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

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.