Polyomino Tessellations: A Class ProjectAuthor(s): Nick MackinnonSource: Mathematics in School, Vol. 18, No. 3 (May, 1989), pp. 8-9Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214587 .

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Polymino

Tessellations

a

Class

protect

by Nick MacKinnon, Winchester College

This article describes a project that 2Mr (a class of 21 thirteen year olds) did recently.

An n-omino is a shape composed of n unit squares each connected to at least one other square along an edge. Readers will be familiar with the 12 pentominos and the various puzzles associated with them.'

2Mr were looking at the question of whether a given polynomial will tessellate.

Given a polynomino how do you decide whether it tessellates? The instinct of some was to get drawing, and be satisfied when a reasonable amount of graph paper had been covered.

"I think this will work but I'm not sure."

I think that my class got a real understanding of what it means to prove something in mathematics when Richard Barrett discovered this tessellation.

"I can make copies of this pattern and slot them together like Lego."

With the insight that a small patch is all that is needed provided there is an obvious pattern in it, they were quickly able to show that all the pentominos will tessellate. On to the hexominos.

There are 35 hexominos so the first job (set for home- work) was to find them all. Then everyone got one hex- omino to test for tessellation. To my surprise they all tessellated! One of the neatest tesselations was of the J- shaped hexomino which curls round itself to make a rectangle (copies of which will then cover the plane).

I thought that the idea of tiling rectangles would be an interesting continuation. There is a discussion of it in Reference 3.

We dropped back to the tetrominoes and discovered these tilings, hereafter called "rectangulations".

8 Mathematics in School, May 1989

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I also showed the class a proof that the other tetromino will not rectangulate. This argument was hereafter referred to as a "corners proof".

"The shaded tile must fit into a corner of the rectangle as shown. Once this is done there is only one way to continue and the other corner can never be filled."

With this argument to use as a template the class then found the three obvious pentominos that would tile a rectangle, and were able to prove that the others wouldn't.

A corners proof. (Rupert Beloff)

There was one rather disturbing surprise however. Several "proofs" that the Y-pentomino would not rectan- gulate were produced before Toby Vereker came up with this tiling.

Most of the hexominos were easily classified as not rectangulating, using a corners proof. The class found 8 easily rectangulating hexominos and the only two that were in doubt were these.

After a hard struggle William Hawkins, Toby Vereker, and Harry Burn found rectangulations, one of which is shown, which just left the Y-hexomino.

After the trouble with the Y-pentomino it was clear that it was going to be difficult to classify the Y-hexomino. After a determined attack from two directions we are still unable to decide whether the Y will rectangulate or not. The problem cannot be solved using the algebraic technique described in [2] either. Perhaps this problem is logically undecidable?

[These tiling problems are very like Diophantine equ- ations. I think the analogy can be made precise and I'm working on this at the moment. For example the Diophant- ine equation x2 + y2

= 15 is found to have no solutions just by trying a few small values of x and y. This procedure is analogous to a corners proof, where a finite number of trials will decide the matter. The Diophantine equation x2 + y2 = 4xy + 3 is proved to have no solutions by working modulo 4. The LHS is congruent to 0, 1 or 2 (mod 4) while the right hand side is congruent to 3 (mod 4). This simple method is analogous to a "parity proof" in tiling problems, the most well known of which is the proof that an 8 x 8 square with two opoosite corners removed cannot be covered with 1 x 2 rectangles. There are undecidable Dio- phantine equations, so perhaps there are undecidable tiling problems?].

There are plenty of polyomino tessellation problems left to consider. All the n-ominos will tessellate up to n = 6. There is one heptomino that certainly won't tessellate which is this rather dubious one.

Ignoring the holey polyominos what is the least value of n for which there is an n-omino that won't tessellate? There is at least one non-holey nonomino that won't tessellate (can you find it?). Is there a heptomino or octomino that won't? And what is the probability that a randomly selected n- omino will tessellate?

References 1. Golomb, S. W. (1965) Polyominoes. Allen and Unwin. 2. MacKinnon, Nick. An Algebraic Tiling Proof. Math Gazette 1989. 3. Gardner, Martin. (1971) Mathematical Magic Show. Penguin.

Mathematics in School, May 1989 19899

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