Question

mn squares of equal size are arranged to form a rectangle of dimension m by n, where m and n are natural numbers. Two square will be called neighbors if they have exactly one common side. A number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighboring squares. Show that this is possible only if all the numbers used are equal.

Answer 1

Let mn squares of equal size are arrange to form a rectangle of dimension m by n. Shown as, from figure.

neighbours of `x_(1)` are `{x_(2), x_(3), x_(4), x_(5)} x_(5)` are `{x_(1), x_(6), x_(7)}` and `x_(7)` are `{x_(5), x_(4)}`
`rArr " " x_(1) = (x_(2) + x_(3) + x_(4) + x_(5))/(4), x_(5) = (x_(1) + x_(6) + x_(7))/(3)`
and `x_(7) = (x_(4) + x_(5))/(2)`
`therefore` `4x_(1) = x_(2) + x_(3) + x_(4) + (x_(1) + x_(6) + x_(7))/(3)`
`rArr " " 12x_(1) = 3x_(2) + 3x_(3) + 3x_(4) + x_(1) + x_(6) + (x_(4) + x_(5))/(3)`
`rArr " " 24x_(1) = 6x_(2) + 6x_(3) + 6x_(4) + 2x_(1) + 2x_(6) + x_(4) + x_(5)`
`rArr" " 22x_(1) = 6x_(2) + 6x_(3) + 7x_(4) + x_(5) + 2x_(6)` where, `x_(1), x_(2), x_(3), x_(4), x_(5), x_(6)` are all the natural numbers and `x_(1)` is linearly expressed as the sum of `x_(2), x_(3), x_(4), x_(5), x_(6)` where sum of coefficients are equal only if, all observations are same.
`rArr" " x_(2) = x_(3) = x_(4) = x_(5) = x_(6)`
`rArr` All the numbers used are equal.