*Thanks to our good friend Giles, and his mathematically-minded family from Canada, for this intriguing puzzle!*

A life insurance agent knocks on a door. A woman answers and he tries to sell her a policy. He asks her how many children she has and their ages. She replies: "I have three children, but I will not even listen to your sales pitch unless you guess their ages."

She gave him a clue: "The product of their ages is equal to 36."

The insurance salesman jots down some figures and replies, "I do not have enough information to answer yet".

The lady replies, "Okay, here is clue number 2: If you go the house next door (points to it), the sum of their ages is the same as the house number".

The salesman goes next door, looks at the number, jots down a few more things and then returns to say, "I still don't have enough information".

The lady then replies, "Okay, your third and final clue: My eldest is learning to play the piano!!"

The salesman replied with the answer, it was correct, he gave his sales pitch, she bought a policy. What were their ages (assume all ages are whole numbers)?

## Comments

## four answers

I have four answers. With distinct sums. All the sums could be house numbers, so I'm not sure how to lower this down to 1 without knowing the neighbor's house number.

## Eight answers, but

This is probably a spoiler ...

There are, I believe, eight possible answers where the product is 36 and the sum might be a house number. For six of those possibilities, the sum is unique; that is, the house number and product would be enough to completely determine the childrens' ages. Because that was NOT enough information, those six can be eliminated, narrowing it down to two. Of those two the finals statement is required to distinguish them.

## The second clue doesn't make

The second clue doesn't make much sense to me. Is there something special about the way that Canadians number their houses?

## hope this helps

The 2nd clue didn't make sense to me at first either, but I think I understand it now. When it says,

`The lady replies, "Okay, here is clue number 2: If you go the house next door (points to it), the sum of their ages is the same as the house number" '

`their' refers to the lady's children. At first I thought `the sum of their ages' meant that the sum of the ages of the people next door, which confused me. But I think the clue is supposed to give you an equation of the form,

a + b + c = # of house next door

where a, b and c are the ages of the lady's children.

## complete information for sales man but incomplete for reader.

May be in Canada a neighbour always numbers there house as 13 :D :D :D :D

## Here is my solution

There are seven distinct ways of factoring 36 into groups of three integers:

1 3 12 (sum is 16)

1 4 9 (sum is 14)

1 1 36 (sum is 38)

1 6 6 (sum is 13)

2 2 9 (sum is 13)

2 3 6 (sum is 11)

3 4 3 (sum is 10)

The 2nd clue is not about the house number but the fact that, even knowing the

house number is insufficient to work out their ages. This means that the sum

cannot be unique.

Of all the possibilities listed above, only (1 6 6) and (2 2 9) provide sums

that are non-unique.

The 3rd clue tells us that there is an eldest. Hence the ages are 2, 2 and 9.

## My solution

There are seven distinct ways of factoring 36 into groups of three integers:

1 3 12 (sum is 16)

1 4 9 (sum is 14)

1 1 36 (sum is 38)

1 6 6 (sum is 13)

2 2 9 (sum is 13)

2 3 6 (sum is 11)

3 4 3 (sum is 10)

The 2nd clue is not about the house number but the fact that, even knowing the

house number is insufficient to work out their ages. This means that the sum

cannot be unique.

Of all the possibilities listed above, only (1 6 6) and (2 2 9) provide sums

that are non-unique.

The 3rd clue tells us that there is an eldest. Hence the ages are 2, 2 and 9.

## Clues 2 and 3 don't make sense

Clue 2

What restriction could there be on the number of the house next door? The only thing that seems to make sense in the context of this problem is that it must be even. But why would that be?

Clue 3

I imagine this is meant to preclude a possibility which has a much older child (36 years?) as the oldest. But I know someone aged 50 who is learning the piano!

## Confusing!

I have many answers, all of which satisfy abc = 36, although what does it have to do with the neighbor? I assumed this clue was to restrain to whole numbers.

Also confused about the piano! Does this just mean one is older (use of the word eldest) than the other 2? If so I still have many solutions!

## An unusual sales pitch

The ages of the children can be 1 1 36, 1 2 18, 1 3 12, 1 4 9, 1 6 6, 2 2 9, 2 3 6, and 3 3 4.

