Question

Сколькими способами можно расставить на шахматной доске черного белого королей так, чтобы они не били друг друга(не стояли на соседних клетках)?

Answer 1

Есть 3 случая:
1) Белый король стоит в угловых клетках: 4 варианта. Для каждого из этих вариантов у чёрного короля 60 возможностей. 4*60 = 240 расстановок.
2). Белый король стоит по краям доски, но не в углах: 24 варианта.Для каждого из этих вариантов у чёрного короля 58 возможностей. 24*58 = 1392 расстановок.
3) Белый король на клетках,не примыкающих к краям: 36 вариантов.Для каждого из этих вариантов у чёрного короля 55 возможностей.36*55 = 1980 расстановок.
Всего возможных расстановок будет 240+1392+1980 = 3612.
Ответ. 3612

Answer 2

Number of Ways to Arrange Black and White Kings on a Chessboard

To determine the number of ways to arrange the black and white kings on a chessboard such that they do not attack each other (i.e., they do not stand on neighboring squares), we can consider the following approach:

1. Position the White King: There are 64 squares on a chessboard, so the white king can be placed in any of these squares. This gives us a total of 64 possibilities for the white king's position.

2. Position the Black King: After placing the white king, we need to position the black king. Since the black king cannot be placed on any of the neighboring squares of the white king, we need to consider the squares that are not adjacent to the white king's position.

- If the white king is placed in one of the corner squares (a1, a8, h1, h8), there are three squares that are not adjacent to it. - If the white king is placed on one of the edge squares (excluding the corners), there are five squares that are not adjacent to it. - If the white king is placed on any other square, there are eight squares that are not adjacent to it.

Therefore, the number of possible positions for the black king depends on the position of the white king.

3. Calculate the Total Number of Arrangements: To calculate the total number of arrangements, we need to multiply the number of possibilities for the white king's position by the number of possibilities for the black king's position, considering the different cases mentioned above.

- If the white king is placed in one of the corner squares, there are 3 possibilities for the black king's position. - If the white king is placed on one of the edge squares, there are 5 possibilities for the black king's position. - If the white king is placed on any other square, there are 8 possibilities for the black king's position.

Therefore, the total number of arrangements can be calculated as follows:

- If the white king is placed in one of the corner squares: 64 (possibilities for the white king) * 3 (possibilities for the black king) = 192 arrangements. - If the white king is placed on one of the edge squares: 64 (possibilities for the white king) * 5 (possibilities for the black king) = 320 arrangements. - If the white king is placed on any other square: 64 (possibilities for the white king) * 8 (possibilities for the black king) = 512 arrangements.

Therefore, the total number of ways to arrange the black and white kings on a chessboard such that they do not attack each other is 192 + 320 + 512 = 1,024 arrangements.

Please note that the calculations provided above are based on the assumption that the kings are indistinguishable (i.e., all white kings are considered the same, and all black kings are considered the same).