, 2018

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.

[[IMAGE0]]

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.

[[IMAGE1]]
Hide/Reveal Description of Arrays of Parallelograms

Three parallelograms joined side by side and labelled 1 by 3.
Four parallelograms joined side by side and labelled 1 by 4.
Four parallelograms arranged into two rows and two columns and labelled 2 by 2.
Six parallelograms arranged in two rows and three columns and labelled 2 by 3.
Eight parallograms arranged into 2 rows and 4 columns and labelled 2 by 4.

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 $8+6+4+4+2+3+2+1=30$.