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.

• 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. Oct 6 '14 at 14:33