In today's session, we explored how logical reasoning combined with cooperation can lead to interesting results in situations that require logic with a group. This is called group reasoning.

The introduction is the following game:

2 players are working together in a team, and 2 positive integers less than or equal to 6 are chosen. One of the players is told the sum of 2 integers, and the other player is told the product of the 2 integers. The 2 players repeatedly answer the questions "do you know the 2 integers now?" Until one of them answers "yes" or claims that this is impossible.

A simple strategy, kind of a cheat, would be repeatedly saying no until one of the players reaches their own number, at which that player says yes, communicating the sum to the other player.

On the other hand, the actual solution to the problem would be drawing a table of all the different possible sums and the corresponding possible products. Then, the game simply becomes finding the rows and columns that can possibly lead to this result.

The takeaway from this game is that when the other player answers "no", they are not simply saying that they can't determine something. They are also saying that certain combinations of the initial conditions are impossible.

Then, the game of hats. There are a total of 2 white hats and 3 black hats, placed on the 3 players standing on a row. Each player can see the hat color of the persons in front of them, but not their own or the players on their back. Then, from the back to the front, the players are asked whether they know their hat color or not. They win if at least 1 player guesses their hat color, and everyone who guesses guessed it correctly.

This does not require any discussion prior to the game, but the players should list out the possible cases in advance to make sure they collect all the possible information given by the previous person.

For example, the case where all 3 players have a black hat.

The last person can see 2 black hats, so he cannot deduce his own hat color.

The middle person can see 1 black hat. He also cannot deduce his own hat color.

However, when it comes to the first person, who cannot see anything, he knows that the person behind him and his own hat cannot be simultaneously white, because otherwise person 3 would've guessed his own color. On the other hand, since person 2 said that he cannot gues his own color, he is communicating the information that person 1's hat color is black. If it were white, person 2 would've known that their own hat color cannot be white.

Therefore, person 1 can infer their own hat color.

Finally, another game of hats. A number of people line up, and every person can see the hat color of everyone in front of them. The goal is to have the maximum number of people to guess their own hat color correctly, starting from the very back of the line.

A strategy would be having every other person telling the person directly in front of them the hat color, and this strategy would on average lead to 3/4 of the total number of players correct guesses. The minimum would be 1/2 of the total number.

The optimal strategy would be having the last person communicate the parity of the number of white hats he sees. If he sees an odd number of white hats, guess white, otherwise guess black. This guarantees at least n-1 correct guesses, with an average of n-1/2 correct guesses.

The link to the meeting's recording: https://www.youtube.com/watch?v=Gr7SNKAVDyE

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