Chess Board
Source: OMM 2008 3
July 19, 2014
geometryrectanglecombinatorics unsolvedcombinatorics
Problem Statement
Consider a chess board, with the numbers through placed in the squares as in the diagram below.\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
\hline
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\
\hline
25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 \\
\hline
33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\
\hline
41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 \\
\hline
49 & 50 & 51 & 52 & 53 & 54 & 55 & 56 \\
\hline
57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 \\
\hline
\end{tabular}Assume we have an infinite supply of knights. We place knights in the chess board squares such that no two knights attack one another and compute the sum of the numbers of the cells on which the knights are placed. What is the maximum sum that we can attain?Note. For any or rectangle that has the knight in its corner square, the knight can attack the square in the opposite corner.