Question 105
Two squids are forced to take part in a game. Before it begins, they are told all the rules and may discuss their strategy freely. Then they are locked in separate rooms, and each is told a distinct positive integer not exceeding $2023$ as its ID. The squids then take turns alternately; on its turn, a squid chooses one of the following actions:
1. announce a positive integer, which will be heard by the other squid;
2. declare which squid has the larger ID. If correct, they win together and are released; otherwise, they lose together and are made into fried squid.
Find the smallest positive integer $N$ such that, no matter which IDs they are given, the squids can guarantee to win within finitely many turns with the sum of the numbers announced during the game not exceeding $N$.
1. announce a positive integer, which will be heard by the other squid;
2. declare which squid has the larger ID. If correct, they win together and are released; otherwise, they lose together and are made into fried squid.
Find the smallest positive integer $N$ such that, no matter which IDs they are given, the squids can guarantee to win within finitely many turns with the sum of the numbers announced during the game not exceeding $N$.