Problems about polyominoes

Problems about polyominoes

Season 2

Episode 14 Time frame 4 periods

Prerequisites :

Objectives :

•

^{Workon afew lassi problems about} polyominoes.

Materials :

•

^{Hand-out withthe solutions.}

•

^{Cut out} tetrominoes and pentominoes.

•

^{Dierent problems.}

•

^Beamer.

1 – Solve some problems 3 hours

Students work inteams of 4 or5.

Some problems about polyominoes are available at the teaher's desk. Eah team piks

any problemand works onit, with the following rules.

•

^{Eah team an pik any problem,randomly orhoosing it onpurpose.}

•

^{Aftera10minutes searhateamandeidetostop workingonthispartiularproblem}

and pik a new one.

•

^{Answers must not be passed along fromone team to another.}

•

^{Eah team must nd the answer toatleast three problems during the available time.}

•

^{Grades are given aording tothe following sheme :}

⋄

^{three problems solved :B (15);}

⋄

^{ve problems solved : A (18);}

⋄

^{eightproblems solved : A}

^⋆

^(20).

2 – Presentation 1 hour

Eah team must present to the lass the result of one problem. The quality of the pre-

sentation may be awarded a grade.

Document Solutions

1 Rep-tiles

A polyomino is a rep-tile if a larger version of itself an be tiled with only

opies of the initialpolyomino.Find fournon-trivial polyominoesrep-tiles.

The easiest way is tond polyominoesthat an beused tomake a square. Then we just

have toreprodue this square as inthe initialpolymino.

The L-triomino The L-tetromino

The T-tetromino

The P-pentomino

2 Tiling a retangle with pentominoes

Find aretangle tileablewith the 12 pentominoes used exatlyone.

One possibiltyisthe

6 × 10

^{retangle pituredbelow. Other}possibilitiesare

5 × 12

^,

4 × 15

or the

3 × 20

^{that is the answerto anotherproblem.}

3 Tiling a retangle with the tetrominoes

Prove that it's impossible to tile a retangle with the 5 tetrominoes used

exatly one.

Any tileable retangle shouold have an area of

5 × 4 = 20

^{. Color the squares of any}

retangle of this area blak and white, as on a hessboard. Obviously, there is the same

number of blak squares and white squares. Now olor the squares of the 5 tetrominoes

in the same way :

BeauseoftheT-tetromino,the numberofblaksquaresandthenumberofwhitesquares

will always be dierent. Soit'simpossible totile aretangle with the tetrominoes.

4 Tiling a speial retangle with pentominoes

Tile a

3 × 20

retanglewiththe 12pentominoesused exatlyone.

There are onlytwo solutions,oneof them ispitured below. Forthe othersolution,leave

the four polyominoes on the left and the polyomino on the right and rotate the whole

middle part.

5 Three ongruent groups of pentominoes

Divide the twelve pentominoes into three groups of four eah. Find one 20-

square region that eah of the threegroups willover.

6 Minimal region for the twelve pentominoes

Find a minimal region made of squares on whih eah of the 1é pentominoes

an t.

Here are two possible answers. Eah of the 12pentominoests inany of these regions.

7 A tromino or a tetromino on a hessboard

We've seen thatyouan tilea hessboard with dominoes.Can youdoit with

the I-tromino, orwith the L-tromino?

A tromino is made of three squares. Soany region tileable with a tromino must have an

area that is amutiple of 3.It's not the ase for ahessboard, asthe area is

8 × 8 = 64 = 3 × 21 + 1

^{. Soahessboard isnot tileablewithtrominoes(Ior L).Afurther questionis :}

what about a hessboard with one square removed?

Whih tetrominoesan be used to tileaomplete hessboard?

The I, L, O and Ttetrominoes tilea

4 × 4

^{square, tothey alsotilea hessboard.}

8 Tripliation of pentominoes

Pik apentomino,thenuse nineofthe otherpentominoestoonstrut asale

model, three times as wide and threetimes as high as the given piee.

Beloware showthe tripliations of the V pentomino and the X pentomino. Other tripli-

ations are possible.

9 Pentominoes on an amputated hessboard

Cover a hessboard with the four orners missing with the 12 pentominoes

used exatly one.

Here is one solution. Other positions of the four missing squares are possible, inluding

the next one, where the four missingsquares makethe square tetromino.

10 Pentominoes and a tetromino on a hessboard

Covera hessboard with the twelve pentominoesand the square tetromino.

A simpletrik isto ombine the square tetrominowith the V pentominoto makea

3 × 3

square. Then all we have to do is tile the remainingportion of the hessboard with the

11 other pentominoes. There are many ways todo so,below isone example.

11 Three 3

×

^{7 retangles}

Divide the twelve pentominoes into three groups of four eah. To eah group

add a monomino and forma

3 × 7

^retangle.

Here istheonlysolution.Theproofthatnoothersolutionispossiblestartsfromthe only

possibility for the U and X pentominoes.

12 Tetrominoes and a pentomino on a square board

Covera 5

×

^{5board with the ve} tetrominoesand one pentomino.

Beloware shown two possibilities.There may be more.

Document 1

^{Cut-out tetrominosand} pentominoes

The 5free tetrominoes.

The 12 freepentominoes.

Document 2

^Problems

Rep-tiles

A polyomino is a rep-tile if a larger version of itself an be tiled

with only opies of the initial polyomino.

Find four non-trivial polyominoes rep-tiles.

Tiling a retangle with pentominoes

Find a retangle tileable with the 12 pentominoes used exatly

one.

Tiling a retangle with the 5 tetrominoes

Prove that it's impossible to tile a retangle with the 5 tetromi-

noes used exatly one.

Tiling a speial retangle with pentominoes

Tile a

3 × 20

retanglewith the 12 pentominoes used exatly one.

Three ongruent groups of pentominoes

Divide the twelve pentominoes into three groups of four eah.

Findone 20-square region that eah of thethreegroups will over.

Minimal region for the twelve pentominoes

Find a minimal region made of squares on whih eah of the 12

pentominoes an t.

A tromino or a tetromino on a hekerboard

We've seen that you an tile a hessboard with dominoes.

Can you do it with the I-tromino, or with the L-tromino?

Whih tetrominoes an be used to tile a omplete hessboard?

Tripliation of pentominoes

Pik a pentomino, then use nine of the other pentominoes to

onstrut a sale model, three times as wide and three times as

high as the given piee.

Pentominoes on an amputated hessboard

Cover a hessboard with the four orners missing with the 12

pentominoes used exatly one.

Pentominoes and a tetromino on a hessboard

Cover a hessboard with the twelve pentominoes and the square

tetromino.

Three 3

×

^{7 retangles}

Divide the twelve pentominoes into three groups of four eah. To

eah group add a monomino and form a

3 × 7

retangle.

Tetrominoes and a pentomino on a square board

Cover a 5

×

^{5 board with the ve} tetrominoes and one pentomino.