MathDB

On an infinite grid, a square with four vertices lie at

(m, n)

(m-1, n)

(m,n-1)

(m-1, n-1)

is denoted as cell

(m,n)

(m, n \in Z)

. Some marbles are dropped on some cell. Each cell may have more than one marble or have no marble at all. Consider a "move" can be conducted in one of two following ways: i) Remove one marble from cell

(m,n)

(if there is marble at that cell), then add one marble to each of cell

(m - 1, n- 2)

and cell

(m -2, n - 1)

. ii) Remove two marbles from cell

(m,n)

(if there is marble at that cell), then add one marble to each of cell

(m +1, n - 2)

and cell

(m - 2, n +1)

. Assume that initially, there are

n

marbles at the cell

(1,n), (2,n - 1),..., (n, 1)

(each cell contains one marble). Can we conduct an finite amount of moves such that both cells

(n + 1, n)

and

(n, n + 1)

have marbles?

Problem Statement