Regular pentagons and squares of the same sides are placed together in a pattern as shown in the diagram. Caleb claims that if he continues the pattern, a closed loop will form. Explain whether his claim is true, showing your working clearly.

1. Find the exterior turn made by each shape

For a regular pentagon, the interior angle is $108^\circ$, so the exterior turn is
$180^\circ-108^\circ=72^\circ.$
For a square, the interior angle is $90^\circ$, so the exterior turn is
$180^\circ-90^\circ=90^\circ.$

2. Find the turn for one repeated pentagon-square unit

One repeated unit consists of one pentagon and one square.
$72^\circ+90^\circ=162^\circ$
So each pentagon-square pair turns the pattern by $162^\circ$.

3. Check whether the total turn can be a multiple of $360^\circ$

A closed loop is possible if the total turn is a whole number of complete turns, i.e. a multiple of $360^\circ$.

Check multiples of $162^\circ$:
$20\times162^\circ=3240^\circ$
$3240^\circ\div360^\circ=9$
Since this is a whole number, the pattern returns to its starting direction after $20$ pentagon-square pairs.