Here is the labeling of the 9 positions from before.
A B C D E F G H I
The last hint suggested that there are only certain conditions for when you want to go from B down to E. Suppose we started with an even number at B, and go down to E. After adding 9, we will have an odd number, so we cannot go to the left or right, but must proceed to H, adding 8. We will still have an odd number, so will will not be able to go to I, but must continue to G, adding 8 again.
At this point we will be stuck UNLESS we happen to have a multiple of 3. Ironically, though, having a multiple of 3 won't help us all that much. For if we do have a multiple of 3, we can proceed to D. But we will still have an odd number, so we cannot go to E, but rather go up to A. After adding 5 we will have an even number, and then we double it to get back to B.
The problem is that, after doubling an even number, we will have a multiple of 4. The last hint told us that we can only go to C if we have an even number that is not a multiple of 4. Hence, we will be forced back to E.
Now the process repeats. We find ourselves forced to go round and round in the loop until eventually we find ourselves at G without a multiple of 3. (There is one exception: if one is at B with a value of 80, and goes to E, he will be forced back to B with a value of 80 again. Thus, he will be forced to go around the same loop over and over again.)
Thus, we find that, if we start at B with an even number, we should not go down to E. This rule is actually the most helpful, since following it will keep you clear of many of the dead ends in the maze. Note that if we start at A, and go right to B, we will have an even number, so this rule forces us to go on to C.
Now that we have analyzed where we can go from the point B, consentrate on the central square E. Because of the nature of the maze, every path must go though the central square on a regular basis. Are there any restrictions for when we can go up from the center square E?
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