Census Taker

Date: 02/20/2002 at 11:54:45
From: Christina
Subject: Census Taker

The census taker says, "I need to know the ages of your children."  
The mother replies, "I have no one-year-olds. The product of my 
children's ages is 90, and the sum of their ages is the same as my 
house number."  

The census taker replies. "I can see the house number but I still need 
more information." 

The mother says, "You're right. You also need to know that the boy 
across the street is older than my oldest child." 

The census taker says, "Thank you, I now know the ages of your 
children." 

What are the ages of the children, and what is the house number and 
the age of the boy across the street?

Date: 02/20/2002 at 12:14:02
From: Doctor Peterson
Subject: Re: Census Taker

Hi, Christina.

Make a list of all ways to factor 90 (with no ones): 2*3*3*5, 2*3*10, 
and so on. One of those sets of factors gives the ages. 

Add the factors in each case; if the house number is that value, the 
census taker knows that those are the ages - unless there are two 
possibilities that give the same sum. So the ages must be one of those 
sets. Which one? 

Since knowing that the oldest child is younger than the boy across the 
street tells the census taker the answer, we can presume that the 
census taker has already been across the street and knows that age. 
What age will give enough information for him to now determine the 
answer uniquely?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/