On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?
How good are you at estimating angles?
Explore patterns based on a rhombus. How can you enlarge the
pattern - or explode it?
A square is labelled clockwise $ABCD$. $P$ and $Q$ are points outside the square such that triangles $ABP$ and $BCQ$ are both equilateral.
How big is angle $PQB$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
This problem is taken from the UKMT Mathematical Challenges.