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?

The last question asks: What numbers are possible by this method?

One can divide a square into smaller portions based on n^2 + m^2 (n > 0 and n>=m>0)
"n" is the number of segments that each side is divided into and   (-n/m) and (m/n) are the slopes of the lines going through the square.

These are some of the possible combinations, with the total squares that could be made:

n   m   tot
1   1    2
2   1    5
2   2    8
3   1    10
3   2    13
3   3    18
4   1    17
4   2    20
4   3    25
4   4    32
5   1    26
5   2    29
5   3    34
5   4    41
5   5    50
.
.
7   3    58
.
.
100 5 10025
.
.

Edited on May 16, 2006, 12:35 pm

Posted by Leming on 2006-05-16 12:34:33