Ages of Three Children puzzle: Difference between revisions
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{{Short description|Logic puzzle}} |
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The '''Ages of Three Children puzzle''' is a [[logic puzzle]] which on first inspection seems to have insufficient information to solve, but which rewards those who persist and examine the puzzle critically. |
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{{Orphan|date=November 2021}} |
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The '''Ages of Three Children puzzle''' (sometimes referred to as the '''Census-Taker Problem'''<ref name="Meyers See 1990 p. 86">{{cite journal | last1=Meyers | first1=Leroy F. | last2=See | first2=Richard | title=The Census-Taker Problem | journal=Mathematics Magazine | volume=63 | issue=2 | date=1990-04-01 | doi=10.2307/2691063 | page=86| jstor=2691063 }}</ref>) is a [[logic puzzle|logical puzzle]] in [[number theory]] which on first inspection seems to have insufficient information to solve. However, with closer examination and persistence by the solver, the question reveals its hidden mathematical clues, especially when the solver lists down all the possible outcomes. |
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This puzzle illustrates the importance of close inspection while approaching a problem in number theory, while enforcing mathematical thinking and rigor, which is a foundational skill in [[Mathematical analysis]] |
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==The puzzle== |
==The puzzle== |
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A census taker approaches a woman leaning on her gate and asks about her children. She says, "I have three children and the product of their ages is seventy–two. The sum of their ages is the number on this gate." The census taker does some calculation and claims not to have enough information. The woman enters her house, but before slamming the door tells the census taker, "I have to see to my eldest child who is in bed with measles." The census taker departs, satisfied.<ref>{{cite web|url=http://mathforum.org/library/drmath/view/58492.html |title=Ask Dr. Math |publisher=Math Forum |date=2008-11-22 |accessdate=2010-09-12| archiveurl= https://web.archive.org/web/20100830232400/http://mathforum.org/library/drmath/view/58492.html| archivedate= 30 August 2010 | url-status= live}}</ref> |
A census taker approaches a woman leaning on her gate, number 14, and asks about her children. She says, "I have three children and the product of their ages is seventy–two. The sum of their ages is the number on this gate." The census taker does some calculation and claims not to have enough information. The woman enters her house, but before slamming the door tells the census taker, "I have to see to my eldest child who is in bed with measles." The census taker departs, satisfied.<ref>{{cite web|url=http://mathforum.org/library/drmath/view/58492.html |title=Ask Dr. Math |publisher=Math Forum |date=2008-11-22 |accessdate=2010-09-12| archiveurl= https://web.archive.org/web/20100830232400/http://mathforum.org/library/drmath/view/58492.html| archivedate= 30 August 2010 | url-status= live}}</ref> |
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The problem can be presented in different ways, giving the same basic information: the product, that the sum is known, and that there is an oldest child (e.g. their ages adding up to today's date, |
The problem can be presented in different ways, giving the same basic information: the product, that the sum is known, and that there is an oldest child (e.g. their ages adding up to today's date,<ref name=dum>{{Citation |
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|page=209 |
|page=209 |
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|title=Math Word Problems for Dummies |
|title=Math Word Problems for Dummies |
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|isbn=978-0-470-14660-6 |
|isbn=978-0-470-14660-6 |
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| accessdate = 2010-09-12 |
| accessdate = 2010-09-12 |
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| url = https://books.google.com/books?id=QLLxVESo2qMC& |
| url = https://books.google.com/books?id=QLLxVESo2qMC&q=%22Ages+of+Three+Children%22&pg=PA209 |
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}}</ref> or the eldest being good at chess<ref>{{citation |
}}</ref> or the eldest being good at chess<ref>{{citation |
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|title=A problem solving approach to mathematics for elementary school teachers |
|title=A problem solving approach to mathematics for elementary school teachers |
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== Solutions == |
== Solutions == |
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=== |
=== For 72 === |
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The [[Integer factorization|prime factors]] of 72 are 2, 2, 2, 3 |
The [[Integer factorization|prime factors]] of 72 are 2, 2, 2, 3 and 3; in other words, {{math|2 × 2 × 2 × 3 × 3 {{=}} 72}} |
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This gives the following triplets of possible solutions |
This gives the following triplets of possible solutions: |
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{| class="wikitable" |
