The Dominoes and the Chessboard

On Monday, I posted a puzzle regarding a chessboard and some dominoes. As promised, I am now going to give the answer (so that my boyfriend's family will stop arguing about it!)

So, is it possible to place 32 dominoes, each covering two squares, on a chessboard with two diagonally opposite corners cut out?

The answer is: No. It is not possible.

But how do you prove it? Most of the modules in my degree this term are all about proving things, and it can get very confusing. Especially Analysis. But there is a simple way to think about this puzzle.

As the two corners cut out of the chessboard are diagonally opposite each other, they must be the same colour. This means that we have 30 of one colour (in the picture above it is black) and 32 of the other colour (obviously, in this case white). As the dominoes only cover two squares, no matter how you place them, they are going to cover one of each colour. So if we pair off the colours, we can match 30 black to 30 white, but then we are left with 2 of the same colour left over. We know that a single domino cannot cover two squares of the same colour, so the puzzle is therefore impossible.

Hopefully I explained that clearly so you aren't left feeling as confused as before!

Monday Maths Madness 14

This week I have another puzzle for you to ponder over (just one this time, I promise!) I think I've heard of this puzzle before, but I read about it recently in Mathematics: A Very Short Introduction by Timothy Gowers and thought I would share.

The basic idea, is you have a chess board with two corners cut out like so:

 

You also have 31 dominoes that each cover two squares. Can you lie all of the dominoes on the chessboard, so they cover it completely?

Answer in a few days!

Edit: Ok, it's not totally obvious from the picture, but the top left and bottom right squares have been cut out.