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### Magic Pentagon

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Date: 12/02/2001 at 07:47:15
From: Ramona

Dear Dr. Math,

There is a pentagon, and on each side of the pentagon, there are three
circles. How can you make the sum of the three circles all the same
as the others? You can only have 4 answers: 14, 16, 17, 19. You can
only use the numbers from 1-10, and you can only use each of them once
because there are only 10 circles.

I figured out all the possible three numbers that could add up to 14.
None of them are the same, so it looks as if there are only a few.

1 + 3 + 10 = 14
1 + 4 +  9 = 14
1 + 5 +  8 = 14
1 + 6 +  7 = 14

2 + 3 +  9 = 14
2 + 4 +  8 =14
2 + 5 +  7 =14

4 + 3 +  7 = 14

5 + 3 +  6 = 14

I looked in the archive and found a similar question but I still can't
figure this one out.

Ramona
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Date: 12/04/2001 at 10:25:29
From: Doctor Peterson

Hi, Ramona.

I can suggest a few tricks that help a lot in problems like this. I
easily found the solution where the sums are 14 using these methods,
and you can probably do the rest almost as easily.

First, what happens if you add all five sides together? The sum will
be 5*14, or 70; but that will be the same as adding all ten numbers,
and then adding the five corner numbers again (since each of them
counts on two sides). This says that 55 (the sum of 1 through 10) plus
the sum of the corner numbers is 70; so the corner numbers add up to
70-55, which is 15. A little thought will show you exactly which
numbers you have to use at the corners!

Now, place the first of those numbers (1) at one corner, say the top.
You will find that the second number (2) can't go next to the 1,
because then the number between them would have to be 11 in order to
get a sum of 14. So you can place the 2 at one of the bottom corners.
Now where can the third number go? If it is next to the 2, you will
find a problem, so that leaves only one place for it. Keep on like
this to fill in the corners.

Now just find what has to go in the middle of each side in order to
add up to 14, and see if it works out.

I recommend trying the sum of 19 next, before tackling what are
probably the hardest cases, 16 and 17. But I haven't tried them myself
yet, so maybe you'll find they're easy!

This puzzle is probably meant mostly to give you lots of practice
adding, but also lets you develop your own ways to solve logical
problems. It won't all be easy, but at the least you will learn how to
go through all the possible arrangements in an orderly way. At best,
you may find an even better way to solve these than I came up with!

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