{| class="wikitable" style="text-align:center;" |
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!Age one !! Age two !! Age three !! Total (Sum) |
!Age one !! Age two !! Age three !! Total (Sum) |
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|1||2||36||39 |
|1||2||36||39 |
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|- |
|-hi |
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|1||3||24||28 |
|1||3||24||28 |
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|2||4|| 9||15 |
|2||4|| 9||15 |
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|- style="color:#000;background:#ffc;" |
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|- |
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|2||6|| 6||'''14''' |
|2||6|| 6||'''14''' |
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|- style="color:#000;background:#ffc;" |
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|- |
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|3||3|| 8||'''14''' |
|3||3|| 8||'''14''' |
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Because the census taker knew the total (from the number on the gate) but said that he had insufficient information to give a definitive answer |
Because the census taker knew the total (from the number on the gate) but said that he had insufficient information to give a definitive answer, there must be more than one solution with the same total. |
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Only two sets of possible ages add up to the same totals: |
Only two sets of possible ages add up to the same totals: |
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A. {{math|big=1|2 + 6 + 6 {{=}} 14}} |
:A. {{math|big=1|2 + 6 + 6 {{=}} 14}} |
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B. {{math|big=1|3 + 3 + 8 {{=}} 14}} |
:B. {{math|big=1|3 + 3 + 8 {{=}} 14}} |
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In case 'A', there is no 'eldest child' |
In case 'A', there is no 'eldest child': two children are aged six (although one could be a few minutes or around 9 to 12 months older and they still both be 6). Therefore, when told that one child is the eldest, the census-taker concludes that the correct solution is 'B'.<ref name=dum/> |
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=== |
=== For 36 === |
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The prime factors of 36 are 2, 2, 3 |
The prime factors of 36 are 2, 2, 3 and 3. |
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This gives the following triplets of possible solutions |
This gives the following triplets of possible solutions: |
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{| class="wikitable" |
{| class="wikitable" style="text-align:center;" |
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|- |
|- |
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!Age one !! Age two !! Age three !! Total (Sum) |
!Age one !! Age two !! Age three !! Total (Sum) |
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|1||4|| 9||14 |
|1||4|| 9||14 |
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|- style="color:#000;background:#ffc;" |
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|- |
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|1||6|| 6||'''13''' |
|1||6|| 6||'''13''' |
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|- style="color:#000;background:#ffc;" |
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|- |
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|2||2|| 9||'''13''' |
|2||2|| 9||'''13''' |
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Using the same argument as before it becomes clear that the number on the gate is 13, and the ages 9, 2 and 2.<ref name=fun/> |
Using the same argument as before it becomes clear that the number on the gate is 13, and the ages 9, 2 and 2.<ref name=fun/> |
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A criticism of this problem is that a household may have two children of the same age in terms of natural numbers, but different birthdays, such as children in a mixed-parent household. |
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==References== |
==References== |
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{{reflist}} |
{{reflist}} |
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==Further reading== |
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* {{cite arXiv |eprint=1204.2071 |last1=Garces |first1=I. J. L. |last2=Loyola |first2=M. L. |title=Revisiting a Number-Theoretic Puzzle: The Census-Taker Problem |date=2012 |class=math.HO }} |
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* {{cite web | title=ISBN 9780805308518 | website=ISBN Search | url=https://isbnsearch.org/isbn/0805308512 | access-date=Apr 14, 2024}} |
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[[Category:Logic puzzles]] |
[[Category:Logic puzzles]] |
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Latest revision as of 16:30, 7 October 2024
The Ages of Three Children puzzle (sometimes referred to as the Census-Taker Problem[1]) is a logical puzzle in number theory which on first inspection seems to have insufficient information to solve. However, with closer examination and persistence by the solver, the question reveals its hidden mathematical clues, especially when the solver lists down all the possible outcomes.
