Configurations and combinations in a chess board

Question

In a chessboard, a pawn needs to move from the lower left to the upper right corner.

a) Find the number of possible paths if one "move" is a combination consisting of 3-steps up and 4-steps right, for example, R-UU-RR-UU-RR-UUU-RR is one possible path

b) Find the number of possible configurations if all pieces are now placed in lines 1&2, and if the only limitation is that the queen can be to a same-color block and that paws can only occupy line 1

For a, I do something like

a) Each path is 14 moves in total but we need to move in total by 7 blocks right(R) in 3 moves and by 7 blocks up(U) in 4 moves. So we have $C(7,3)\cdot C(7,4)$ possible paths.

b) I would say that we have for pawns 8! ways, and then 2! for each of the towers, rooks, and horses, 1 way for the king, and 1 way for the queen. However, I am troubled by the condition given for the queen, which I guess indicates that I should consider all configurations in the 1rst line, not necessarily the "correct one". In such a case what the solution would be?

Answer 1

Part A is a Stars and Bars problem.

For Stars and Bars theory, see this article and this article.

Consider the number of solutions to the following:

• $x_1 + x_2 + x_3 + x_4 = 7.$

• $x_1, x_2, x_3, x_4 \in \Bbb{Z_{\geq 1}}$.

You make the change of variable $y_i = x_i - 1.$

Then, you are computing the number of solutions to

• $y_1 + y_2 + y_3 + y_4 = (7-4).$

• $y_1, y_2, y_3, y_4 \in \Bbb{Z_{\geq 0}}.$

By Stars and Bars theory, the number of solutions is

$$\binom{[7-4] + [4-1]}{4-1} = \binom{6}{3}.$$

This takes care of the $4$ right moves.

For the $3$ up moves, you want the number of solutions to

• $x_1 + x_2 + x_3 = 7.$

• $x_1, x_2, x_3, \in \Bbb{Z_{\geq 1}}$.

You make the change of variable $y_i = x_i - 1.$

Then, you are computing the number of solutions to

• $y_1 + y_2 + y_3 = (7-3).$

• $y_1, y_2, y_3, \in \Bbb{Z_{\geq 0}}.$

By Stars and Bars theory, the number of solutions is

$$\binom{[7-3] + [3-1]}{3-1} = \binom{6}{2}.$$

So, the total number of variations is the cross product represented by

$$\binom{6}{3} \times \binom{6}{2}.$$

---

For Part B, I interpret it to mean that you are supposed to regard the pawns as distinguishable. This suggests that (for example) the two rooks are also distinguishable from each other. So, the only constraints are that

• All the pawns go on the 1st rank, and the pieces go on the 2nd rank.

• The queen is supposed to get her color, which means that the white queen must be placed on a white square.

Under this very simplistic (and quite possible erroneous) interpretation, the computation is simply

$$\frac{1}{2} \times 8! \times 8!. \tag1 $$

In (1) above, the first factor represents that, by symmetry, exactly half of the configurations will result in the Queen on a white square.

In (1) above, the 3rd factor represents that the rooks, knights, bishops, king and queen are all distinguishable, so you are computing the number of ways of positioning $8$ distinguishable pieces.

---

Part B continued:

It is very doubtful that the my first interpretation above is the one intended by the problem composer. Presumably, the reason that chess pieces are involved is that you are supposed to assume that (for example) the two rooks are indistinguishable from each other.

However, you can't have it both ways. Given the problem as it is currently written, if (for example) the rooks are indistinguishable from each other, then so are the pawns.

This implies that there is only one way of positioning the 8 pawns, all in one row.

Consequently, under this new interpretation, the computation is

$$\frac{1}{2} \times \binom{8}{2} \times \binom{6}{2} \times \binom{4}{2} \times (2!). \tag2 $$

In (2) above, the first factor (again) uses a symmetry argument to consider that the queen must be on a white square. Then, the next three factors represent the number of ways of positioning, in order, the rooks, knights, and bishops. Then, the last factor represents the number of ways of positioning the king and queen.