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PrintSAUDI ARABIAN MATHEMATICAL COMPETITIONS
Saudi Arabia counting and probability
Problem
Chess horse attacks fields in distance . Let several horses are put on the board such, that every square of size contains at least one horse. Find the maximal possible number of cells that are not under attack (horse doesn't attack its own cell).
Solution
Let's note, that if we put a horse in any green cell, then it will attack a grey cell. Since green cells form a square , so one of them contains a horse, so at least one grey is under attack.
Now let's split the board into 72 pairs like in the figure. According what we said above, at least 72 cells are under attack. To get example with exactly 72 cell under attack let's paint board as chessboard and put horses in all black cells. Then all black cells will be free of attack.
Hence, the answer to this problem is 72.
Now let's split the board into 72 pairs like in the figure. According what we said above, at least 72 cells are under attack. To get example with exactly 72 cell under attack let's paint board as chessboard and put horses in all black cells. Then all black cells will be free of attack.
Hence, the answer to this problem is 72.
Final answer
72
Techniques
Coloring schemes, extremal arguments