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Problem 608

Geometry Difficulty 2.7 Find the answer CEMC Gauss (Grade 7) · Canada · 2018

The figure consists of 8 identical small parallelograms, joined as shown.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

The figure formed by combining a pair of adjacent small parallelograms, is also a parallelogram.

For example, each of the two figures shown is a parallelogram.

The reason for this is that opposite sides of these new figures are equal in length and they are parallel. We use the notation $a \times b$ to mean that the new figure has $a$ rows of the small parallelograms and $b$ columns of the small parallelograms.

Similarly, more than 2 small parallelograms can be combined to form new parallelograms.

In addition to the small parallelogram ( $1\times1$ ) and the $1\times2$ and $2\times1$ parallelograms shown above, the sizes of the remaining parallelograms that appear in the figure are shown below.

In the table below, the number of parallelograms of each of the different sizes is shown.

Size	$1\times1$	$1\times2$	$2\times1$	$1\times3$	$1\times4$	$2\times2$	$2\times3$	$2\times4$
Number of Parallelograms	8	6	4	4	2	3	2	1

The number of parallelograms appearing in the figure is $8+6+4+4+2+3+2+1=30$ .

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