Alice and Bob have two positive integers, x and y respectively, glued to their foreheads, so that each can read the other's number but not their own. They also know that |x - y| = 1. The following conversation is overheard between Alice and Bob:

Alice: I don't know my number.

Bob: I don't know my number.

Alice: I don't know my number.

Bob: I don't know my number.

They say this to each other 200 times. And then we overhear them say: Alice: I know my number! Bob: I know my number too!

Can you explain the conversation and figure out the numbers x and y. You may assume that Alice and Bob each know that the other is extremely smart - so each is confident that if the other has sufficient information to deduce the answer then he/she definitely will.

Step-by-step explanation: |x - y| = 1, ok lets play as Alice, my number is y, and the bob number is x.

the condition says that x-y = 1 or x-y = - 1.

so, if you know x, then y = 1 + y or y = y - 1. so you have two possibilities.

let's see two cases : first, let's suppose there are no code in the conversation. Then the only way of being shure of your number, is if one of them have the lowest positive number, so the other should have the next one. So if Bob have the number one, Alice knows for shure that she has the 2. Bob knows that she has a 2, but that means he could have a 1 or a 3, but when he sees that Alice is shure about her number, he knows that his number is the 1.

the second case is where the conversation may be a sort of code, saying a phrase x times and changing when x = the number of the other person, in this case, bob will have the 201 and alice the 202.