How many times are the hands of a clock at right angles between 1 p.m. and 5 p.m.?

Explanation

1. 01

   Normally the hands are at right angles twice in each hour, which would suggest $4 \times 2 = 8$.

2. 02

   But 3:00 is itself a right angle, and it is shared by the hour 2–3 and the hour 3–4. The two right angles of 2 to 3 o'clock are at $27\frac{3}{11}$ minutes past 2 and at 3:00.

3. 03

   List: $21\frac{9}{11}$ and $54\frac{6}{11}$ past 1; $27\frac{3}{11}$ past 2; 3:00; $32\frac{8}{11}$ past 3; $5\frac{5}{11}$ and $38\frac{2}{11}$ past 4. That is 7.