Question

Count the total number of parallelograms in the given figure. 

Hint:

In counting problems on tilings, partition the search by size class. A shape is a parallelogram only if both pairs of opposite sides are continuous straight segments—any internal “step” breaks eligibility.

Answer 1

The tiled rhombus is made from unit parallelograms (all congruent). To avoid double–counting we will count by size:
Step 1: Count all unit tiles
Every small brick-shaped tile visible is itself a parallelogram. A careful sweep (from the upper left, row by row) gives \[ \text{Unit parallelograms} = 9. \] (There are three along the top band, three in the middle band, and three along the bottom band.)
Step 2: Count all $2\times1$ composites
Two adjacent unit tiles can fuse to form a larger parallelogram if they share a long edge. In this layout there are four such fusions: two along the top half (left and right), and two along the bottom half (left and right). Hence \[ \text{$2\times1$ parallelograms} = 4. \] (Other neighbouring pairs meet at a corner only and therefore do not form a single parallelogram.)
Step 3: Count the largest spanwise parallelograms
There are two long parallelograms spanning the figure from the left boundary to the right boundary (one in the upper half and one in the lower half). Each of these is made from three unit tiles chained end-to-end. Thus \[ \text{Largest parallelograms} = 2. \] No $3\times2$ or $2\times2$ shapes fit because the interior joints break the opposite sides from being single straight lines.
Step 4: Total
\[ \boxed{9\;(\text{unit}) + 4\;(\text{$2\times1$}) + 2\;(\text{largest}) = 15}. \]