A teacher gives three clever students in her class a challenge: she writes down 3 *different *numbers on 3 index cards, and has each student hold up one of the cards to their forehead such that they can’t see their own card but everyone else can.

She tells them each card has a different number, and that two of the numbers add up to the third number, and asks them to figure out their number without sharing the numbers they see.

Ava sees Sid has 40 on his forehead and Vlad has 60 on his forehead.

Ava says “I don’t know my number.”

Vlad says “I don’t know my number.”

Before Sid can say anything, Ava realizes she is now able to figure our her number! What is Ava’s number?

#### Solution

100

Ava sees 40 and 60, so she knows her number must be 20 or 100, but she doesn’t know which.

Vlad sees Ava’s number and 40. If Ava’s number were 20, Vlad would see 20 and 40, which would mean his number was 20 or 60 – except he knows the numbers are different, so his number must be 60. Since Vlad could not figure out his number, Ava’s number couldn’t have been 20, so Ava’s number must be 100.

Ava knows Vlad has 60 and Sid has 40, so she can figure out that she has either 20 or 100. So unable to say her number correctly. Now Vlad say she does not know her number.

There are 4 possibilities 20, 20, 40; 20, 60, 40; 100, 60, 40; 100, 140, 40 (as sum of two add up third). First is not possible because two of them can not have same number. So if Ava has 20 then Vlad will say that I have 60. But she does not know that means Ava has 100.