The sum of their ages is respectively 38, 21, 16, 14, 13, 13, 11 and 10. So the number of the house next door is 13, otherwise he would have known their ages. He needs the third clue; there is an eldest one. So the ages of the chidren are 2 2 and 9.

Hub

## Making assumptions you only need the 1st clue

I could not work out how clue 2 helped at all. You don't know the house number the agent calls at or next door...

a+ b + c = n is all very good, but only confuses the issue by adding a 4th unknown. So I ignored it.

There are 3 possible answers, but assuming the 3 children are all different ages & being told they are all whole numbers, it leaves 1 possible answer.

I won't spoil it for anyone else but would like to know if I'm correct & if all 3 clues were used.

## An unusual sales pitch

Clue No. 1: a x b x c = 36

Clue No. 2: The salesman not having enough information after the second clue implies more than one a + b + c sums to the same number - i.e: a1 + b1 + c1 = a2 + b2 + c2

Clue No. 3: "My eldest is learning to play the piano!!" implies there is only one eldest child (not twins).

Therefore 2, 2, 9 is the only possible solution.

a b c Sum Product

1 2 18 21 36

1 3 12 16 36

1 4 9 14 36

1 6 6 13 36

2 2 9 13 36

2 3 6 11 36

3 3 4 10 36

## Answer?

I believe that the children are aged 2, 2 and 9.

The first clue gives a variety of answers (any combination of the divisors of 36 = 2x2x3x3)

The second clue would then give the insurance salesman the opportunity to narrow these down substantially. Although, not knowing the number doesn't give us any insight. BUT the fact that the salesman is still unsure means that at this point, he must still have more than one possibility. After working through a few combinations of the divisors, I realised that if the next door neighbours house number was 13, then this would give the insurance salesman two choices, [2,2,9] and [1,6,6] and thus he would be unable to answer.

The final clue isn't the fact that the child is learning piano, but the fact that she has an eldest child implies that her eldest child is not a twin and thus eliminates the [1,6,6] possibility leaving only [2,2,9] as a possible result. (I'm aware that the woman may have two children aged 6 that are not twins, but ignoring this is the only way I can gleam any insight)

## lucky for some

Assuming twins for simplicity: let the children's ages be x, x and y.

Clue 1: x.x.y=36

Or rewritten: y=36/(x^2) x^2 denotes x squared!

but 36 is a square number and can be written as 6.6 or 6^2.

Now we get y=6^2/x^2 = (6/x)^2

Get the square root of both sides: sqrt(y) = sqrt((6/x)^2) = 6/x or x=6/sqrt(y)

so the age of the eldest child, y>x and must be a square number i.e. 1,4,9,16,25,36,...

let y=1 then x=6 but this does not satisfy the eldest child clue (y>x) so discard.

let y=4 then x=3 accept. So 3+3+4=10 the number on the house next door.

let y=9 then x=2 accept. So 2+2+9=13 and there is no way anyone lives in a number 13 house so discard!

let y=16 then x=6/4 which is not a whole number.

let y=25 then x=6/5 which is not a whole number.

let y=36 then x=1 which is possible also but unlikely lets be honest!

So I reckon I have to lean towards the twins being 3 with the eldest being 4!

## The 2nd Clue does make Sense

The answer is Ages 1, 1 and 36!

Clue 1 makes sure that all the possible ages must have a product of 36. Many threads below have the right idea.

Clue 2 ensures that next door (always either+2 or -2) from your own number. Therefore the sum of the ages must be either +2 or -2 from the product. The only combination possible is 1, 1 & 36 with a product of 36 and a sum of 38.

Clue 3 cannot see the need for it! (But at 36 you could still be learning the piano!)

Youngest Middle Eldest Product Sum Diff

1 1 36 36 38 2

1 2 18 36 21 -15

1 3 12 36 16 -20

1 4 9 36 14 -22

1 6 6 36 13 -23

2 2 9 36 13 -23

2 3 6 36 11 -25

2 3 6 36 11 -25

mbrown1@lipson.plymouth.sch.uk

#MrBrown_Lipson

## I do not agree

if it was 1 1 36 and the next houses number WAS 38

The salesman should have known, but he couldn't!!

