VERY HARD RIDDLE!

Wait I get it

the reason 2,3,6 doesn’t work is because if it did you wouldn’t need the red head hint in order to know

Damn that’s a fire riddle

It doesnt

the fact he has one unique youngest child does

one sec lemme make a full explanation

Two Russian mathematicians meet on a plane
“If I remember correctly, you have three sons,” says Ivan.
“What are their ages today?”
The product of their ages is thirty-six,” says Igor, “and the
sum of their ages is exactly today’s date.”
“I’m sorry, Igor,” Ivan says after a minute, “but that doesn’t tell
me the ages of your boys.”
“Oh I forgot to tell you, my youngest son has red hair.”
“Ah, now it’s clear,” Ivan says. “ I now know exactly how old
your three sons are.”
How did Ivan figure out the ages?

1. we know the product of all three boys’ age is 36

this means that they have to be one of the following configurations (from young to old):

1,1,36

1,2,18

1,3,12

1,4,9

1,6,6

2,2,9

2,3,6

2. we know the three combined have to be the date

we can remove 1,1,36 as 38 is not a valid date

1,2,18

1,3,12

1,4,9

1,6,6

2,2,9

2,3,6

3. He can’t figure it out with this information

this means that there is more than one configuration that has the same sum, otherwise he could immediately tell (if the date was the 11th he’d know it’s 2,3,6)

1,2,18: 21

1,3,12: 16

1,4,9: 14

1,6,6: 13

2,2,9: 13

2,3,6: 11

we know that there must be more than one of the date, so it has to be the 13th, that’s the only sum that appears twice

the list becomes:

1,6,6: 13

2,2,9: 13

4. The youngest child is a red head

this means that there is one unique youngest child, and there can only be one of the lowest number in the configuration

as 2,2,9 doesn’t work, the only option left is 1,6,6

The beggar is a women

Strictly with the info the riddle gives you, you cannot without the red head hint