... You 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).
However, 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.
So 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.
Of 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|
The 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.
This 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...
The 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.
With 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.
The 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.