In the diagram below, there is a rectangle made of 9 squares, each of a different size. If the dimensions of the two smallest squares in the figure are 1x1 and 4x4, can you determine the dimensions of all the other squares?

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Solution to the Problem:

The dimensions of the squares are 1x1, 4x4, 7x7, 8x8, 9x9, 10x10, 14x14, 15x15, and 18x18.

To solve, let x = the length of the segment indicated in the diagram below.   Then represent the lengths of other segments in the diagram in terms of x.



Since the opposite sides of a rectangle are congruent, you can solve for x by setting the two expressions for the top and bottom sides equal to each other:
      (x + 8) + (x + 12) = (x + 3) + (x + 2) + (2x + 3)
      2x + 20 = 4x + 8
      2x = 12
      x = 6

Now go back and substitute 6 for x in all the expressions for the sides.

Here is the complete solution.



The dimensions of the rectange are 33 x 32.

Here is Kimberly Howe's very compact solution:


Many thanks to Sharon K. Miller and Vanessa Revelli for correcting my explanation.



Correctly solved by:

1. James Alarie Flint, Michigan
2. SreeRoopa Sankararaman Singapore
3. Morganne Johnson Mountain View High School,
Mountain View, Wyoming
4. Eliza Sheffield and Anna Tetzlaff Tuscaloosa, Alabama
5. Kimberly Howe Vienna, Virginia