A man asks his father about his age.

Instead of giving a direct answer the father wrote down two positive integer numbers on a piece of paper and says:

- Sum these two numbers and evaluate the square root of the total. Doing this, youŽll get my age.

The son takes his pocket calculator and inadvertently types the two numbers, one after the other, without pressing the "+" button (e.g. if the numbers were 124 and 357, he types 124357, instead of the sum of the numbers).

After this he makes a second mistake by pressing the square root button, not once, but twice.

Finding an integer number as the result he shows it to his father:

- This is your age.

- No, youŽre wrong, but the number you found is precisely the age of your mother, who is older than me.

How old is the father?

Note: While a solution is trivial with the aid of a spreadsheet, can you derive it without one?

I apologize if I seem to have broken the conventions of this puzzle site by proposing a variation of a given problem.  I (and "Charlie" I believe) are active on another problems site, where variants or extensions of problems are often proposed when the stated problem has been sufficiently answered (as this "Father and Son" has been).  That site has well vetted weekly puzzles, tested for consistency and lack of ambiguity before posting; the interest there is not solely, or even primarily, in "getting the right answer" (or getting it first) but rather in discussions of methods of approach, and (after the initial problem is solved), in suggesting related matters, pending the next puzzle. 

It would be of interest to me to see if anyone responds to the "Note" at the end of this original text, seeking derivations which do not rely on computer assistance.  Why would it be better to have that discussion on another puzzle's thread?

My variant "challenge" kept all of the original conditions except that the mother be older than the father.  The father could be 20 and the mother 14, from the pair (384, 16); or the father could be 45 and the mother 21, from the pair (1944, 81).

The phrase "older than me" is no doubt colloquial, even though a dubious example for one's son.  May I merely plead guilty to pettifoggery?

Posted by ed bottemiller on 2008-08-06 11:43:17