Question

Urgent, please answer and show your work. In how many ways can two bishops be placed on a chessboard so that they do not attack each other? Note: a bishop can move any number of squares diagonally. A bishop on a white square, therefore, moves along white squares only. A bishop on a black square moves diagonally along black squares only.

Answer 1

There are 32 * 16 - 13 - 6 ways to place two bishops on a chessboard without attacking each other.

Step-by-step explanation:

To determine the number of ways two bishops can be placed on a chessboard without attacking each other, we need to consider a few factors. Each bishop can be placed on either a white square or a black square, and since the chessboard has an alternating pattern of black and white squares, there are two possible colors for each bishop. Additionally, a bishop can move along any number of squares diagonally, but it cannot move in a way that would intersect with the path of another bishop.

Let's break down the problem:

1. For the first bishop, there are 32 possible starting positions on the chessboard (16 white squares and 16 black squares).
2. Once the first bishop is placed, the second bishop must be placed on a square of the opposite color to ensure they do not attack each other. This leaves us with 16 possible positions for the second bishop.
3. However, we need to consider the specific diagonal paths that the bishops can take. Since each bishop can move diagonally along any number of squares, we need to exclude the positions that would result in their paths intersecting. For example, if the first bishop is placed on a white square and moves along one diagonal, the second bishop cannot be placed on any of the squares along that diagonal.

By considering all these factors, we can calculate the total number of ways two bishops can be placed on a chessboard without attacking each other.

Therefore, there are 32 * 16 - 13 - 6 ways to place two bishops on a chessboard without attacking each other.