proof_pile.txt

This example tells you the story (I've coded it up, no idea how to solve it deductively):
(1, 37, 220)
(2, 36, 220)
(4, 36, 218)
(8, 32, 218)
(16, 32, 210)
(32, 32, 194)
(0, 64, 194)

It's always possible. Proof:
- Let our state be (A, B, C) where A <= B <= C.
- By the remainder theorem write B = A*x + r where r < A.
- It is enough to find a process that sets the minimum to r,
  hence lowers the minimum of the state in each step.
- Let h be largest int such that 2^h <= x.
- We double A in each step (A, 2^1 A, 2^2 A, ..., 2^h A)
  - the only question is where do the chips come from (B or C)
  - if in step 2^k we have that (x>>k)&1 is set,
    we substract it from B, otherwise from C
  - what's the maximum amount of chips we take from C?
    - answer: 1 + 2 + ... + 2^(h-1) < 2^h <= B <= C, so we won't take too much
- After the process A is replaced with r, so the minimum shrinks (strictly)
- Repeat process until A = 0