MathDB

A polyomino is a figure which consists of unit squares joined together by their sides. (A polyomino may contain holes.) Let

n\ge3

be a positive integer. Consider a grid of unit square cells which extends to infinity in all directions. Find, in terms of

n

, the greatest positive integer

C

which satisfies the following condition: For every colouring of the cells of the grid in

n

colours, there is some polyomino within the grid which contains at most

n-1

colours and whose area is at least

C

.
Proposed by Nikolai Beluhov, Stara Zagora, Bulgaria and Stefan Gerdjikov, Sofia, Bulgaria

A. 783