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Get one half by squaring (Posted on 2004-09-20) Difficulty: 4 of 5
Can you solve the following equation?

= 1/x + 1/y +...+ 1/z

All variables must be different, positive integers, and there must be a finite number of terms.

See The Solution Submitted by Federico Kereki    
Rating: 4.2500 (12 votes)

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Solution An eleven terms solution | Comment 18 of 31 |
I set up an spreadsheet so I could enter x, y, z... values and see the difference between 0.5 and the sum of the inverses of the squares of the numbers.

I decided starting with the numbers 2, 3, 4, 5, 6 and 12, since I wasn't convinced that 11 coudl/would provide a solution.

After 12, I needed a number greater than or equal to 25. I tried 25, but after many experiments, couldn't make it work, so I decided to try 30 instead.

I needed a number greater than 43 now, so I first tried 45, but couldn't find a solution. Then I tried 50, and that required a number above 80 -- and I tried 100, which then required a new number above 134, and when I tried 150, the spreadsheet showed that 300 would finish the job!

Thus, the numbers x, y, z... are 2, 3, 4, 5, 6, 12, 30, 50, 100, 150, 300.

I guess there can be many other solutions, for I wasn't methodical; I just went by trial and error, picking the numbers that looked right to me and discarding others.
  Posted by e.g. on 2004-09-21 19:39:13

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