How many initial positions?

Barring rotations and reflections, how many initial positions of 4 white checkers and 4 black checkers are possible on a chess board, if no two checkers may occupy the same row or column?

I'm playing with and idea for a game that has an opening procedure that would result in such positions, but I'm not sure about the math.

Thank you :)

Taking the information form http://blog.chess.com/kurtgodden/8-lonely-rooks we have a lower bound of 5282, which is the answer for 8 checkers of the same color.

There are (5*6*7*8)/(1*2*3*4) = 70 ways to apply the 2 different colors to your 8 checkers, so 70*5282=369740 gives an upper bound. The real answer is lower than that, because of symmetry again.

If you'd force me to guess I'd say 300000. But maybe someone else will actually calculate the correct answer.

Oh, another point: From your question it is not entirely clear if 2 positions that can be transferred to each others by color flipping should be counted as identical or not. If yes, divide the upper bound and my guess by 2.

What do you mean with barring rotations and reflections?

Could you specify with examples which solutions are supposed to be equavalent?

I got 354,844 positions in the case where colour flipping is not "legal".

In the other case, I got 177,810 positions, but I am not quite sure I did it right.

Hi guys, thanks so far. At least I have an indication from richy's post of the order in which to think.

"The real answer is lower than that, because of symmetry again"

Given that the 5282 positions are unique, how could symmetry arise when half of the men are made black?

"From your question it is not entirely clear if 2 positions that can be transferred to each others by color flipping should be counted as identical or not."

That relates to the above - no, interchanging a black and a white checker will as far as I can see always give a different position.

"Could you specify with examples which solutions are supposed to be equavalent?"

Any two positions that can be mapped one-on-one after a reflection and/or rotation of one of them would be considered the same position.

May I say that in any case the number of different starting positions arising from an opening procedure that allows the two players to alternately place a man, not on the same row or column than a man already present on the board, would be sufficient for the purpose.

So thanks for that for starters :)

Correction: 178,022 in the second case.

Checking richy's link and assuming that the number 5282 stated there is correct I second the total number of 369740.

The rotational and reflectional symmetries were already taken into account in the lonely rooks problem. Colouring the checkers can not introduce symmetry, thus there are 70 distinct ways for colouring each of the 5282 rook configurations.

The symmetry between white and black is usually broken by the fact that one colour defines the first player. If there is no first player (i.e. both players are moving simultaneously) then richy also has already remarked correctly that the number of configurations is halved.

> Given that the 5282 positions are unique, how could symmetry arise when half of the men are made black?

Very simple:

Position 1:

White on: a1, b2, c3, d4

Black on: e5, f6, g7, h8

Position 2:

White on: e5, f6, g7, h8

Black on: a1, b2, c3, d4

Both these Positions are counted in the 369740. But surely they are the same when you take symmetry into account. The devils results are within the bounds, so he may well be right.

Given that the 5282 positions are unique, how could symmetry arise when half of the men are made black?

Put black pieces on A2, B8, H7 and G1, and white pieces on B1, A7, G8 and H2. Rotational symmetry is preserved.

Without taking any symmetries into account you have 8!*70=40320*70=2822400 different starting positions.

@TasmanianDevil: You are putting for example A2 _and_ A7, this is not allowed. ;-)

I guess the last is not a legal example anyway, is it?

Hehe, that's true Thomas.

This was what it was all about.

Eventually 7x7 seemed better, if only because of the 'integrated' swap rule with the 4w/3b division. Makes for a slightly different calculation, but there seem more than enough possible opening arrangements to keep things interesting.