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Algebra/"Magic Pentagon"


QUESTION: a pentagon has 10 empty spaces around its perimeter; 1 on each corner and 1 in the middle between each corner. i'm having trouble placing digits 1 through 10 (each only used once in each empty space) so that each side of the pentagon has the same sum. i've seen that there are four (4) different solution sums; in other words, these are the four (4) sum-of-the-sides solutions: one sum-of-the-sides solution will = 14; another sum-of-the-sides solution will = 16; a third sum-of-the-sides solution will = 17; and the final sum-of-the-sides solution will = 19. thank you for your assistance. - ricardo

I'm also Ricardo - Richard
I solved it for 14 in a few minutes.  First, I recognized that the big numbers had to be in the middles and not at the corners.  10 can only be paired with 1 & 3 so that gives you one side.  9 can only be paired with 2 & 3 or 1 & 4 so the next two positions from 1 are 9 & 4.  The next two positions have to be 8 & 2 and then 7, 5 and 6 to complete the ring.  
For 16, 10 can be paired with 1 & 5 or 4 & 2.  Work on it for an hour or so and get back to me if you still have trouble.  


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Richard J. Raridon


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