Hi, this is Presh Talwalkar.
A census taker visits the home of a mathematician
and asks for the ages of his children.
The mathematician offers up a riddle:
"If you add up the ages of my 3 children,
you get my house number,
and if you multiply the ages, you get the result 72."
The census taker thinks about it
and replies he still doesn't know the ages of the children.
The mathematician says, "Of course not."
"I forgot to mention my oldest child loves chocolate chip cookies."
Suddenly the census taker says he gets it,
and he writes down the 3 ages of the children.
What are the ages of the children?
Give this puzzle a try, and when you're ready,
keep watching the video for the solution.
Like any good puzzle, you have to deduce
some of the details from the story.
First you have to deduce that the census taker
knows the house number of the mathematician.
This is a census taker who should know the address
and he's also visiting the mathematician at his home.
The other detail, which is a little more controversial,
but is generally common sense,
is that you have to conclude the mathematician has an oldest child with a distinct age,
so not twins,
and not two children that are born almost a year apart.
This comes from the detail where the mathematician says his oldest child
loves chocolate chip cookies.
This implies the mathematician does not have two oldest children who are twins,
because you wouldn't generally refer to twins as
being the oldest child,
even though one of the children
is probably older by a few minutes.
And you could also have a child who's born, say,
9 to 12 months apart,
and they could possibly be the same age,
even though they're not exactly born in the same...
you know, they're not twins.
But commonly, this doesn't happen.
So in this story we're going to say the mathematician has an oldest child
with a distinct age,
and that's gonna give the census taker
the detail he needs.
So how can the census taker figure this out?
The mathematician says the three children
have a product of their ages of 72.
We can factor 72 into the primes
2 times 2 times 2 times 3 times 3.
There are only a few prime factors
and so when we consider the way that you can get 3 different ages to multiply to be 72,
there are actually only 12 different possibilities.
These are written in increasing order.
You could have 1, 1, 72, 1, 2, 36,
1, 3, 24,
1, 4, 18,
1, 6, 12,
1, 8, 9,
2, 2, 18,
2, 3, 12, 2, 4, 9,
2, 6, 6, 3, 3, 8,
or 3, 4, 6.
The census taker could work out the sum of the three ages
to get these results.
So now,
here's what goes through the census taker's mind.
The mathematician says if you add up the three ages,
you get the house number.
So if the house number was anything besides 14,
the census taker would uniquely be able to identify
the three ages of the children.
If the house number was,
uh, 22 for example,
the census taker would know
the children have to be age 2, 2, and 18.
Or if the house number was 15,
the census taker would know
the ages would have to be 2, 4, 9,
because there's no other way you could get to that sum.
So when the census taker says
he's tried to work it out, but he still can't figure it out,
knowing the house number,
that means the house number has to be 14.
So the census taker is trying to figure out
whether the 3 kids are age 2, 6, and 6,
or 3, 3, and 8.
And this is where the detail that the oldest child
loves chocolate chip cookies comes into play.
The "oldest child" implies there's one child who has a distinct age
that's the oldest child.
So we can rule out the possibility of 2, 6, and 6,
and that means the answer is 3, 3, and 8.
Did you figure it out?
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I make videos on math and game theory.
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