# What are the three numbers in this puzzle?

There are three people in a room without any equipment.

• Each one has a number on his forehead
• Every person knows the numbers the other two have,
• Every person knows that his number is either the sum of the two, or the difference (but higher than 0)
• All numbers are positive and non-decimal
• No two numbers have the same value

The problem is that nobody of them knows what is the number they have on their forehead (but sees the other two numbers).

The first person is asked, what number he has on his head. He replies, "I don't know."

The 2nd person and after that the 3rd person are asked the same question, and they reply the same.

Again, they ask the 1st person, what number does he have, and again, he doesn't know.

But then, the 2nd person is asked again, and he replies, "50"

How did he find out and what are the other two numbers?

• The numbers in the puzzle are: 1, 2, 3, and 50. – Ian MacDonald Aug 23 '16 at 18:24
• This is tagged as a riddle so it should be sandboxed. However I'm not sure its a riddle – Beastly Gerbil Aug 23 '16 at 18:25
• @BeastlyGerbil he was interpretting the title "What are the three numbers in this puzzle?" Literally by reading the numbers in the puzzle. – gtwebb Aug 23 '16 at 18:27
• @scott Usually you expect the people to be good at mathematics and correct in these puzzles... – yo' Aug 24 '16 at 1:44
• And, at the risk of splitting hairs — if each player sees the other two numbers (which are integers), and knows that his number is either the sum or the difference of the numbers he sees, then he will immediately realize that his number must be an integer.  That doesn’t need to be stipulated as given knowledge.  (But the fact that each participant knows that his number is positive is important.) – Peregrine Rook Apr 23 '17 at 21:57

The numbers on each of their foreheads are 20, 30, and 50.

My reasoning:

So, the number on the first person and third person's forehead should be 20 and 30. Probably what the second person is thinking is that his number has to be either 50 or 10 because it could be the sum or difference. Here's the catch: 30 minus 20 is 10, so if the second person's number was 10, then either the first person or the third person would know because the difference between 20 and 10 is 10 again. It can't be, meaning the person with the number 30 would realize 10+20 is equal to their number. The only other option he has is 50 which is his number.

How I got the numbers:

This part took a little bit of guess and check with reasoning. I predicted that the sum would be 50 only if the differences of the other two numbers equaled 1 of the numbers they had. I don't know if there are any other solutions, but this is the only one I could think of.

• Not only is your answer not right; it’s wrong. Even before the first and second person spoke, the third person would see that they have $25$ and $50$ on their foreheads, and so he would think, “I must have either $75~(25+50)$ or $25~(50-25)$ on my forehead.  But I can’t have $25$ on my head, because the first person has $25$ and we all have different numbers.  So I must have $75$.”  So he would know the number on his forehead when he was asked. – Peregrine Rook Aug 24 '16 at 3:05