Hi folks,

Anyone know the answer to Q.3 from today's Maths Olympiad? It defined a tetromino as a plane figure made of 4 non-overlapping squares joined by common edges (or something), and asked:

1. Prove that there are only seven (if rotations are not distinct but reflections are)
2. Is it possible to tile a 7x4 square with one of each of the 7?

The last part threw me totally.

David

Colouring the 7x4 as a chessboard. The T-shape has 3 of one colour and 1 of the other, whilst the other 6 contains 2 of each, so it is not possible.

Ah, you just beat me to it Kerwin.

This is very similar to the more standard problem of showing that if you remove two diagonally opposite corner squares of a chessboard, then you can't tile the remaining squares with 31 2x1 rectangles.

James.

Kerwin,

Thanks for the solution.

David

Here's another nice chessboard puzzle (it is quite a bit harder than the BMO one above). It was given to us by our Director of Studies to mull over.

We are on an 8x8 chessboard. You have to make 64 moves (horizontally or vertically) in succession, so that you end up on your starting square, and you pass every other square on the chessboard exactly once. You must make the same number of vertical moves as horizontal moves. Can this be done?