N denotes a minimum number of bishops to guarantee full coverage of a standard chessboard.
How much is N, and where should these bishops be placed?

To cover a square means either to stay on it or to attack it.

Convert bishop moves into rook moves:

...O...

..OOO..

.OOOOO.

OOOOOOO

OOOOOOO

.OOOOO.

..OOO..

...O...

Turn the board 45 degrees. Remove the black squares. Rotate each white square by 45 degrees and squash them together sideways.  The diagram above is a poor representation of the result.  

Within the diagram is a 4x5 rectangle.  The four bishops must be placed in this rectangle with one in each row and at least one in each of the middle 3 columns.

To count them, consider where the fourth bishop goes, then count the number of choices by column:  

In the first column: 4!=24

In the second column: 4x3/2x2x1=12

In the third column: 4x3x2/2x1=12

In the fourth column: 4x3x2x1/2=12

In the fifth column: 4!=24

Total: 84

Posted by Jer on 2021-09-02 14:14:08