Pentagon PuzzleDate: 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/ |
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