There are exactly eight possibilities for the product of three ages to be 36.

Son 1 Son 2 Son 3 Product Sum

1 1 36 36 38

1 2 18 36 21

1 3 12 36 16

1 4 9 36 14

1 6 6 36 13

2 2 9 36 13

2 3 6 36 11

3 3 4 36 10

Since Ivan could not solve the problem when he knew the sum of the three numbers – the date of the encounter – that meant the sum had to be 13, for which there are 2 possibilities. The added information about the youngest son means that one of the possibilities can be eliminated – the 9 year old and the two 2 year olds – since in this case there is no youngest son. Igor’s sons are 1 years old with twins of 6 years old.

In case you didn’t get that. xP :

So, the first thing you’re probably wondering is: how can we solve this puzzle if we don’t know the date when they meet?

Let’s look at it this way: there are 8 possibilities for the product of the three ages to equal 36. If you look at the sum, you can see that 13 is the only sum to be repeated twice. If the sum was, say, 14, Ivan wouldn’t have any trouble solving the problem. But because he had trouble, it showed us that the date of the encounter was the 13^{th}, because he couldn’t solve the problem – there were two possible answers.

Now that we nailed the date, we can concentrate on the ages of the sons. We have two possibilities as to what the ages could be:

First possibility: Twins of 2 years old and a boy that’s 9 years old.

Second possibility: A one year boy and twins of 6 years old.

At this point, Ivan told Igor he couldn’t solve the problem. Igor realized this and told him that his youngest son has red hair.

So you’re probably thinking – that’s irrelevant. What does that have to do with anything?

Take a look at the ages again – if there were twins of 2 years old and a 9 year old boy, doesn’t that mean there is no youngest son? (And no, the whole “one twin is born before the other” doesn’t count).

That leaves you with the last choice.

Igor has a one year old boy and 6 year old twins.