Find the ages of mathematician's children

Question

Mathematician: "It was the August of 1997 when they barged into my house..." - Here is his story:

Students surprised their mathematics teacher with a cake on his birthday. The teacher was delighted when he noticed that every single year of his life is accounted for with a candle. He repeatedly tried to blow out the candles, but he never got all of them. Eventually, he gave up and said: "The youngest of my children, my youngest son, had more luck on his birthday cakes!". One of the students followed up: "How many children do you have and how old are they?"
The teacher decided it is time for a riddle: "The product of their ages is equal to the total number of candles on the cake, while the sum of their ages is equal to the number of candles still left burning on the cake. What can you tell me about my children?"
The students took some time and eventually replied: "We don't know their ages, but we do know that you don't have twins among your children."
The teacher added: "Oh, I forgot to mention that the age of my youngest son is not a cube number."
They replied: "We now know their exact ages!"

Question. What are the ages of the children?

Bass · Answer

After trying some

I managed to find at least one possible solution:

This would make for the following deductions:

This gives a conceivable age for the teacher, somewhat high but by no means impossible age for getting the last child (the teacher was referred to as a "he"), his age explains why the candles were too many to blow out, and the youngest child has also had more than one birthday, as implied.

Alexey Burdin · Answer

I'll "strip" the problem to pure mathematical context:

Given product and sum of some positive integer numbers it's impossible to tell what the numbers are, but it's possible to tell that none of them are equal -- unless it's given that the least number is not a perfect cube -- then it's possible to tell what the numbers are.
What are the numbers?

Also we assume that the product of the numbers is not more than $200$ (for sure, it's the professor's age).
So we consider two totally different possibilities (not much of a spoiler):

as mentioned in the Bass's answer. Only the difference is that I have python script to show it rigorously than no other possibilities left.

Find the ages of mathematician's children

2 Answers

Add your own answers!

Ask a Question