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Consider a rectangular grid m×n consisting of mn squares such that we call two squares neighbours of each other if they share exactly one side. Then if all the squares are filled each with a number such that the number in each and every square is the mean of the numbers in its neighbours, prove that it is necessary for all squares to be filled with the same number.

Now if I start up with a few variables around corner and proceed, I end up taking a lot of variables and stuck. How do I solve this? Just even a hint can help me solve it.

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2 Answers 2

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HINT: Suppose towards a contradiction that the grid is filled with different numbers. Consider a square whose number is highest. (If there is more than one, think about which square to consider.)

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Let $m$ be the square with the smallest number in it. It's neighbors must either be equal, or have a smaller number (and a larger, to take the mean or average), which contradicts $m$ being the smallest. Therefore all the squares must contain the same number.

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