Finding the Ages of the Farmer's Daughters

Date: 12/21/95 at 21:55:5
From: Anonymous
Subject: Age problem

A census taker came to a house where a man lived with
three daughters.  "What are your daughters' ages?" he asked.
The man replied, "The product of their ages is 72, and the
sum of their ages is my house number."
"But that's not enough information," the census taker insisted.
"All right," answered the farmer, "the oldest loves chocolate.

What are the daughters' ages?

I have spent many hours over this problem, but to no avail.
All I know is that there is an answer to this.

Date: 4/28/96 at 19:21:55
From: Doctor Steven
Subject: Re: Age problem

We can look at all the possibilities this way - the oldest may be 
the same age as the middle, and the middle may be the same age as 
the youngest:

   OLD        MID        YOUNG              SUM

    72         1           1                 74
    36         2           1                 39
    24         3           1                 28
    18         4           1                 23
    18         2           2                 22
    12         3           2                 17
    12         6           1                 19
     9         8           1                 18
     9         4           2                 15
     8         3           3                 14
     6         6           2                 14
     6         4           3                 13

These are all the possibilities, since if we give the oldest a 
lower age, there is no way the product can equal 72.

We know their ages add up to the farmer's house number.  The sum 
column gives all possibilities for the house number.  When the 
farmer gave the census person the information about the product 
of their ages being 72 and the sum of their ages being his house 
number the census person said this was not enough information.  
So there must have been at least two different possibilities that 
were still viable options for the census taker to choose from.  
This means the house number must have had two sums equal it.  So 
the daughters are:

       OLD       MID      YOUNG            SUM
        8         3         3               14
        6         6         2               14

since they both add up to 14, the only number that appeared twice 
in the sum of the ages.  The fact that the farmer had an oldest 
daughter says that the daughters must in fact be ages 8, 3, and 3.

Hope this helps.

-Doctor Steven,  The Math Forum