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Could you please help me find the answer to this question. I think it has something to do with grouping or pairing some numbers.I would appreciate easy-to-understand solutions. Thank you.

There are 99 sticks of lengths 1,2,3...,99 respectively, the width of the sticks can be ignored. Is it possible to form the perimeter of a rectangle using all the sticks?

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1 Answer 1

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Hint: Use $1,2,$ and $3$ for the short ends of the rectangle. That leaves $4,5,\ldots,99$ to divide into two equal sets. Note that $4+99=5+98=\cdots=51+52$. How many such pairs are there?

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  • $\begingroup$ Are their 43 pairs? Once I find the number of pairs, what do I do? I am confused. $\endgroup$
    – anonymous
    Feb 20, 2015 at 23:00
  • $\begingroup$ No, there are not 43 pairs. $\endgroup$
    – TonyK
    Feb 21, 2015 at 7:24
  • $\begingroup$ Are there 48 pairs of 103 then? $\endgroup$
    – anonymous
    Feb 21, 2015 at 10:29
  • $\begingroup$ Yes. Now what can you do with these pairs? $\endgroup$
    – TonyK
    Feb 21, 2015 at 16:30
  • $\begingroup$ I can divide it by 2 so that there are 24 pairs of 103 on either long side of the rectangle. $\endgroup$
    – anonymous
    Feb 21, 2015 at 22:04

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