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There is a puzzle, it goes something like this:

Someone talks to a guy, and asks, Give me the age of my three sons, The other guy asks for some clues:

  • The product of the age of the three sons (of someone) is equal to 36.

"I can't figure out their ages." says solver...

  • The sum of the ages of the three brothers is the same as the number of windows you can see in this building (points to some building).

"I still can't figure out their ages." says solver...

  • The oldest has blue eyes.

"Now I know their ages." says solver!.

So, I do not have any clue of how to solve this logic puzzle...

How to link the number of windows in a building with the product of their ages?, or how would be the approach?...

Any idea?

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    Either my brain just broke or this question is nonsensical.2011-05-19
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    Raymond Smullyan has a number of books of meta-puzzles that explore this. Key to many of them is knowing that somebody can't solve it with some of the data and can solve it when the last piece comes in.2011-05-19
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    This is a variation of the [census-taker puzzle](http://en.wikipedia.org/wiki/Ages_of_Three_Children_puzzle)2011-05-20

2 Answers 2

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Well, first off, let's list all the possible combination of ages (and their sum):

$1,1,36; 38$
$1,2,18; 21$
$1,3,12; 16$
$1,4,9; 14$
$1,6,6; 13$
$2,2,9; 13$
$2,3,6; 11$
$3,3,4; 10$

I'm not sure what to make of the building one, but note the specific wording in the third clue: "older". The only reason you would say "older" when referring to THREE people (you would typically use "oldest") means that two of them must be twins. So, you now have three possibilities left:

$1,1,36; 38$
$2,2,9; 13$
$3,3,4; 10$

I don't know how to use the building clue to pare the choices down to one.

That help?

EDIT: Apparently, "older" should be "oldest". In that case, the solution could be any of them but one. In addition, the missing piece is that if the person solving the puzzle knows the number of windows in the building but still cant figure it out, then the two possibilities are:

$1,6,6; 13$
$2,2,9; 13$

At this point, the remark about "oldest" rules out the first one and leaves only $2,2,9$ as the correct answer.

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    I'm not sure if the OP meant that the oldest son had blue eyes or if it is your interpretation. If it is the oldest, the question changes.2011-05-19
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    You also missed one combination 1,6,6;13. Are there any others?2011-05-19
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    Very good thinking, I guess you almost solve it... I am thinking to how to use the building sentence.2011-05-19
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    The puzzle is misquoted. And in your analysis you left out $1$, $6$, $6$. The second sentence of the puzzle should be "is the same as the number of windows you can see in this building" and he points. The responder says at this point (s)he still does not know the ages. Then person says "the oldest has blue eyes." After the second statement, the only way the other person can fail to know is if there are two (or more) equal sums. This happens with sum $13$ only, and then "**the** oldest" settles things.2011-05-19
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    @El'endia : a building has several floors with the same number of windows at each floor, so the number must be composite and not prime.2011-05-19
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    @svenkatr : You are right that this way was not listed. But if there is (one) older son, 1,6,6;13 is not possible.2011-05-19
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    @user6312: Ok, I updating it2011-05-19
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    @darkcminor: You forgot to put in the **crucial** bit of dialogue after the second question. Might as well do it this way. After first question: "I can't figure out their ages." After the second "I still can't figure out their ages." After the third "Now I know their ages."2011-05-19
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    @svenkatr: Good catch. Editing...2011-05-19
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    @user6312 : Absolutely right. This changes everything.2011-05-19
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    @ogerard: It never occurred to me that a building always has the same number of windows at each floor...is that an architectural standard, or simply what's typical and as such, safe to assume? Not nit-picking, just curious...2011-05-19
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    If the answer were 1,6,6 and the two 6-yr-olds were born 11 months apart, wouldn't there still be an "oldest"?2011-05-20
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    @Fixee: [chuckle] Yes, but would you SAY that? :P Actually...you'd probably put in a comment about birthdays... :P2011-05-20
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    There is no reason why the building must have more than one floor, or that there must be the same number of windows on each floor. The Empire State building has 6500 windows and 102 floors, and 6500 is not divisible by 102. It's also doubtful that you should rule out having two 6-year-olds because of the word "oldest". Even with twins, one is generally born first. Moreover it's quite possible to have two 6-year-olds that are not twins.2011-05-20
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    There are implicit conventions to this kind of "logic" problem, and if the "blue eyes" comment was sufficient but the earlier two were not, "blue eyes" must have distinguished between $2$, $2$, $9$ and $1$, $6$, $6$. And one should not worry about the matter, it is not mathematics, despite the presence of numbers.2011-05-20
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    @Robert Israel: I agree with you, and I should not have presented my interpretation as if I believed that buildings are somehow figurate numbers. Just a hint or a trick toward a possible solution. As user6312 said, we are not doing mathematics here. We are just trying to outguess/reverse engineer a very vague and inconsistent fantasy world which is more or less shared among variants of these kinds of problem. This riddle is only a support for successive hypothesis and manipulation of integers, not a scientific slice of reality.2011-05-20
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    @Amy : see my answer to R. Israel. No this is not a standard, just a possible thought process of the problem's author. As these problems are supposed to make youngsters manipulate properties of integers, this made-up rule is a way to introduce the composite/prime distinction in the problem.2011-05-20
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    Actually, El'endia words the situation well: **If**, after the second clue, the person trying to solve the puzzle *still* can't figure it out, then since the sum of ages/number of windows are distinct, save for the triples (1, 6, 6) -> sum 13; (2, 2, 9) -> sum 13, we can infer the number sum of ages must be 13. The last clue decisively pointing to (2, 2, 9).2011-05-20
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    @Fixee If you allow rational (instead of just positive integral) ages, then you can't solve the problem absent more clues.2018-06-30
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    @RobertIsrael There is only hope of solving the problem as it stands only if the ages are integral. Also, the **the** in *the oldest* settles the uniqueness of the **positive integer** sought.2018-06-30
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I'll be refering to the answer given by El'enidia Starman, just a bit more changes and additional explaination.

