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Maths Problem : Infinite grid

Imagine that there is an infinite grid, within which each box contains an integer. Prove that the only way each box has a mean that is less than or equal to the 4 surrounding it, is if every box is equal.

or

Imagine that there is an infinite grid, within which each box contains an integer. How many way to put the number so that each box has a mean of the 8 surrounding it.


I got this type of questions twice, just cant get my head around this type of questions.
Help please!
Original post by bigcrazysix
Imagine that there is an infinite grid, within which each box contains an integer. Prove that the only way each box has a mean that is less than or equal to the 4 surrounding it, is if every box is equal.

or

Imagine that there is an infinite grid, within which each box contains an integer. How many way to put the number so that each box has a mean of the 8 surrounding it.


I got this type of questions twice, just cant get my head around this type of questions.
Help please!

my advice is to draw a cross shape with 5 squares label the integers with letters and then find an expression for the mean and then consider the mean of a square next to the middle square

just out of curiosity, where is this question from?
Original post by bigcrazysix
Imagine that there is an infinite grid, within which each box contains an integer. Prove that the only way each box has a mean that is less than or equal to the 4 surrounding it, is if every box is equal.

I got this type of questions twice, just cant get my head around this type of questions.
Help please!


I presume you mean each box has a value less than or equal to the mean of the four boxes immediately to the left, right, up, down.

If that's the case, then the assertion isn't true.

If we label our grid points (x,y), where x,y are integers - assuming this is a regular square grid.

Then let f(x,y) = x be the value of the box at (x,y).

The mean of the four surrounding boxes is [f(x-1,y)+f(x+1,y) + f(x,y-1) +f(x,y+1)]/4

= [x-1 + x+1 + x + x]/4 = x which is <= x = f(x,y)

Contradicting the fact that f(x,y) is supposed to be constant.
(edited 10 years ago)

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