# 2×2 squares inside a 9×9 square

You are given an empty $9\times9$ grid (sort-of like a blank Sudoku grid, or sort-of like a chessboard, but with one extra row and one extra column).
You fill in these cells randomly with all the integers between 1 and 81 (inclusive).
Prove that there is at least one $2\times2$ square where the sum of the numbers is more than 137.

• Sidenote: a chessboard has 8x8 squares – fondor May 10 '16 at 7:58
• @fondor. I know...that's why i put chessboard in brackets. Just so people can visualize it better. – Marius May 10 '16 at 8:10
• @fondor - see chessboard description; size and shape may be altered from the classical 8 by 8 square. – Jonathan Allan May 10 '16 at 15:53