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Square dissection (Posted on 2006-05-16) Difficulty: 3 of 5
Take a square and place two equally spaced points on each side (trisecting the sides.) Starting at one corner label the points and corners around the perimeter A, B, C, D, , L.
Connect with straight lines the pairs AI, BH, CG, DL, EK, and FJ. The resulting figure has four squares and pieces around the edge that can be rearranged to make 6 more (for a total of 10.)

How could you use a similar method to dissect a square into twenty-nine squares? How about 58? What numbers are possible by this method?

See The Solution Submitted by Jer    
Rating: 3.6667 (3 votes)

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Solution Solution | Comment 1 of 4

For the initial square the length of any of the lines becomes

sqrt(3^2 + 1^2) = sqrt(10). All the lines are perpendicular and make squares sqrt(9/10) long.

For 29 squares:

Divide each side into 5 segments. So the first side would have A,B,C,D,E and F. Then F,G,H,I,J and K make the second side. K,L,M,N,O and P for the third, and P,Q,R,S,T and A for the fourth.

Connect AN, BM and similar parallel lines. Connect FS, GR, and their parrellel lines.  Each point should have a parallel line drawn from it.  

Line AN will be sqrt(5^2 + 2^2) = sqrt(29) long. Each square will be sqrt(25/29) units long, creating 29 squares of area 25/29.

58 Squares

Divide each side into 7 segments and connect Point A to the point 4 segments from the opposite corner. Then draw all the other lines parallel and perpendicular to this line through all the other points. This will create lines sqrt(7^2 + 3^2) =sqrt (58) long and squares that are 49/58 in area. 

Edited on May 16, 2006, 9:49 am
  Posted by Leming on 2006-05-16 09:48:09

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