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I found this teaser recently, anyone will suggest a clue to trackle it. Forgive me if it has bee posted before.Alice and Bob play a game in which they take turns removing stones from a heap that initially has n stones.The number of stones removed at each turn must be one less than a prime number. The winner is the player whotakes the last stone. Alice plays first. Prove that there are infinitely many n such that Alice has a winning strat-egy. (For example, if n = 17, then Alice might take 6 leaving 11; then Bob might take 1 leaving 10; thenAlice can take the remaining stones to win.)

There might be something you need to add to your puzle statement. Because we have infinite primes, 2,3,5,7,11,...For every n of form prime-1 = 1,2,4,6,10, ..., Alice takes all of the stones on her first turn itself (because they are prime -1) and wins.

Goldbach's conjecture ?