El'enidia Starman's Answer: Well, first off, let's list all the possible combination of ages (and their sum):

1,1,36;381,1,36;38 1,2,18;211,2,18;21 1,3,12;161,3,12;16 1,4,9;141,4,9;14 1,6,6;131,6,6;13 2,2,9;132,2,9;13 2,3,6;112,3,6;11 3,3,4;103,3,4;10 I'm not sure what to make of the building one, but note the specific wording in the third clue: "older". The only reason you would say "older" when referring to THREE people (you would typically use "oldest") means that two of them must be twins. So, you now have three possibilities left:

1,1,36;381,1,36;38 2,2,9;132,2,9;13 3,3,4;103,3,4;10 I don't know how to use the building clue to pare the choices down to one.

That help?

EDIT: Apparently, "older" should be "oldest". In that case, the solution could be any of them but one. In addition, the missing piece is that if the person solving the puzzle knows the number of windows in the building but still cant figure it out, then the two possibilities are:

1,6,6;131,6,6;13 2,2,9;132,2,9;13 At this point, the remark about "oldest" rules out the first one and leaves only 2,2,92,2,9 as the correct answer.

Additions: Now some of you must be wondering how did he end up with one correct answer from two, so let me simplify.

Firstly, 1,1,36;381,1,36;38 1,2,18;211,2,18;21 1,3,12;161,3,12;16 1,4,9;141,4,9;14 1,6,6;131,6,6;13 2,2,9;132,2,9;13 2,3,6;112,3,6;11 3,3,4;103,3,4;10 In the above List, when the second clue is provided, the solver would know that the window has a certain number of windows. This number may be 38, 21, 16, 14, 13, 13, 11, 10 now if the number of windows was 38, 21, 16, 14, 11 or 10, then the solver would have come to know the ages as there are no other combination with the same number of windows. But the solver says, that he is still not able to find the answer. This means that the number of windows is 13 and there are two combinations with the sum of ages as 13 which are:

1,6,6;131,6,6;13 2,2,9;132,2,9;13

Now the third clue says that the oldest son has blue eyes. Now, in the first age combination 1,6,6 there are two oldest son's ages which is 6. Thus, there is no oldest son in this case. However, in the second combination there is only one oldest son with age 9. That is why the solver was able to conclude that 2,2,9 is the correct age combination.

Hope I could clarify the doubts if any.