# "Guess the number" game

two logicians decided to play a lovely well-known game , number-guessing from the least significant informations , the juror proposed to create the challenge :

the juror told the logicians two numbers secretly , unbeknownst to eachother , all they know , is that the product of eachother's number is a number divisible by 2 , and between 2 and 10.

juror said that , no action is permitted , no gestures , no word , except when they know numbers , they are permitted to announce it!

• A suspensious silence was been hovering arround the place , no word! everyone was staring at the other wierdly .

After this few minutes silence both jumped suddenly and declared : We know these numbers!

Question: What are these numbers ?

• Can both logicians be given the same number and are the numbers integers? Apr 19, 2015 at 16:23
• the numbers are integers 1 isnt included , 10 is included . DOT. Apr 19, 2015 at 16:32
• I am just double checking... the product is between 2 and 10? their numbers are integers? Apr 19, 2015 at 16:33
• yes........this is my easiest logic puzzle Apr 19, 2015 at 16:33

Assuming that both of their numbers (a,b) are integers. Neither can be 0 and if either are negative, they know the other has a negative number so it is trivial to reduce it to the same problem.

Both players know that all possible pairs contain at least a 1 or a 2. If either player has a 10, 8, 6, 5, or 3, then they immediately know to combine it with the 1 or 2 to find the 1 possible product within that range. As no one immediately finds the solution, neither player has these numbers.

The only numbers a player may have, therefore, is 1, 2, or 4. It has been revealed since I starting writing the answer that neither player can have a 1. The possible pairs are, therefore: (2,2) and (2,4).

If either player had a 4, they would realise that the other has a 2 and say the result. This means that they both know it is (2,2).

• this answer was first . Apr 19, 2015 at 16:35
• @Agawa001 Just remember that "first" doesn't always correspond to "best". It's a good habit to leave a question open for a while, even if a correct answer has been posted, unless you're sure that no better answer could be posted. Btw, nice answer kaine! Apr 19, 2015 at 16:39
• do you think i would choose it if it wasnt corrent ? Apr 19, 2015 at 17:02

Lets assume that we are one of the logicians and get a number, how much can you reason out from that number?

Numbers 6+ are impossible because the minimum is two and the product can't be higher than 10. Number 5: The other must have two, otherwise the product would not be divisible by 2. Number 4: The other must have two. Number 3: The other must have two, otherwise the end product is not divisible by two. Number 2: The other must have 2 or 4.

The only combination that carries uncertainty for both parties is if both have the number 2, hence this is the number that both logicians had.

So the product of the two numbers lies between 2 and 10 and has to be divisible by two 2. The following are the possible pairs:

1. 10:(2,5)
2. 8:(2,4)
3. 6:(2,3)
4. 4:(2,2)
5. 2:(2,1)

In cases 1,2,3 and 5, the logician with the number other than 2 can blurt out his number and correctly guess that the other logician has the number two.

But in case 4, since both of them fall silent as either one has the value two. Therefore, both the logicians know they have the same number, i.e., 2.