This puzzle illustrates the importance of close inspection while approaching a problem in number theory, while enforcing mathematical thinking and rigor, which is a foundational skill in Mathematical analysis
The puzzle
[edit]A census taker approaches a woman leaning on her gate, number 14, and asks about her children. She says, "I have three children and the product of their ages is seventy–two. The sum of their ages is the number on this gate." The census taker does some calculation and claims not to have enough information. The woman enters her house, but before slamming the door tells the census taker, "I have to see to my eldest child who is in bed with measles." The census taker departs, satisfied.[2]
The problem can be presented in different ways, giving the same basic information: the product, that the sum is known, and that there is an oldest child (e.g. their ages adding up to today's date,[3] or the eldest being good at chess[4]).
Another version of the puzzle gives the age product as thirty–six, which leads to a different set of ages for the children.[5]
Solutions
[edit]For 72
[edit]The prime factors of 72 are 2, 2, 2, 3 and 3; in other words, 2 × 2 × 2 × 3 × 3 = 72
This gives the following triplets of possible solutions:
| Age one | Age two | Age three | Total (Sum) |
|---|---|---|---|
| 1 | 1 | 72 | 74 |
| 1 | 2 | 36 | 39 |
| 1 | 3 | 24 | 28 |
| 1 | 4 | 18 | 23 |
| 1 | 6 | 12 | 19 |
| 1 | 8 | 9 | 18 |
| 2 | 2 | 18 | 22 |
| 2 | 3 | 12 | 17 |
| 2 | 4 | 9 | 15 |
| 2 | 6 | 6 | 14 |
| 3 | 3 | 8 | 14 |
| 3 | 4 | 6 | 13 |
Because the census taker knew the total (from the number on the gate) but said that he had insufficient information to give a definitive answer, there must be more than one solution with the same total.
Only two sets of possible ages add up to the same totals:
- A. 2 + 6 + 6 = 14
- B. 3 + 3 + 8 = 14
In case 'A', there is no 'eldest child': two children are aged six (although one could be a few minutes or around 9 to 12 months older and they still both be 6). Therefore, when told that one child is the eldest, the census-taker concludes that the correct solution is 'B'.[3]
For 36
[edit]The prime factors of 36 are 2, 2, 3 and 3. This gives the following triplets of possible solutions:
| Age one | Age two | Age three | Total (Sum) |
|---|---|---|---|
| 1 | 1 | 36 | 38 |
| 1 | 2 | 18 | 21 |
| 1 | 3 | 12 | 16 |
| 1 | 4 | 9 | 14 |
| 1 | 6 | 6 | 13 |
| 2 | 2 | 9 | 13 |
| 2 | 3 | 6 | 11 |
| 3 | 3 | 4 | 10 |
Using the same argument as before it becomes clear that the number on the gate is 13, and the ages 9, 2 and 2.[5]
A criticism of this problem is that a household may have two children of the same age in terms of natural numbers, but different birthdays, such as children in a mixed-parent household.
References
[edit]- ^ Meyers, Leroy F.; See, Richard (1990-04-01). "The Census-Taker Problem". Mathematics Magazine. 63 (2): 86. doi:10.2307/2691063. JSTOR 2691063.
- ^ "Ask Dr. Math". Math Forum. 2008-11-22. Archived from the original on 30 August 2010. Retrieved 2010-09-12.
- ^ a b Mary Jane Sterling (2007), Math Word Problems for Dummies, For Dummies, p. 209, ISBN 978-0-470-14660-6, retrieved 2010-09-12
- ^ Rick Billstein; Shlomo Libeskind; Johnny W. Lott (1997), A problem solving approach to mathematics for elementary school teachers (6 ed.), Addison-Wesley, ISBN 978-0-201-56649-9
- ^ a b "Math Puzzle - Census - Solution". Mathsisfun.com. Archived from the original on 3 September 2010. Retrieved 2010-09-12.
Further reading
[edit]- Garces, I. J. L.; Loyola, M. L. (2012). "Revisiting a Number-Theoretic Puzzle: The Census-Taker Problem". arXiv:1204.2071 [math.HO].
- "ISBN 9780805308518". ISBN Search. Retrieved Apr 14, 2024.