MathDB

Let

n

be a natural number. A tiling of a

2n \times 2n

board is a placing of

2n^2

dominos (of size

2 \times 1

1 \times 2

) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two sepearate tilings of a

2n \times 2n

board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours.
Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that

n

is divisible by

3

Proposed by Tejaswi Navilarekallu

Problem Statement