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?

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