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Maximum number of 2x2 tiles on a chessboard with 1,2,...,64

Source: XI Olimpíada Matemática del Cono Sur (2000)

July 26, 2011
pigeonhole principlecombinatorics unsolvedcombinatorics

Problem Statement

The numbers 1,2,…,641,2,\ldots,64 are written in the squares of an 8×88\times 8 chessboard, one number to each square. Then 2×22\times 2 tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than 100100. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers 1,2,…,641,2,\ldots,64 into the squares of the chessboard that admits this maximum number of tiles.
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