Sunday, 27 September 2015

Dominoes, Triangles, Squares and Tetris

My maternal grandfather was a coal miner from the North West of England. Unsurprisingly, for a working class man of that era, he played dominoes. Dominoes is often thought of as a children's game; but that's very unfair. Indeed there are championships played throughout the world. By subtle play, an experienced player can discern which tiles are in their opponents' hands. For a double-six deck, this is supposedly easy; for a double-nine deck, it is somewhat more difficult.

Dominoes are a favourite of mathematicians. Consider the number of tiles in a deck. If we arrange the tiles into rows where the value of the greater "side" is zero, one, two, three, four, five and six in turn, we get:

01 11
02 12 22
03 13 23 33
04 14 24 34 44
05 15 25 35 45 55
06 16 26 36 46 56 66

This obviously forms a triangle: the number of tiles in a complete double-n deck is a triangular number:

tiles in a double-n deck, D(n) = T(n+1) = (n + 1) × (n + 2) ÷ 2

For a double-six deck, D(6) = 28. For a double-nine deck, D(9) = 55.

In terms of construction, dominoes are two squares bolted together with a common edge:

There's no real choice about how we join the two squares, but when we go to three squares, there are two arrangements (ignoring rotations):

These are known as trominoes, not to be confused with triominoes.

When we get to four squares, we get the tetrominoes, The one-sided variants (allowing reflection) also being known as the Tetris pieces:

If we take the seven Tetris pieces and add the two trominoes and the single domino, we get ten pieces made up of a total of 36 squares (7×4 + 2×3 + 1×2):

Thirty-six is interesting: it is the first number after one that is both a square and a triangular number. Therefore, we can arrange the ten pieces above into both a square:

And into a triangle:

You can also arrange them into 2-by-18, 3-by-12 and 4-by-9 rectangles:

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