Rocks on the chessboard
Task number: 4207
There are 33 rocks on the chessboard \( 8 \times 8 \). Prove that we find 5 rocks between them that do not endanger each other.
Solution
The rocks must occupy at least five rows, we choose the five fullest.
By analyzing the cases, we find that the fifth fullest has at least one rock: \( 8 + 8 + 8 + 8 + 1 \), fourth fullest at least two: \( 8 + 8 + 8 + 2 + 2 + 2 + 2 + 1 \), third at least three: \( 8 + 8 + 3 + 3 + 3 + 3 + 3 + 2 \), the other at least four: \( 8 + 4 + 4 + 4 + 4 + 3 + 3 + 3 \) and the first at least five: \( 5 + 4 + 4 + 4 + 4 + 4 + 4 + 4 \).
From the fifth most complete we will choose any rock, then from the fourth most complete we will choose a rock so as to avoid the column with the already selected rock, etc. up to the fullest row.
