MathDB

Given two positive integers

n

and

m

and a function

f : \mathbb{Z} \times \mathbb{Z} \to \left\{0,1\right\}

with the property that \begin{align*} f\left(i, j\right) = f\left(i+n, j\right) = f\left(i, j+m\right) \qquad \text{for all } \left(i, j\right) \in \mathbb{Z} \times \mathbb{Z} . \end{align*} Let

\left[k\right] = \left\{1,2,\ldots,k\right\}

for each positive integer

k

. Let

a

be the number of all

\left(i, j\right) \in \left[n\right] \times \left[m\right]

satisfying \begin{align*} f\left(i, j\right) = f\left(i+1, j\right) = f\left(i, j+1\right) . \end{align*} Let

b

be the number of all

\left(i, j\right) \in \left[n\right] \times \left[m\right]

satisfying \begin{align*} f\left(i, j\right) = f\left(i-1, j\right) = f\left(i, j-1\right) . \end{align*} Prove that

a = b

Problem Statement