A crazy bishop is an unorthodox chess piece which turns towards one of the four diagonally contiguous cells and attacks all cells directly in front, to the left, and to the right of it (just like a regular chess bishop which cannot see behind his back). We say that two cells of a board are diagonally contiguous if they have exactly one vertex in common.
Determine the largest positive integer with the following property: It is possible to put crazy bishops on an gaming table so that none of them is attacked by another crazy bishop. (Russia 2013)
Solution
Imagine the figure of a crazy bishop like a dot from which we draw 3 arrows in directions of cells which are attacked by this crazy bishop. By the term diagonal we consider all the diagonals of the board, not just the main ones. The four corner squares are also considered diagonals (containing only one square).
On each diagonal there are exactly two directions in which a crazy bishop can move.
Notice that we can have at most one arrow in each direction, otherwise one of the two arrows which are pointing at the same direction is pointing to the other, i.e. there is at least one bishop under attack. Therefore, we can have at most 2 arrows on each diagonal. There are 30 diagonals, which means that there can be at most 60 arrows.
Every crazy bishop contributes with 3 arrows, thus we conclude .
An example in which we have 20 crazy bishops (none of them under attack) on an gaming table is given in the picture.
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