Case 14 · strategy · Hard · About 10–15 minutes
A keeper sets three bowls on a table and challenges you to a small game. The rules fit in a sentence, but the first move matters. A careless opening hands the keeper a winning reply.
Your task is to find a move that wins against perfect opposition, not merely against one friendly response.
The bowls are labeled A, B, C and begin with 1, 2, 4 stones, respectively.
You move first. Players alternate turns. On each turn, remove either one or two stones from exactly one bowl. You may not remove more stones than that bowl contains. Removed stones do not return.
The player who takes the final stone wins. There are no passes, transfers, or moves affecting two bowls at once. An empty bowl stays empty.
Which first move guarantees a win, whatever legal replies the keeper chooses? Identify both the bowl and the number removed. Explain why your plan works.
There are five legal opening moves. A convincing solution must survive every possible reply after its chosen opening. To establish uniqueness, you should also be able to explain why each other opening gives the keeper a winning strategy.
Hint 1 - reveal when you want a nudge
Try removing both stones from B. The remaining bowls contain 1 and 4. There are only three possible replies.
Hint 2 - reveal when you want a nudge
After that opening, answer a removal from the 1-stone bowl by leaving 3 in the other. If the keeper removes 2 from the 4-stone bowl, leave two separate single stones.
Reveal the solution and explanation
Remove both stones from bowl B. This is the only winning opening.
The position becomes (1,0,4). If the keeper removes A's stone, remove one from C, leaving a single bowl of three. If the keeper removes one from C, remove A's stone, again leaving a single bowl of three. From three, answer a removal of one by taking two, or a removal of two by taking one.
If the keeper instead removes two from C, the position is (1,0,2). Remove one more from C, leaving two separate single stones. The keeper takes one, and you take the last.
The other openings allow these keeper replies:
- You remove one from A: the keeper removes two from C, leaving (0,2,2).
- You remove one from B: the keeper removes one from C, leaving (1,1,3).
- You remove one from C: the keeper removes one from B, leaving (1,1,3).
- You remove two from C: the keeper removes one from A, leaving (0,2,2).
At (0,2,2), the keeper mirrors your removal in the other equal bowl. At (1,1,3), the keeper mirrors a move on the single stones, or completes a total removal of three from the last bowl. Each paired response finishes the affected part. The keeper therefore takes the final stone in every case.
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