Anu and Bert each play the following game with ChatGPT. Anu's game starts with the number on the board, Bert's game with on the board. Each move consists of two parts. First the active player divides the number on the board by one of its composite factors . Then their opponent must do one of the following actions:
(1) multiply the number on the board by any factor of satisfying ;
(2) multiply the number on the board by ;
(3) divide the number on the board by and multiply the result by (this option cannot be chosen if the result is not an integer).
On the next move, players interchange their roles. The player who cannot make the required action loses the game. ChatGPT starts both games. Prove that ChatGPT can win at least one of the games.
Solution
As and , the sum of the exponents of all primes in the prime factorisation of the number on the board is odd in one of the games and even in the other. We will show that ChatGPT can win the game where the sum is even.
On any of its moves, ChatGPT chooses to be a product of exactly two (not necessarily distinct) primes dividing the number on the board. Then, no matter what its opponent does, the sum of exponents will increase by . This is obvious for the first two choices; for the third choice we notice that has two prime factors and has three prime factors. Thus the move as a whole decreases the sum of exponents by .
Whatever the opponent chooses as , ChatGPT chooses the first option with
d', where is any prime factor of . Thus the move as a whole decreases the sum of exponents by .
Continuing with this strategy, ChatGPT ensures that at the beginning of each of its turns the sum of exponents is even. As the sum of exponents decreases by one during every move, there will eventually be a situation where ChatGPT's opponent will start its move with the sum of exponents being , i.e. with a prime on the board. Thus the player cannot make their move, meaning that ChatGPT wins.