Tuesday 29 September 2015


Was it George Bernard Shaw who said that England and America are two countries divided by a common language?

Some words have very different meanings on either side of the pond. So while an Englishman puts his trunks in the boot of his car before driving off on holiday, an American would put his boots in the trunk.

Similarly, an Englishman puts his pants on before his shorts.

But my all-time favourite [or should that be favorite?] only really works when spoken aloud:
In England, we pay our bills with a cheque. In America, they pay their checks with a bill.

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: