MathDB

Given an integer

n>1

. The board

n\times n

is colored white and black in a chess-like manner. We call any non-empty set of different cells of the board as a figure. We call figures

F_1

and

F_2

similar, if

F_1

can be obtained from

F_2

by a rotation with respect to the center of the board by an angle multiple of

90^\circ

and a parallel transfer. (Any figure is similar to itself.) We call a figure

F

connected if for any cells

a,b\in F

there is a sequence of cells

c_1,\ldots,c_m\in F

such that

c_1 = a

c_m = b

, and also

c_i

and

c_{i+1}

have a common side for each

1\le i\le m - 1

. Find the largest possible value of

k

such that for any connected figure

F

consisting of

k

cells, there are figures

F_1,F_2

similar to

F

such that

F_1

has more white cells than black cells and

F_2

has more black cells than white cells in it.

Problem Statement