Start with a non-empty pile of N counters. A move consists in splitting a pile into two non-empty parts. Repeat the process by applying the moves to thus obtained piles. The process ends when there is no pile left with more than a single counter.
Every process described above that starts with a pile of n counters ends in n - 1 steps.
Proof #1 (by induction)
In the proof by induction the quantity N - 1 might appear to have come out of the blue. However, there is a justification. Realization that the number of splits only depends on the size of a pile leads to the equation
which I rewrite as g(m + n) = g(m) + g(n), where g(n) = f(n) + 1. g is an additive integer function and thus is bound to be linear.
Proof #2 (detecting an invariant)
Let start counting how many piles we have after a number of splits. The important observation is that every time we split a pile the total number of piles is grows by 1. (For one bigger pile is being replaced with two smaller ones.) When there is no piles to split, each pile consists of a single counter. At the beginning (after 0 splits) we had 1 pile. After 1 split we got 2 piles. As I said earlier, increasing the number of splits by one increases the number of piles by 1. Therefore, the latter is always greater by one than the former.