On October 28, 2002, a mathematical riddle was posted on a newsgroup
for a University of Illinois (Urbana-Champaign) Computer Science
course. The only information I have as to the origin of the riddle is
the title of the thread, "Interesting Problem from Iowa State." The
newsgroup server has since been shut down, so I cannot access the
original post or any follow-up replies. I still have not figured out
the answer to this riddle, over two years later. Here's the riddle:

---

Three mathematicians are talking.
"How many kids do you have?" The first asks the second mathematician.
"Three," she answers.
"How old are they?" he asks.
"Well if you multiply their ages, you get 72. But if you add them, you get
your office number." She says.

The first mathematician thinks for a few seconds and says, "I do not think
there is enough information for me to solve this!" he says.

The mother mathematician immediately says, "Oh, of course, I forgot to tell
you that the eldest plays the violin!"

---

Please end my torment, and find for me:

1) The ages of the children.
2) The office number of the first mathematician.

I already attempted the shotgun approach of simply saying "any
combination of ages such that their product is 72 and one is bigger
than the others; it's 'probably' 2, 4, 9, or maybe 3, 4, 6," and that
was shot down. There is supposedly one pair of answers, and that's
what I'm looking for.

The children's ages (including a pair of twins!) are 3, 3, and 8. The
office number is 14.

Substantially the same puzzle, with superficial changes, is posted on
a University of Calgary site:

"A census taker comes to a man's home. Upon inquiring about the ages
of his three children, he receives the following reply. 'If you
multiply their ages, you get 72. Furthermore, the sum of their ages is
the house number' (which the census taker knows).

The census taker considers the situation for a moment, then says,
'Okay. I need one more piece of information.'

'Sure thing,' replies the man. 'My oldest child loves chocolate.' The
census taker smiles, thanks the man, and walks away. How old are the
man's three children?

Answer: factor 72 to 1*2*2*2*3*3 

1-2-3: 
1,2*2,2*3*3 = 1+4+18 = 23 
1,2*3,2*2*3 = 1+6+12 = 19 
1,3*3,2*2*2 = 1+9+8 = 18 

1-1-4: 
1,2,2*2*3*3 = 1+2+36 = 39 
1,3,2*2*2*3 = 1+3+24 = 28 

1-1-5: 
1,1,2*2*2*3*3 = 1+1+72 = 74 

1-1-3: 
2,2,2*3*3 = 2+2+18 = 22 
2,3,2*2*3 = 2+3+12 = 17 
3,3,2*2*2 = 3+3+8 = 14 

1-2-2: 
2,2*2,3*3 = 2+4+9 = 15 
2,2*3,2*3 = 2+6+6 = 14 
3,2*2,2*3 = 3+4+6 = 13 

We don't have to know the house number. 

The fact that the census taker knows the house number and that's not
enough for him to determine the childrens' ages tells us that it must
be ambiguous.

There is only one sum that comes up more than once, it's 14. 

Therefore, that must be the one. Because there *is* an oldest child,
their ages are 3,3,8."

University of Calgary: A Collection of Riddles and Puzzles For DMs
http://www.ucalgary.ca/~ammaster/d_n_d/puzzles.html

Here you'll find another version, also mathematically identical to
your puzzle, with some students' solutions:

Math Forum: Strawberry Ice Cream, April 20-24, 1998
http://mathforum.org/elempow/fullsolutions/19980420.fullsolution.html

And yet another:

Winchester Public Schools: Answer to November 9, 1998 Problem 
http://www.pen.k12.va.us/Div/Winchester/jhhs/math/probweek/a110998.html

My Google search strategy:

Google Web Search: "if you multiply their ages, you get 72"
://www.google.com/search?hl=en&q=%22if+you+multiply+their+ages%2C+you+get+72

I hope this is precisely what you need. If it is not, please request
clarification; I'll gladly offer further assistance before you rate my
answer.

Best regards,
pinkfreud

Hah! Such a ridiculously simple answer, and after all this time I
still never figured it out. This is why I didn't do too well on my
theoretical CS courses. ;-)

Exactly what I was looking for. Thanks for the prompt response -- I
may now rest easy at night.

Thank you for the five stars and the nice tip!

~pinkfreud

I'm more puzzled now. Please forgive me if I'm slow or naive but, can
anyone tell me how the fact that a child plays violin or likes
chocolat is relevant to solve the riddle?

samuelb,

The remark about the oldest child playing violin or liking chocolate
is relevant only because it establishes that there is one "oldest"
child. This eliminates the 2, 6 and 6 possibility.

Maybe, I am just dumb but why can't the ages be:

9 8 1

9*8*1 = 72

Just wondering?

djones913,

The ages must add up to 14, since that is the only ambiguous
possibility that would require the additional info about the eldest
child. If the children were 9, 8, and 1, the mathematician (knowing
his own office number - in this case, 18) would not have required the
hint about one child being the eldest, and would have been able to
give an answer immediately upon learning that the product of the ages
was 72 and the sum of the ages was the same as his office number.

Dear Pink,

Got it. Thanks!

This reminded me of a somewhat similar, albeit much more complicated,
puzzle that can be found at
http://www.greylabyrinth.com/puzzles/puzzle.php?puzzle_id=puzzle086. 
It's very interesting.