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?

My description of the lines left room for interpretation. The easiest way for me to describe how to draw the lines is as follows:

  Divide each side of the square into 5 equal segments.

  Starting at the top left corner and continuing clockwise label the points A,B,C, . . . S,T. The corners should be A, F, K and P.

Draw line AN.

  From all points on the top and bottom lines draw lines parallel to AN.

  From all points on the sides, draw lines perpedicular to AN.

  Most, but not all of these lines will cross another labeled point. Parallel lines AN,BM,CL,DK, and perpendicular lines FS, GR, HQ, IP will cross two points, but parallel lines from E and O, and perpendicular lines from J and T will not.

This will divide the larger square into smaller sections that can be rearanged into 29 smaller squares.

Edited on May 16, 2006, 11:51 am

Posted by Leming on 2006-05-16 10:45:05