A regular polygon is drawn around a circle so that every one of its sides touches the circle. P and Q are the points at which two adjacent sides touch, and O is the centre of the circle. If ∠POQ = 36°, how many sides has the polygon?

Each radius drawn to a point of contact is perpendicular to its side, and by symmetry the n points of contact are spread evenly round the circle. The n angles like ∠POQ therefore fill one complete turn, so each is 360°/n. Setting 360/n = 36 gives n = 10. The trap 5 imagines that consecutive contact points span two of those sectors, 20 halves the sector angle before dividing, and 12 comes from confusing the given 36° with the 30° that a twelve-sided polygon would produce.