Clocks

At a glance 9 rules

1. 1Hand speeds$\text{minute: } 6^\circ/\text{min},\ \text{hour: } 0.5^\circ/\text{min}$$\text{the minute hand gains } 5.5^\circ \text{ a minute}$
2. 2Angle at H hours M minutes$\theta = |30H - 5.5M|$$\theta > 180^\circ \Rightarrow \text{use } 360^\circ - \theta$
3. 3Hands mirrored across 12-6$\theta_h + \theta_m = 360^\circ$$\theta_h = 30H + 0.5M,\ \theta_m = 6M$
4. 4Together after H o'clock$M = \dfrac{60H}{11}\ \text{min past } H$$\text{every } 65\tfrac{5}{11}\ \text{min: } 11 \text{ times in } 12 \text{ hours}$
5. 5Hands θ apart after H o'clock$M = \dfrac{2}{11}(30H \pm \theta)\ \text{min past } H$$\theta = \text{angle wanted; keep } 0 \le M < 60$
6. 6Right angles in a dayRight angles come every $32\tfrac{8}{11}$ min: 22 in 12 hours, 44 a day, not 48. The one at 3 o'clock serves both 2 to 3 and 3 to 4, and the same at 9.
7. 7True time on a fast clock$\dfrac{\text{true time passed}}{\text{clock time passed}} = \dfrac{60}{60 + g}$$g = \text{min gained per true hour},\ g < 0 \text{ if slow}$
8. 8Faulty clock from its handsAny clock's hands meet every $65\tfrac{5}{11}$ min of its own time. If they meet every T true minutes, a smaller T means it runs fast: it gains $65\tfrac{5}{11} - T$ min per T.
9. 9Striking clockn strokes have n - 1 gaps, and the time goes with the gaps: 5 strokes in 8 seconds means 2 seconds a gap.