### Brain Teasers

# Mad Ade's Chess Moves

One evening after enjoying a slap up meal at the "Sweaty Chef" kebab shop Mad Ade found himself at a loose end whilst waiting for his favourite TV program, Pro-celebrity nude underwater longdistance Monkey hurdling, to begin. He turned to his chess set (a full normal chess set) and numbered each piece uniquely. He wondered to himself,

"I wonder how many ways (permutations) are there to set up the board to begin a normal game?"

Being Mad Ade he never actually got round to working it out.

Can you work out the answer for the lazy swine?

"I wonder how many ways (permutations) are there to set up the board to begin a normal game?"

Being Mad Ade he never actually got round to working it out.

Can you work out the answer for the lazy swine?

### Answer

It is best the begin with working out the number of possible permutations for one side of the board, White or Black, it doesn't matter. We'll start with White.First we start with the back row. The King and Queen can be safely ignored, since they can not occupy any other square in the standard

arrangement. However, the two castles can be arranged in one of two ways, as can the two knights and the two bishops. There are two of

each of these pieces, and for each piece there is a choice of two possible squares. So we write:

Castle Permutations = 2! = 2

Knight Permutations = 2! = 2

Bishop Permutations = 2! = 2

It is now a simple matter to calculate the number of permutations for the back row:

Back Row Permutations = 2! * 2! * 2! = 8

Now we can work out the pawns. There are 8 pawns, and there are eight possible squares which each pawn might occupy. So we write:

Pawn Permutations = 8! = 40,320

So, the total number of permutations for White is given by:

White Permutations = Back Row Permutations * Pawn Permutations = 8 * 40,320 = 322,560

Given the symmetry of the chess board, or indeed the fact that our choosing White initially was arbitrary, we shall assume that there must also be 322,560 permutations for Black. So we write:

Black Permutations = White Permutations = 322,560

It is now a simple matter to calculate the number of permutations for both White and Black:

White and Black Permutations = White Permutations * Black Permutations = 322,560 * 322,560 = 104,044,953,600

The final step is to rotate the board through 180 degrees, since it is arbitrary which side of the board White should start from. Note that we do not rotate the board through 90 degrees, since the corner square to our left must always be black. Thinking about this problem in terms of rotations is not the only method. We could simply treat White and Black as two entities, which might occupy either of two

positions ( sides of the board).

Rotary Permutations = 2! = 2

Finally:

Total Permutations = White and Black Permutations * Rotary Permutations = 104,044,953,600 * 2 = 208,089,907,200

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