# Problem of the MonthProblem 8: May 2024

This monthâ€™s problem is an extension of Question 4 from the 2024 Galois contest. It is not necessary to try the problem before attempting the questions below.

In an $$m\times n$$ rectangular grid, we say that two cells are neighbours if they share an edge. The neighbourhood of a cell is the cell itself along with its neighbours.

An $$m\times n$$ grid is called a Griffin Grid if each of its $$mn$$ cells contains either a $$1$$ or a $$-1$$ and the integer in every cell is equal to the product of the other integers in its neighbourhood.

For example, the $$4\times 9$$ grid below is a Griffin Grid. The three shaded regions are the neighbourhoods of the cells in Row 1 and Column 1, Row 1 and Column 8, and Row 4 and Column 4.

 $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$

The Galois problem restricted this definition to $$m=3$$. Here we want to explore what happens more generally. If a question is marked with an asterisk $$(*)$$, it means I was unable to solve it. Solutions will not be provided for these problems, but I would love to hear if you solve one!

1. Show that an $$m\times n$$ grid with $$-1$$ or $$1$$ in every cell is a Griffin Grid if and only if the cells in every neighbourhood have a product of $$1$$.

2. For every $$n$$, determine the number of $$2\times n$$, $$3\times n$$, and $$4\times n$$ Griffin Grids. Determining the number of $$3\times n$$ Griffin Grids in general is essentially what is required to answer part (c) of the Galois question.

3. Show that the number of $$m\times n$$ Griffin Grids is of the form $$2^k$$ for some integer $$k$$ with $$0\leq k\leq m$$.

4. $$*$$ For general $$m$$, determine for which $$k$$ there exists $$n$$ with the property that the number of $$m\times n$$ Griffin Grids is exactly $$2^k$$.

5. Show that for all $$m$$ there exist infinitely many $$n$$ for which there is exactly one $$m\times n$$ Griffin Grid.

6. Show that for all $$m$$ there exist infinitely many $$n$$ for which there are $$2^m$$ distinct $$m\times n$$ Griffin Grids.

7. * Find a simple general way to calculate the number of $$m\times n$$ Griffin Grids.