538’s Riddler Express this week


The Riddler Express presented there (and quoted below) is a reasonably common puzzle I’ve seen before:

From Trevor Ferril, some cafeteria multiplication:

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 solution is pretty straightforward; as far as I know the best way is to just solve this by trying out the possible combinations.

The first student can’t have any card that would mean the second student must have a specific card.  Therefore, the first student can’t have 18, 15, 12, 9, 5, or 4–and the second student now knows this also.

The second student also can’t have any of those cards, for the same reason, but the first student only learns this as the second student makes her statement.  Of course, this also means neither student can have 1. The possible cards remaining are only 2, 3, and 6, so one student has the 6 card and the other has either the 2 or 3.

In this problem, the first student has either the 2 or 3 and the second student (the loser) has the 6.

It can’t be the other way because if the second student had either 2 or 3 she would have known the first student’s number after the first student’s statement.


This is of course similar to (but simpler than) the more-famous “Blue Eyes” logic problem. You can see Randall Munroe’s solution here.