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Cinq Carve (Posted on 2014-04-25) Difficulty: 4 of 5
Consider a rectangular piece of paper ABCD with AB ≠ AD

Show how you can fold this piece of paper into five pieces of equal area.

*** Using any given instrument like straightedge, compass, protractor, ruler etc. is NOT permissible.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution a solution | Comment 1 of 7
Paper folding is hard to describe in words, easier with diagrams, and easiest if you can just show someone.  I'll do my best:

All you need is 5 equal spaces on part of one side and then you can scale this to 5 equal spaces along an entire other adjacent side.

Fold B and C less than a fifth of the way to A and D (creates a narrow rectangle.)  
Repeat 4 more times and unfold.
Call the points where these creases meet edge CD by E, F, G, H, I
Make crease IB.
Creases parallel to IB through H, G, F and E will meet edge BC at H', G', F', and E' which split BC into 5 equal segments.

(For greater accuracy in the four folds, I can be folded to points on the crease IB such that the new creases formed pass through each of H, G, F and E.  Then when creating the parallel creases described above each point will have a creases to fold to.)

Finally crease you four folds C and D toward B and A through each of E', F', G' and H'.  These rectangles formed are the fives equal area pieces sought.

(This method can be used to create and number of equal area rectangles.)

  Posted by Jer on 2014-04-26 11:28:00
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