Answer to March 10, 2003 Problem

The Rectangle / Squares Problem

In the figure below, a rectangle has been cut into eleven squares of various sizes. The smallest square measures 9 cm x 9 cm. Determine the dimensions of the large rectangle.

 

Solution to Problem:

The answer is 176 cm x 177 cm.

Recall that the square shaded in grey is 9 cm x 9 cm.
Let x represent the small length on the right just above the 9 x 9 cm square (it is marked in red in the diagram below).
Then the square shaded in green has sides of length (9 + x) cm.
Keep working from square to square to represent the length of each side: the next square would be 18 + x, then 27 + x, then 27 + 2x, then 36 + x, then 36 + 3x, then 3x, then 36 + 6x, then 63 + 2x, and finally 36 + 9x (the large square in the upper left).

Since the large figure is a rectangle, equate the top length and the bottom length and solve for x:
(36+9x) + (36+6x) = (63+2x) + (27+x) + (18+x) + (27+2x)
so, 72 + 15x = 135 + 6x
therefore, x = 7.
Now substitute 7 for x in the two sides of the rectangle to get
176 cm x 177 cm.



Correctly solved by:

1. Richard Johnson La Jolla, California
2. Rick Jones Kennett Square, Pennsylvania
3. Keith Mealy Cincinnati, Ohio
4. Sue Mealy Cincinnati, Ohio
5. Jeff Gaither Winchester, Virginia
6. George Gaither Winchester, Virginia