Two intelligent, honest students are sitting together at lunch one day when their math teacher hands them each a card. “your cards each have an integer on them,” the teacher tells them. “the product of the two numbers is either 12, 15 or 18. the first to correctly guess the number on the other’s card wins.” the first student looks at her card and says, “i don’t know what your number is.” the second student looks at her card and says, “i don’t know what your number is, either.” the first student then says, “now i know your number.” what number is on the loser’s card?

The loser's card shows 6.

Explanation: Let's start by naming the first student A and the second student B.
Since the product of A and B are either 12, 15, or 18, let's list every single possibility, the first number being A's number and the second number being B's number.

1     12
1     15
1     18
2     6
2     9
3     4
3     5
3     6
4     3
5     3
6     2
6     3
9     2
12  1
15  1
18  1

Now, the information says that A doesn't know what B has, so we can immediately cross off all of the combinations that have the integer appearing once and once ONLY off, because if it happened once only, A would know of it straight away. Now, our sample space becomes much smaller.

1     12
1     15
1     18
2     6
2     9
3     4
3     5
3     6
6     2
6     3

Using this same logic, we know that we can cross off all of the digits that occur only once in B's column.

2     6
3     6

Now, A definitely knows what number B has because there is only one number left in B. Hence, we can conclude that the loser, B, has the integer 6.