MathDB

Some unit squares of

2007\times 2007

square board are colored. Let

(i,j)

be a unit square belonging to the

ith

line and

jth

column and

S_{i,j}

be the set of all colored unit squares

(x,y)

satisfying

x\leq i, y\leq j

. At the first step in each colored unit square

(i,j)

we write the number of colored unit squares in

S_{i,j}

. In each step, in each colored unit square

(i,j)

we write the sum of all numbers written in

S_{i,j}

in the previous step. Prove that after finite number of steps, all numbers in the colored unit squares will be odd.

Problem Statement