## House number not needed for us, just him.

We do not know the house number of the lady or the next door. We only know that after he looked at the next door, he still needed more information. Most of the combinations have unique sums, but two of them 1,6,6 and 2,2,9 add up to 13, so he must have seen a 13, which is why he stated that more information is necessary. When she makes a statement about the 'eldest' that tells him there is one who is older than all of the rest. Other people have given examples arguing for the 1,6,6. There are cases where children could be separated by less than a year and they would have the same whole number but one is older, or if they were born far enough apart that a parent would consider one 'the eldest'. But this is a guessing game and if you have enough information to be 90% sure, you could say the answer with confidence, right? Either he was going to give the right answer and get the sales pitch or walk away without trying. Or there's the rare case which this logic fails, and he guesses wrong.

## The third clue only works if you assume ...

... that the children with equal ages are twins - to eliminate 1,6,6 - but, take today (24 April 2013) and suppose the children's birthdays are:

1 Jan 2012

1 Apr 2007

1 May 2006

Ages 1,6, and 6 - but clearly an eldest to learn the piano.

## you don't have to separate them by 9 months.

You don't even have to go that far. Just have two people born on two consecutive days.

1 January 2012

16 December 2006

17 December 2006

Even though they are both twins, it is clear which one is the older one.

## Find Solutions Automatically

Found a neat script that solves this puzzle automatically, written by a high-school student: A Welcome Back Challenge - Yasyf Mohamedali's Blog.

## 2, 2 and 9

As far as I can tell (doing this pretty quick)... there are 6 ways to arrange 3 numbers for a product of 36. This is as follows:

- 1x6x6

- 1x4x9

- 2x2x9

- 2x3x6

- 3x3x4

- 1x1x36

WE DON'T KNOW THE NUMBER OF HER HOUSE... so these combinations, when added give:

- 1+6+6=13

- 1+4+9=14

- 2+2+9=13

- 2+3+6=11

- 3+3+4=10

- 1+1+36=38

The reason he needs more information is because the house number is 13, and there are TWO combinations that result in 13. Therefore the final clue indicates that there is a definitive eldest child, meaning that the combination includes two 2 year olds and one 9 year old child (learning to play piano).

## wat abt - 1,2,18 and 1,3,12

???

## 2, 3, 6 are age of

2, 3, 6 are age of childern

house no is 11

## 2 2 9 and using ALL three clues

Well as you all may know.

Clue one tells us 8 probabilities like the below

1 1 36(I know this one doesn't make sense, but.. math doesn't include common sense!)

1 2 18

1 3 12

1 4 9

1 6 6

2 2 9

2 3 6

3 3 4

Now we could think "Damn it, I'm stuck!", but the second clue is actually the best clue of the three!

The salesman knew the total of the children's age by seeing the other houses number.

BUT the important thing is he didn't know, after that clue.

Let's say 1 1 36 was the answer, and the number was 38.

Then when the salesman jotted down the 8 figures he could easily find out which combination was right!

So.. if the sum only made right a particular value. He could know!

In other words, the fact that he didn't know means there were more than two combinations who had the same total.

That gives us only 2 left.

2 2 9 and 1 6 6 (they sum up to 13)

But, the third clue tells us that there is an Elder one, which means there is only one which is the biggest,

and finally the problem is solved with an elegant answer

2 2 9

Jay Kim

12 years old(I'm not joking, I really am a 12 years old Korean boy)

Korea Yong-in

jaysarang20@naver.com

## 'eldest'

Surely the word 'eldest' simply refers to the oldest of the three, which could perfectly well mean one which was 6 years 11 months old, compared with one which was 6 years and 1 month old.

## 'eldest'

The theory whereby age could be further delineated in months is not be possible since a) the woman revealed only that the children's ages would be whole numbers and b) that the mystery she posed would presumably be solvable with only three clues (the lack of any information indicating differences in age by months precluded that possibility.)

## I disagree

Clue two is arbitrary to the puzzle as the number is not given. That the salesman asked for 3 clues does not cancel the belief that the woman thought one clue sufficient to begin with. Assuming twins in the population are a rarity and most families have a small spread of years between siblings, I would guess the answer at 1 4 9. This would be the greatest spread that allows clue 3 to make sense when the liklihood of piano lessons being more common for nine year olds than 6 year olds.

## 4 Solutions

These are excluded by clue 2 narrows the answer down to 2 possibilities because they are the only duplicates.

1 6 6

2 2 9

the only one with an eldest is 2 2 9