# Unknown Consecutive Integers Puzzle

Two people, Anne and Bob, each hold an index card. A positive integer is printed on each index card, and these integers are consecutive (i.e. Anne's card may display 8 and Bob's card 7). Anne and Bob are aware of this information, and in addition they each know the number printed on their own card, but not the number printed on the other's card.

The following dialogue transpires between Anne and Bob:

1) Anne tells Bob, "I do not know the number on your card."

2) Bob then tells Anne, "I do not know the number on your card."

3) Anne then says to Bob, "Now I know your number. It is divisible by four."

How did Anne determine this, and what numbers were printed on each card? Note that this is a logic puzzle, and can be solved by logical deduction. There is no wordplay involved, extenuating circumstances not stated in the problem, nor coded messages of any kind passed between Anne and Bob.

From 1 we know that Anne does not have 1. From 2 we know that Bob does not have 1 or 2. From 3, we know that Anne has 3 and knows Bob has 4.

I might have to disagree with the above answer being the only solution. What would prevent the numbers from being 11 and 12? You cannot prove that 4 is the number based on the information as 11 and 12, or even 13 and 12 would answer the questions the same way. In fact i would say any numbers in the form of X modulo 4 = 0, and x-1 = y would solve this system. This problems does not limit the integers used outside of being positive. 100 and 99 would solve this with the same dialogue expressed above, there fore there is not a single solution for the question as it is stated.

• Please read the answer, it is all stated there, completely logically correct. If the numbers were 11 or 12, Anne would not (yet) know the number of Bob, and would have to go four through more exchanges of "I don't know your number". But since Anne has 3, she knows Bob has to have 2 or 4. If Anne had 2, Bob would not have answered "I don't know your number", because Bob would have known that Anne can't have 1, so he'd answer "Your number is 3". – Alexander Oct 6 '14 at 14:33
• If Alice has $11$, she knows Bob $10$ or $12$ For either of those, Bob doesn't know Alice's number, but when Bob says that Alice still doesn't know Bob's number. She therefore cannot say it is a multiple of $4$. It could be, but she doesn't know. – Ross Millikan Oct 6 '14 at 14:33
• However my comment is that 3 and 4 are not the only solution. I would like to see a reason that 4 is the only solution. IF the numbers were 11 and 12 the conversation would not change at all. Each players know that other must be within 1 digit of their number. will get even worse when I point out 0 is an integer. – Anthony Leese Oct 6 '14 at 19:56
• @AnthonyLeese Could you explain how the numbers being 11 and 12 would lead to the same solution? I don't think its possible. Also, 0 is not a positive integer, as the question states – Ben Aaronson Oct 6 '14 at 20:44
• If the numbers were 11 and 12, Anne cannot know that Bob's number is a multiple of 4. For all she knows, his number is 10. After a bunch more exchanges, she can rule that out, but not this quickly, so 11 and 12 is not a solution. The point of these puzzles is what does each person know at each point of the conversation, which you are missing. – Ross Millikan Oct 6 '14 at 21:16