Problem 604

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