Problem of the Day #94

POTD March 12, 2024

Consider a 10\times10 lattice grid which has coordinates (0,0) to (9,9). Each point (x,y) has a number m_{x,y} on it. The numbers satisfy:

|m_{x,y}| = \frac{|m_{x-1,y}|+|m_{x+1,y}|+|m_{x,y-1}|+|m_{x,y+1}|}{4}

For each (x,y) (indices are taken mod 10).

Also, x_{0,0} = 1, and the product of the numbers in each row or column is the same. If the total number of possible grids is a \cdot 2^b, where a,b are non-negative integers and a is odd, find a+b.