Guess my number!

I was trying to figure out the "thinking patterns" of Ann and I'd found a solution, but it is only possible if 0 is not one of the possibilities. Are we talking about strictly positive integers or not? In any case, if we do: the only way that Ann and Bill could possibly figure something out is using the fact that 1 only has one valid possibility for the other person to have as a number: 2. Suppose Ann has 3. Bill could have 2 or 4, but she doesn't know, as she says. Suppose Bill has 2: for him, Ann could have 1 or 3, but if she had 1, she would've immediately said Bill had 2 in the beginning. Therefore, since she didn't and in the case Bill would have 2, he would then say he knew Ann has 3, which he doesn't. Therefore, Bill has 4 and she knows that as she'll state in her reply.

We can solve be a process of elimination. We know Bill's number is a divisor of 20 - therefore Ann's number minus one and plus one must give a divisor of 20. -10 is not an option because 9, and 11 do not divide 20. 5 isn't either. and so on. 1 is an option because of 2 and 0 divides both. 3 is also an option because 4 and 2 divides 20 as well. Rereading the question, you will realize that 1 isn't a solution because 0 isn't positive. Therefore 3 is the correct solution.

First of all we can notice that if Ann's number was bigger than three then she couldn't have known which number her friend have. If you still don't get it lets suppose that Ann has four. She obviously don't know which number has her friend. There are two options: Bill has 3 or Bill has 5. Information that Bill doesn't know Ann's number is useless here,because even with that she can't determine which number Bill has. Situation becomes different if we suppose that Ann's number is 3 or less.

Let's start with assumption that Ann has 1. We can immediately reject this, because if she had 1 then she would know which number Bill has (he would have 2 of course).

If Ann had 2, she would know that Bill has 1 or 3. As he says he don't know which number Ann have, so he can't have 1 in this case, because then he would be sure that Ann had 2, so he can have only 3, but three is not divisor of 20 so we also reject this case.

Now let's suppose that Ann has 3. She know that Bill has 2 or 4. If the Bill had 2, he would know which number Ann's have after she claims that she don't know which number Bill have (he has 2 so he knows that Ann has 1 or 3, but if she had one she would say that she know Bill's number so he would be certainly sure that Ann has 3), but he claims that he don't know the number. Assuming that Ann is perfect logician she noticed the same thing that we did and she claims that she knows Bill's number which is four. Thus Ann must have number three.

1. If Ann's number were 1, she would know from the outset that Bill's number was 2. Since she doesn't know Bill's number at first, her number cannot be 1.
2. If Bill's number were 2, he would know that Ann's number was either 1 or 3. After Ann's first statement, Bill would know by the above reasoning that her number wasn't 1, so it must be 3. Since he didn't arrive at this conclusion, his number can't be 2.
3. Upon hearing Bill's statement, Ann knows by parts 1 and 2 that his number isn't 2. Since this is enough information for Ann to deduce Bill's number, her number must be adjacent to 2, i.e. either 1 or 3. By part 1, her number cannot be 1, so it must be 3 , making Bill's number 4.

Note 1: The revelation that Ann knows Bill's number divides 20 is not needed for us to deduce her number.

Note 2: This pattern of reasoning can be continued any number of steps: for example, if Ann were to announce that she still didn't know Bill's number after his first statement, and then Bill said he knew Ann's number, he would have 4 and she would have 5, etc.

Since Bill's number is a divisor of 20,Ann's number would be odd and not a divisor of 20.So the answer is 3.