Twenty-four possible answers against 3^3 = 27 outcome sequences leaves just enough room, and four against four splits those answers into exactly 8, 8 and 8. Six against six always tips, so it wastes the level outcome and leaves twelve answers for nine remaining sequences, which makes halving impossible rather than merely slow. The counting argument bounds outcome sequences rather than strategies, so it rules out every adaptive continuation at once, and the explicit schedule closes the positive half.
Seven pieces cost six cuts and the schedule pays correctly, so the six-cut answer breaks one constraint and nothing else. Because the worker can hand pieces back, the contract is on his holding rather than on the transfer, and the ledger turns out to be a three-bit counter. Brute force finds 1-2-4 is the only three-piece solution.
The reflex answer is around 180, half the calendar. The real threshold is 23, because a match needs a pair and 23 people carry 253 of them. The same reasoning puts the answer to "does anyone share MY birthday" at 253 people, eleven times the crowd, and the square-root threshold behind both is why a 64-bit random id collides after five billion draws rather than eighteen quintillion.