Here is the labeling of the 9 positions from before.
A B C D E F G H I
If we are at B, when can we go to the right to C? We will then have a multiple of 3, and will be forced to go down to F. But after adding 2, we will not have a multiple of 3 anymore, so we cannot go to I, but must divide by 4 to get to E. If we start with an odd number at B, we will be stuck with an odd number at F. On the other hand, if we start with a multiple of 4 at B, then we will no longer have a multiple of 4 after adding 2, so again we will be stuck. Hence, to go to the right from B, one must start with an even number that is not a multiple of 4. In other words, if x is the value at B, x+2 must be a multiple of 4 to go to C. You will wind up at E with the value (3x+2)/4.
Thus, if you are at B, you can only go to the left if x+1 is a multiple of 3, and can only go to the right if x+2 is a multiple of 4. Can you find any restrictions for when you can go down?
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