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    • Hand using knocker on white door

      An unusual sales pitch

      11 April, 2013
      Knock knock

      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)?

      Solution link
      An unusual sales pitch: solution
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      Anonymous

      11 April 2013

      Permalink
      Comment

      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.

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      Anonymous

      13 April 2013

      In reply to four answers by Anonymous

      Permalink
      Comment

      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.

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      Anonymous

      12 April 2013

      Permalink
      Comment

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

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      Andrew Irving

      12 April 2013

      In reply to The second clue doesn't make by Anonymous

      Permalink
      Comment

      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.

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      AMIT KUMAR SINGH

      2 December 2019

      In reply to The second clue doesn't make by Anonymous

      Permalink
      Comment

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

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      Anonymous

      12 April 2013

      Permalink
      Comment

      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.

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      hunterdust

      16 March 2022

      In reply to Here is my solution by Anonymous

      Permalink
      Comment

      Since when can't mothers refer to the twin that came first as the "oldest"? In fact, what if they are Irish twins and her children are 1 years old, 6 years 2 days old and 6 years 11 months and 4 days old? While the latter is unlikely, it is very common for a parent to refer to their children as older and younger even when they are twins.

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      Anonymous

      12 April 2013

      Permalink
      Comment

      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.

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      Anonymous

      12 April 2013

      Permalink
      Comment

      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!

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      Anonymous

      12 April 2013

      Permalink
      Comment

      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!

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      Anonymous

      12 April 2013

      Permalink
      Comment

      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

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      Anonymous

      13 April 2013

      Permalink
      Comment

      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.

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      Anonymous

      13 April 2013

      Permalink
      Comment

      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

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      MyKo101

      14 April 2013

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      Comment

      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)

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      Anonymous

      22 April 2013

      Permalink
      Comment

      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!

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      Anonymous

      24 April 2013

      Permalink
      Comment

      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

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      Anonymous

      3 July 2013

      In reply to The 2nd Clue does make Sense by Anonymous

      Permalink
      Comment

      if it was 1 1 36 and the next houses number WAS 38
      The salesman should have known, but he couldn't!!

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      Anonymous

      4 July 2014

      In reply to The 2nd Clue does make Sense by Anonymous

      Permalink
      Comment

      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.

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      Anonymous

      24 April 2013

      Permalink
      Comment

      ... 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.

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      Anonymous

      27 April 2013

      In reply to The third clue only works if you assume ... by Anonymous

      Permalink
      Comment

      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.

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      Anonymous

      30 April 2013

      Permalink
      Comment

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

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      Anonymous

      3 May 2013

      Permalink
      Comment

      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).

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      Anonymous

      18 March 2014

      In reply to 2, 2 and 9 by Anonymous

      Permalink
      Comment

      ???

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      Anonymous

      11 May 2013

      Permalink
      Comment

      2, 3, 6 are age of childern
      house no is 11

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      Anonymous

      3 July 2013

      Permalink
      Comment

      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

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      Anonymous

      15 July 2013

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      Comment

      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.

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      Anonymous

      8 December 2013

      In reply to 'eldest' by Anonymous

      Permalink
      Comment

      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.)

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      Anonymous

      4 May 2015

      In reply to 'eldest' by Anonymous

      Permalink
      Comment

      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.

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      Anonymous

      14 January 2016

      Permalink
      Comment

      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

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      Rebecca Walker

      31 August 2021

      Permalink
      Comment

      The factors of 36 are listed and if you add the numbers together, the reason why he said he need more info was because there was more than on answer
      1:1:36=38
      18:2:1=21
      9:2:2=13
      4:3:3=10
      12:3:1=16
      6:6:1=13
      9:4:1=14
      6:3:2=11
      She said her oldest played piano
      6:6:1=13
      9:2:2=13
      there cant be two oldest so the answer is 9:2:2=13

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