Maths Olympiad Prep

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Problem 267

Combinatorics Difficulty 1.5 Multiple choice CEMC Gauss (Grade 7) · Canada · 2014

In the diagram shown, the number of rectangles of all sizes is

Figure 0

Pick one

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Official solution

We systematically count rectangles by searching for groups of rectangles that are of similar size.

The largest rectangles in the diagram are all roughly the same size and overlap in pairs.
There are 3 of these; each is shaded black and shown below.

Figure 1

Rectangle 2 (shown above) consists of 4 small rectangles.
We shade these rectangles black and label them 4,5,6,74,5,6,7, as shown below.

Figure 2

Together, Rectangle 4 and Rectangle 5 (shown above) create Rectangle 8, shown below.
Similarly, Rectangle 6 and Rectangle 7 together create Rectangle 9, shown below.

Figure 3

Finally, Rectangle 4 and Rectangle 6 (shown above) together create Rectangle 10, shown below.
Similarly, Rectangle 5 and Rectangle 7 together create Rectangle 11, shown below.

Figure 4

There are no other rectangles that can be formed.
In total, there are 11 rectangles in the given diagram.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.