Can squares with sides equal to consecutive values 1 to
tile the plane? An arbitrarily large patch can be covered by a method found by Adam Ponting .
Start with an
gray and white checkerboard, where
is odd and the corners are gray. Fill in the gray squares from left to right, bottom to top, with the numbers from 1 to
. Fill in the white squares top to bottom, left to right, with the numbers from
. At each vertex, a unit segment is drawn, vertical if the gray squares around that vertex are sloping down and horizontal if they are sloping up. This matrix has the property that digit pairs on each side of a segment have the same sum, which allows each number
to be converted to a square of side
. The segments indicate the squares are flush on that side.