# Logic Puzzle of the age of three sons

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. – Stijn May 19 '11 at 21:44
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. – Ross Millikan May 19 '11 at 22:53
This is a variation of the census-taker puzzle – Arturo Magidin May 20 '11 at 3:49

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. – svenkatr May 19 '11 at 21:47
You also missed one combination 1,6,6;13. Are there any others? – svenkatr May 19 '11 at 21:49
Very good thinking, I guess you almost solve it... I am thinking to how to use the building sentence. – cMinor May 19 '11 at 21:50
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. – André Nicolas May 19 '11 at 21:52
@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." – André Nicolas May 19 '11 at 22:06

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.

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