with dominoes

Season 02

Episode 12 Document Solutions

Problem 1

Is it possible toover awhole hessboard with dominoes?

A hessboard is made of 64 squares, in eight lines of eight squares. By putting four

horizontaldominoeson eah line, we overthe board with 32dominoes.

Problem 2

One orner has been removed from a hessboard. Is it possible to over the

remaining portion of the board with dominoes so that eah domino overs

exatly two squares? What if two opposite orners are removed?

If one orner is removed from the hessboard, then the number of squares is odd. As a

domino is made of two squares, itwill then be impossible toover the amputated board

with dominoessothat eahdomino overs exatlytwosquares : one would foredly have

one half outside of the board.

If two opposite orners are removed, the amputated board is made of 62 squares, so

the problem stated above doesn't arise. Now, paint the squares blak and white as in a

regularhessboard.The opposite ornersarethe sameolor,sothe amputatedboardwill

be made of, say 30 blak squares and 32 white squares. But a domino will always over

a blak square and awhite square, so it'simpossible toover the amputated board with

Season 02 • Episode 12 • Covering a chessboard with dominoes

Problem 3

Two arbitrary but adjaentsquares have been removed from a hessboard. Is

itpossibletooverthe remainingportion ofthe boardwith dominoessothat

eah dominopiee overs exatly two squares?

By studying dierent positions for the removed squares, it's easy to gets onvine that

it's always possible. But this isnot asatisfatory way toproveit. So, onsider, the basi

tilingofthe hessboard onegets with 32horizontaldominoes. Threesituationsmayarise

for the two removed squares.

1

board is obviously tileablewith dominoes.

2

two horizontally adjaent dominoes, then we just have to

plae one horizontal domino on the row below or above

those two squares (depending onthe parityof the number

of the row), one vertial domino on the right and one of

the left. We get the onguration shown on the ri ght or

its horizontal mirrorimage, whih llsa

4 × 2

board, therest ofthehorizontaldominoesbeing unmoved.

2

dominoes,then we justneed toplae a vertial dominoover the two other halves of

these dominoes.

Problem 4

Twoarbitrarysquaresofdierentolorshavebeenremovedfromahessboard.

Is it possible to over the remaining portion of the board with dominoes so

that eah dominopiee overs exatly two squares?

It's alwayspossible.Tosee this, drawtwo forksas

on the diagram. This splits the hessboard into a

hainofalternatingsquares.Thehainan betra-

versed by startingat any square and overing two

squares atatime withapiee ofdomino.The two

removedsquaresmustbeadierentolor,sothere

willalwaysbeanevennumberofsquaresalongthe

hainbetween them. Startfrom one side of one of

the removedsquaresandplaedominoesalongthe

hain untilyou get to the other one. Do the same

forom the other side. Note that this wouldn't be

possible if the two removed squares are the same

olor,asthehainsbetween themwillhaveanodd

Season 02 • Episode 12 • Covering a chessboard with dominoes

Problem 5

Two arbitrary pairs of squares of dierent olors have been removed from a

hessboard. Is it always possible to over the remainingportion of the board

with dominoessothat eah dominopiee overs exatly two squares?

Theproblemhastwodierentanswersif weallowtheremovedsquares toisolateonepart

of the board or not. If we allow this, then the tilingis not always possible. Consider for

exemple the situationwere one ornerissurrounded by two removed squares of thesame

olor, the two other ones being anywhere else onthe board. Then, the isolated orner is

obviously not tileable with dominoesand so the amputated board is not.

If we restrit the situation to the ases where the board doesn't fall apart, itseems that

the tilingis always possible, but the proof is not easy towrite.

Problem 6

Three arbitrary pairs of squares of dierent olors have been removed from a

hessboard, so that the hessboard does not split into two or more separate

piees. Is it always possible toover the remainingportion of the board with

dominoessothat eah dominopiee overs exatly two squares?

This is not always possible to do, aswe willshow with

one speial onguration.

The gure onthe rightdepitsthe lowerrightornerof

the board.Removethethreeblaksquares markedwith

arightross, and thethree whitesquares anywhereelse

on the board, so that itdoes not split into two ormore

piees.Thenthe onlywaytooverthelowerrightwhite

square istoput adominoasshown witharosshathed

retangle. It will then be impossible to over the right

Season 02 • Episode 12 • Covering a chessboard with dominoes

Problem 7

Adominohastwoedges,alongedgeandashortedge.Twoadjaentdominoes

must be in one of the only three possible ongurations : long edge to long

edge, shortedge toshortedge and longedge toshort edge.In adominotiling

ofahessboard,whatistheminimumnumberoflong-edgetolong-edgepairs?

The tilings shown on the right as only one

long-edgetolong-edgepair.Thisisthesmal-

lest possiblenumberof suh pairs,asit'sim-

possible to do a tiling with none. Indeed, if

youstart with adomino inone orner, avoi-

dinglong-edgetolong-edgepairswillimpose

the plaement of all the dominoes as shown

onthegure,exeptthefourmiddlesquares.

Thenweneedtoplaetwodominoesonthese

squares,whihwillgiveusthreelong-edgeto

long-edge paris in one ase and one in the

other.

Problem 8

Provethatinanyoverofhessboardwithdominoes,thenumberofhorizontal

dominoeswithablakleftsquareandthenumberofhorizontaldominoeswith

a white leftsquare are equal.

The proof of this property is given by E. W.Dijsktra :

For a given partitioning, onsider one of the seven lines that separate two adjaent o-

lumns; itdivides the board into aleft and a rightpart. Weobserve that

1

equal numbers of blak and white squares, and;

2

the left of the lineomprise equalnumbers of blak and white squares.

Combiningthetwoobservations,weonludethat,amongthehorizontaldominoesrossed

by the line, there are as many with a blak left square as with a white one. Sine eah

horizontaldominoisrossedbyexatlyone ofthe seven lines,weonlude by summation

that the number of horizontal dominoes with a blak left square and the number of

Season 02 • Episode 12 • Covering a chessboard with dominoes

Document 1

Season 02 • Episode 12 • Covering a chessboard with dominoes

Document 2