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Square of Triangles (Posted on 2006-02-04) Difficulty: 4 of 5
Given a partition of a square into n triangles of equal area, prove that n is even.

No Solution Yet Submitted by Josh70679    
Rating: 2.0000 (2 votes)

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Solution re(3): Equal area, congruent triangles. | Comment 10 of 21 |
(In reply to re(2): Equal area, congruent triangles. by Richard)

"It is not possible to dissect a square into an odd number of equal area triangles.  If we can dissect a unit square into an odd number, n, of rectangles, then each rectangle has area 1/n, and since n is an odd integer, |1/ n|2 <= 1.  We will show that we must have |1/n|2> 1, so this will prove the result."

Here's a link to the rest of the proof by Paul Monsky:

Edited to replace "" with "<=". Apparently, the greater than or equal to symbol from the format tool above the comment area does not always display properly.

Edited on February 5, 2006, 12:52 pm
  Posted by Mindrod on 2006-02-05 02:07:14

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