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... **Y**ou
you are ready to give a answer just after (don't forget — you are in the shoes
of the second mother!) counting "number of windows in the house across the
street" (you **are** in the street and **can** see the house across
the street).

**H**owever,
counting windows will give you an immediate solution under one condition only
— number of windows should identify the row of the sum table **uniquely**.
This is a case for all sums except **13**.

**S**o
the fact that (after deriving the factor and sum tables and counting number of
windows) the second mother still found that "what had been told to her"
so far was "not sufficient to solve the problem" can mean only one thing —
the sum of sons' ages (and the "the number of windows across the street")
was exactly **13**.

**O**f
those two factorings left as a possible solutions, only the second has the
"oldest son." After a fact about the "oldest son" was provided, the
second mother was able to identify solution uniquely.

Factor Table |
Sum Table |
||||||||||||||

36 |
= | 1 |
x | 1 |
x | 36 |
1 |
+ | 1 |
+ | 36 |
= | 38 |
||

36 |
= | 1 |
x | 2 |
x | 18 |
1 |
+ | 2 |
+ | 18 |
= | 21 |
||

36 |
= | 1 |
x | 3 |
x | 12 |
1 |
+ | 3 |
+ | 12 |
= | 16 |
||

36 |
= | 1 |
x | 4 |
x | 9 |
1 |
+ | 4 |
+ | 9 |
= | 14 |
||

36 |
= | 1 |
x | 6 |
x | 6 |
1 |
+ | 6 |
+ | 6 |
= | 13 |
||

36 |
= | x | x | 2 |
+ | 2 |
+ | 9 |
= | 13 |
|||||

36 |
= | 2 |
x | 3 |
x | 6 |
2 |
+ | 3 |
+ | 6 |
= | 11 |
||

36 |
= | 3 |
x | 3 |
x | 4 |
3 |
+ | 3 |
+ | 4 |
= | 10 |

**T**he
fact that the oldest son is "red-haired" was obviously irrelevant to the
problem — I hope that was clear to you from the very beginning.
What's important that **any** fact was given about the "oldest son", thus
stating that there **was** an oldest one. "The oldest one plays chess"
or "the oldest one is sick today", etc. are equivalent to "the oldest one
is red-haired" in terms of the problem.

**I**'ve
acquired this problem by word of mouth back in 1970 in this exact form
that combines both challenge and simplicity. I have never seen this
problem described elsewhere in the same way.

**T**his
problem was thrown sometimes as a tooth-breaker to know-it-all "smarties" at
entrance exams to the Math faculty of Leningrad University. It
was definitely a punch "below the belt" since, considering the inevitable
exam stress involved, and severe time limitations, it was next to impossible
to solve it on the spot despite its almost obvious simplicity. Well,
it was (and still is) jungle out there...

**T**he
only similar problem I've ever encountered was in one of the problem-solving
books by Poya. I don't remember the details now, but can recollect
that product of ages was 72, and that problem setting was different — two
friends were talking in the house and the sum of ages was the street number
of the house. It was stated __explicitly__ that one of them went downstairs
and/or outside to look at the number plate, and after that declared the
data to be insufficient.

**W**ith
all due respect to Poya as an unquestionable authority in problem-solving,
I still consider this to be too much of a hint, stressing upfront the fact
that is a key to the solution.

**T**he
version of problem presented here leaves it to you to figure out what is
both obvious and crucial, yet not directly prompted, giving thus a rare
opportunity
to experience the joy of a sudden insight.

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