# Problem 132 Solution

Cosinder all the ways you can define the length of an edge of the outer square, where each variable represents the length of the side or the corresponding square in the image:

1. a+b+c
2. a+b+d+e
3. a+f+g+h+e
4. a+f+g+i+j
5. k+l+m+n+g+h+e
6. k+l+m+n+g+i+j
7. k+l+m+o+p+j
8. k+l+q+p+j
9. k+l+q+r
10. k+s+t+r
11. u+t+v
12. a+k+u
13. a+l+s+u
14. a+l+t
15. b+f+l+s+u
16. b+f+l+t
17. b+f+m+q+t
18. b+f+n+o+q+t
19. b+g+o+q+t
20. b+g+q+r
21. b+h+i+p+r
22. b+h+j+r
23. c+d+h+i+p+r
24. c+d+h+j+r
25. c+e+j+r
Next take 21 of these equations and set them equal to 1. You can not just use any 21, I would suggest taking out four from various places among the list as opposed to four in a row. After taking out four of the above equations I was left with the following matrix:

 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1

Then use Cramer's Rule to solve the equations. In Excel the MDETERM(a1:u21) command will give you the determinant of the matrix bounded by a1 and u21, making this process quick and mostly automated. If the determinant used for the denominator is zero then choose a different set of 21 equations.

When solving for a..u ignore the deonominator determinant since it will be the same for all 21 variables. After solving for all 21 looking for common factors among the solutions and divide.

Thanks to Terry W. Ryder of San Leandro, CA and New Scientist magazine for the problem and Terry for the graphic.

Michael Shackleford, A.S.A., February 23, 1999