PolyominoesAuthor(s): Bruce HenrySource: Mathematics in School, Vol. 7, No. 2 (Mar., 1978), p. 13Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213364 .

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Miscellany

Polyominoes

by Bruce Henry, State College of Victoria, Rusden

I read with interest Mr Millington's article "Polyominoes" (Mathematics in School, May 1977). It immediately raised many new questions on this topic which is well known to most teachers of mathematics. I have managed to answer a few of these and I pose others which your readers may be able to help me with. 1. The frieze design using the 12 pentominoes is inter- esting; perhaps there exists a design for a tessellation with similar properties. I found one, shown in Figure 1, which tessellates without rotation. Are there others of this kind? Are there other patterns for tessellation which do involve rotation or reflection?

Figure 1

2. Is the poster design using 56 polyominoes unique? I have made several different ones, including some where I fixed certain polyominoes (in a fairly random way) first and built the design with this restriction. Figure 2 shows one with all the straight polyominoes and all those made with four or less squares placed in a 5x 8 rectangle in one corner of the poster. I have used this

poster to help me to prove that a poster is possible with the monomino in any nominated position. I suspect that it is possible to make a poster with much more of a restriction than that.

3. I am sure that puzzles such as the poster design discussed above are made easier when

(i) more pieces are available, (ii) the smaller polyominoes are included.

Thus I consider it harder to place the 35 hexominoes on a 21 x 10 rectangle (I have yet to solve this one).

it is evident that sets of polyominoes in the mathe- matics classroom can be very useful in contributing to those spacial experiences so necessary to the develop- ment of the child's spacial perception. I hope that the article by Mr Millington will spark off new ideas for the classroom.

Teachers planning to use polyominoes may find it advisable to make them by gluing half-inch wooden cubes together with polyvinyl acetate glue. They last longer, they are easier to manipulate (especially for small fingers) and they are useful for packing problems (for example, pack the 12 pentominoes into a 5 x 6 x 2 cuboid.

Postscript I have now solved the problem of 35 hexominoes in a 21 x 10 grid (and other rectangular shaped grids) - by showing it to be impossible!

Imagine the grid to be a checkerboard and each hexomino shaded in a check pattern. There are 24 hexominoes with three squares of each colour, and 11 hexominoes with four squares of one colour and two of the other (note that these latter hexominoes must cover an EVEN number of squares of either colour). The 24 hexominoes account for 72 squares of each colour no matter where they are placed, leaving 33 squares of each colour to be covered. But the remaining 11 hexominoes must cover an even number of squares of each colour. So it cannot be done!

Figure 2

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