I have a math problem, and I have no idea how to solve it. I first saw it over on the Code Golf StackExchange, here. The poster there hinted at knowing that there is a proof, but no proof was provided.
We start with a pair of integers $a$ and $b$. We double one and add one to the other. We have the power to decide which to double and which to increment. We repeat this "doubling/+1" process, until the two integers are equal.
For example, starting with $(2, 5)$ we can double the 5 and increment the 2 to give $(3, 10)$. Then, we can double the three and increment the 10 for $(6, 11)$. We can double the six and increment the 11 for $(12, 12)$ - and now we have made the numbers equal.
Given any pair of integers, it it always possible to make them equal using these steps?
Partial Proof All pairs of negative numbers will terminate. This relies on the fact that doubling 0 is 0. If we start with a pair of negative numbers, we can repeatedly add one to one of the numbers until it hits 0. While the other number will have been doubled several times, we can repeatedly decrement it until it too, hits 0.
$(-6, -3) \to (-12, -2) \to (-24, -1) \to (-48, 0) \to (-47, 0) \to (-46, 0) \to \cdots \to (0, 0)$