Penguin and Icebear

Hi folks. This is a quite hard one, took me long time to figure it out. Hope not everybody knows it yet. Have fun.


1. The Penguin says to t he Icebear: I know a sum X which is calculated of a and b so a+b=X , but I don't know what a and b are.

2. The Icebear says: I know a product Y which is calculated by the factors a and b so a*b=Y . I don't know which numbers a and b are, but I knew that you wouldn't know either.

3. The Penguin replies: Well if you say that, than I know the solution which numbers a and b are.

4. The Icebear says: OK, so if you say that, than I also know what a and b are.


Which numbers are a and b?

Yes it's the same a and b in a+b=X and a*b=Y and no, it can not be any number, and yes, Penguins and Icebears may meet somewhere between Arctic and Antartic to talk about stupid riddles. 😉 Sorry if the english isn't the best, but should be no matter to the riddle...

Originally posted by Kandinsky
Hi folks. This is a quite hard one, took me long time to figure it out. Hope not everybody knows it yet. Have fun.


1. The Penguin says to t he Icebear: I know a sum X which is calculated of a and b so a+b=X , but I don't know what a and b are.

2. The Icebear says: I know a product Y which is calculated by the factors a and b so a*b=Y . I don't kn ...[text shortened]... tupid riddles. 😉 Sorry if the english isn't the best, but should be no matter to the riddle...

In English we call them 'polar bears'.

I think the answer is a=b=2 (assuming they have to be counting numbers). Statement 2 indicates that Y is composite (otherwise the polar bear would know the numbers). This is sufficient for the penguin to work out a and b, so it must be that there is exactly one set of numbers which sum to X and have a composite product. The only number for which this is true is X=4, and hence a=b=2.

OK thank you, it's 'Polar Bear' than. 🙂

Reading you're answer made me think some minutes because remembering the brain cracking work I had with this riddle I couldn't imagine it to be solved so easily (and differing from my solution), and after reading my previous post over and over again I now have to seriously apologize for 2 major mistakes in the riddle as I wrote it down. 😳

First of all I (shame on me) completely missed to say the integers are to be >1 (and <100 if that is for any help) so 1<a,b<100. I'm sorry.

The second and even worse thing is that I mixed up the order in which the Polar Bear and the Penguin appear.
The final and correct version of the riddle I sadly messed up 😞 before is:


1. The Polar Bear is telling the Penguine: I know a product X which is calculated by the factors a and b so a*b=X , but I don't know what a and b are.

2. The Penguin says: I know a sum Y which is calculated by a and b so a+b=Y . I don't know which numbers a and b are, but I knew that you wouldn't know either.

3. The Polar Bear replies: Well if you say that, than I know the solution which numbers a and b are.

4. The Penguin says: OK, so if you say that, than I also know what a and b are.


I apologize for any inconvenience. :'(

Originally posted by Kandinsky
OK thank you, it's 'Polar Bear' than. 🙂

Reading you're answer made me think some minutes because remembering the brain cracking work I had with this riddle I couldn't imagine it to be solved so easily (and differing from my solution), and after reading my previous post over and over again I now have to seriously apologize for 2 major mistakes in t ...[text shortened]... if you say that, than I also know what a and b are.


I apologize for any inconvenience. :'(

Ok, this time assuming that the animals know that a and b are at least 2:

Statement 2 indicates that Y is not the sum of two primes, otherwise the Penguin would not know that the Polar Bear did not know what the numbers were. (😕)

This is sufficient for the polar bear, so X must be such that exactly one of its factorisations into two numbers >1 does not have a sum which is the sum of two primes. In addition this must give an a and b whose sum is not shared by any other a and b that satisfy the previous conditions, as otherwise the Penguin wouldn't be able to work out the answer. (😕😕)

I think that the answer is 4 and 13. However, it's quite likely I made a mistake on the way, and even if it's correct, I don't know if the answer is unique.

Good puzzle, Kandinsky!

Originally posted by Acolyte

I think that the answer is 4 and 13. However, it's quite likely I made a mistake on the way, and even if it's correct, I don't know if the answer is unique.!

Very good, that's correct! 🙂
Neither am I completely sure, but as far as i checked it that should be unique at least for all integers <100.