# 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. Aug 23, 2016 at 18:24
• @BeastlyGerbil he was interpretting the title "What are the three numbers in this puzzle?" Literally by reading the numbers in the puzzle. Aug 23, 2016 at 18:27
• @scott Usually you expect the people to be good at mathematics and correct in these puzzles...
– yo'
Aug 24, 2016 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.) Apr 23, 2017 at 21:57
• This should be the chosen answer : puzzling.stackexchange.com/a/93496/57212 Mar 10 at 14:03

Since it's been years since the question was asked, I'm not using spoilers.

There are 4 possible answers, and we cannot deduce which is correct.

Let the numbers be $$a, b,$$ and $$c$$. Each player knows that their number is either the sum or the positive difference between the other two numbers, a total of two options. To conclude what the other two numbers are, they must eliminate one of the options.

Let $$a = ix$$, $$b = jx$$, $$c = kx$$, where $$x$$ is the greatest common factor of $$a, b,$$ and $$c$$. It must be that $$i, j,$$ and $$k$$ are all relatively prime to one another, since any number that is a factor of two numbers is also a factor of their sum and their difference.

The players all know what $$x$$ is; it's the GCF of the two numbers they see. So they're all trying to deduce the triad $$(i,j,k)$$.

Which patterns (triads) can we rule out?

Rule 4 (all numbers are positive integers) rules out the patterns

$$\begin{array}\ (1,1,0) &(1,0,1) &(0,1,1)\ \end{array}$$

and all patterns containing negative numbers.

Rule 5 (no two numbers are the same) rules out the patterns

$$\begin{array}\ (1,1,2) &(1,2,1) &(2,1,1)\ \end{array}$$

What do we deduce from each person's answer?

Person A, first round

If person A saw the partial pattern $$(-,1,2)$$, they would know the only possible patterns are $$(1,1,2)$$ and $$(3,1,2)$$. They know the first one is ruled out, so if they saw that, they would know the pattern had to be $$(3,1,2)$$, and they would know their number.

To determine which patterns a "Don't know" answer eliminates, take each already-eliminated pattern and find the complement pattern for the person who answered.

Person A said "Don't know", so the new patterns that are eliminated by that answer are:

$$\begin{array}\ (3,1,2) &(3,2,1) \end{array}$$

Person B, first round

Likewise, person B's "I don't know" answer will eliminate

$$\begin{array}\ (1,3,2) & (2,3,1)\\ (3,5,2) & (3,4,1) \end{array}$$

The first two were eliminated based on the initial ruled out patterns; the last two were eliminated based on the patterns ruled out by A's answer.

Person C, first round

$$\begin{array}\ (1,2,3) & (2,1,3) \\ (3,1,4) & (3,2,5) \\ (1,3,4) & (2,3,5) & (3,5,8) & (3,4,7) \end{array}$$

Person A, second round

Person A says "Don't know" again, eliminating more patterns based on the patterns eliminated in the first round by B and C. The newly eliminated patterns are:

$$\begin{array}\ (5,3,2) & (4,3,1) & (7,5,2) & (5,4,1) \\ (5,2,3) & (4,1,3) & (5,1,4) & (7,2,5) \\ (7,3,4) & (8,3,5) & (13,5,8) & (11,4,7) \end{array}$$

Person B, second round

This time, B said "Yes, I do know the answer." So we proceed as above, identifying complementary patterns to the recently eliminated ones, except one of these must be the correct pattern.

The complementary patterns for B at this round are:

$$\begin{array}\ (1,4,3) & (2,5,3) & (3,7,4) & (3,8,5) \\ (1,5,4) & (2,7,5) & (3,11,8) & (3,10,7) \\ (5,7,2) & (4,5,1) & (7,9,2) & (5,6,1) \\ (5,8,3) & (4,7,3) & (5,9,4) & (7,12,5) \\ (7,11,4) & (8,13,5) & (13,21,8) & (11,18,7) \end{array}$$

Since B declared their answer is $$50,$$ the pattern must have a factor of 50 as the second number. The patterns satisfying that are:

$$\begin{array}\ (2,5,3) & (1,5,4) & (3,10,7) & (4,5,1) \end{array}$$

Thus, the three numbers must be one of:

$$\begin{array}\ (20,50,30) & (10,50,40) & (15,50,35) & (40,50,10) \end{array}$$

Side note: of these four possible solutions, only for $$(40,50,10)$$ does B need to rely on A's second-round answer. For the other three, B concluded their own number after C's first-round answer.

• This should be the chosen answer. Mar 10 at 14:02
• To be honest, I completely missed this answer! Hundred apologies to you. In retrospect, the question suffers from some imprecision (for example, are the other players "perfect" too?), probably due to my then bad English. After some review, I'm convinced your answer is actually correct. Thanks many times and also thanks to @Hermant. Mar 13 at 19:27

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.

• There are actually 4 possible answers, of which this is one. Feb 7, 2020 at 20:39

• 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. Aug 24, 2016 at 3:05