Pentagon Puzzle

Date: 09/18/2002 at 21:43:13
From: Zoee
Subject: Pentagon puzzle

Zoee and her whole family are stumped!

Question: Write the numbers 1 through 10 in the boxes below
(the diagram shows a pentagon in the shape of a house with 5 boxes 
from peak to floor on each side of the house, and three along the 
bottom) so each side of the pentagon has a sum of 14.

We would appreciate your input - please!

The Davidsons

Date: 09/18/2002 at 22:51:44
From: Doctor Peterson
Subject: Re: Pentagon puzzle

Hi, Zoee.

I think it looks something like this, with three numbers on a side, 
right?
           _
        _     _
     _           _

      _         _

       _   _   _

The first thing I do with a problem like this is to think about the 
total of all the numbers. The sum of 1 through 10 is 55. The sum of 
the five sides is 5*14=70. But if I add those five sums, I have added 
the five corner numbers twice each, since each is on two sides, so 
the sum of the corners is the difference between 55 and 70, or 15.

But what five numbers add up to 15? Their average will be 3, so they 
have to be 1,2,3,4,5. Put those in the corners in some order:

           1
        _     _
     5           2

      _         _

       4   _   3

Now try to fill in the remaining numbers so that each side adds to 
14. If you find that is impossible, try rearranging the corners; you 
may be able to see why it was impossible, which will give you ideas 
for the best way to arrange the corners. I had no trouble finishing.

For further discussion, see the Dr. Math archives:

   Magic Pentagon
   http://mathforum.org/library/drmath/view/56732.html

If you have any further questions, feel free to write back.

- Doctors Peterson and Sarah, The Math Forum
  http://mathforum.org/dr.math/