Invariants much?
Imagine a circle divided into sections (this is done by drawing lines through the center). The number is placed in of these sections, and in the rest the number is placed.
In any move, you may add to any two sections that share a side. In how many of these arrangements can you get the same number in every section?
HINT: An invariant is something that remains constant. There is one in particular that will be very useful!
The answer is 0.
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1 solution
Color the sections black and white, alternating colors so that no black section touches another black section and no white touches another white. In any move we add one to the sum of the numbers in the black sections and white sections, and therefore their positive difference is constant.
Notice that for all digits to be equal, the difference must be 0 and so there must be the same number of 1's in black sections in white sections. This implies the number of 1's you start with must be even in order to make every section contain the same number, so the answer is 0.