The following description considers the manipulation of a $3\backslash$times $3$ array of integers, which can be considered

a kind of simple game. The description will start by describing the rules of the game, and then you$$ll be

told what your task will be$.$ It is highly unlikely that you will discover anything sensible on the internet about

this game. It was invented by one of your instructors.

Consider an $3\backslash$times $3$ array of integers $\{0,,8\}$ such as:

$012$

$345$

$678$

This is a kind of a puzzle, where we allow rows to be shifted left or right, and columns to be shifted up or

down. We will call these actions unit rotations, for reasons that will be clarified immediately.

A unit rotation is simply a shift of all the elements of a row $($or column$)$ by one position. For example, we

can rotate the top row of the previous example one position to the right to get the following:

$201$

$345$

$678$

Notice that the integer $2$ that was on the end of the row moved to the front after the rotation to the right.

Because the number $2$ moved to the front of the row, we call it a rotation. Similarly, we can rotate the

bottom row one position to the left to get the following:

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