Question 2
Every cell of an $m\times n$ board is filled with a nonnegative integer. Two cells are adjacent if they share a side. A filling is a garden if (i) any two adjacent numbers differ by $0$ or $1$, and (ii) a number less than or equal to all its adjacent numbers must equal $0$. Determine the number of gardens in terms of $m,n$.
Proposed by Sungyoon Kim (credited in the supplied notes).
Proposed by Sungyoon Kim (credited in the supplied notes).