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Omni-MATH / The numbers 1,2,…,64 are written in the squares of an 8×8 chessboard, one number to each square. Then 2×2 tiles are placed on the chessboard (without overlapping) so that each tile…

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problem

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.
Plain-text mathematical notation (without MathML)
The numbers 1,2,…,64 are written in the squares of an 8×8 chessboard, one number to each square. Then 2×2 tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than 100. 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,…,64 into the squares of the chessboard that admits this maximum number of tiles.
Original LaTeX notation
The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. 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 $1,2,\ldots,64$ into the squares of the chessboard that admits this maximum number of tiles.

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Source and history

task_id
eda746a4-9f4d-5ef5-9539-1c69af3c80d3
task_key
test--eda746a4-9f4d-5ef5-9539-1c69af3c80d3
task_revision_id
1
config / split
/ test

Official source

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