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### Age of Children

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Date: 02/11/97 at 02:37:23
From: Anonymous
Subject: Re: Thanks for your question!

I have a logic question for you:

I meet up with an old friend on the street and ask her how old her
three kids are. She says the product of their ages is 36. I say I
still don't know how old they are. She says the sum of their ages is
the house number across the street. I still don't know how old they
are. She then says the oldest one has red hair. I say OH, that's how
old they are. How old are the kids and how do I know?
```

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Date: 02/16/97 at 03:29:53
From: Doctor Mike
Subject: Re: Thanks for your question!

Hello LIgirl777,

This is a great problem!  Your first reaction is probably "What the
heck does red hair have to do with it?".  Let's find out.

The special properties of 36 are VERY important here.  You need to see
that the complete factorization of 36 is 2*2*3*3. You need this to get
the most out of these clues.  Let's use A, B, and C for the 3 ages,
and assume they are in order from younger to older.  They could all be
different like A = 2, B = 3, C = 6 or some could be the same like
A = 3, B = 3, C = 4 (she could have 3 year old twins).  Now make a
table of ALL the possibilities.  Don't forget the ones with the first
age being one (1).  Also make a column for the sum of the childrens'
ages:

A    B    C    sum
---  ---  ---  -----
1    1    36   38
1    6     6   13
2    3     6   11
3    3     4   10

The table continues on. I got a total of 8 possibilities in my table.
Remember A <= B <= C.  Now, my first row, with a grown child 36 years
old and 1-year-old twins is unlikely, but we have to consider all the
possibilities so we don't miss anything.  OK?

Now on to clue No. 2. For each line in the table, you should add up
the ages and enter that value in the right-hand most column. You will
get a different sum of ages A + B + C for most of the rows, BUT two of
the rows will come out with the same sum. That must be the house
number across the street, because otherwise the second clue would be
enough to show which set of ages was right. Before you go on to clue
No. 3, be sure you have all eight rows of numbers in the table
completely filled out, and that you see which 2 rows have the same sum
in the last column.

Now on to clue No. 3. This third clue could just as well be "the
oldest one has 12 toes". The real clue here is the word "oldest" which
eliminates the possibility of one younger child and two older twins.
That's because the phrase "the oldest one" implies only one.  If you
did all the work along with me, you know the answer.

If you like this kind of problem, you might like to find more of them.
They are great for developing your clearness and accuracy of thought.
You probably can find some books with puzzles like this in the math
section of your public library.  One good one has a title something
like _Games for the Super-intelligent_, by an author whose name I
can't recall right at the present.  The reference librarian can

Good luck, and I hope this helps.

-Doctor Mike,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Logic

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