Clock & Calendar

Clocks

• Angle between hands at $H{:}M$: $\theta = \lvert 30H - 5.5M\rvert$ (if $>180°$, subtract from 360°).
• Minute for a given angle: $M = \dfrac{2}{11}\,(30H \pm \theta)$ — try both signs.
• Faulty clock (hands coincide every $x$ min instead of $\frac{720}{11}$): gain / loss per day $= \left(\dfrac{720}{11} - x\right)\times \dfrac{60\times 24}{x}$ (positive = fast).

Alignments per 12 h: hands coincide (0°) 11 times, are opposite (180°) 11 times, at a right angle (90°) 22 times.

Image tricks: mirror image $= 11{:}60$ − time; water image $= 18{:}30$ − time.

Calendars — Odd Days

Solve by the remainder when total days ÷ 7. An ordinary year has 1 odd day; a leap year 2.

$$\text{Total} = \text{Date} + \text{Month code} + \text{Year} + \left\lfloor \frac{\text{Year}}{4} \right\rfloor + \text{Century code}$$

Divide by 7; the remainder is the day code.

• Month codes: 1, 4, 4 (Jan–Mar) · 0, 2, 5 (Apr–Jun) · 0, 3, 6 (Jul–Sep) · 1, 4, 6 (Oct–Dec). (Subtract 1 from Jan & Feb in a leap year.)
• Century codes: 1600s = 6, 1700s = 4, 1800s = 2, 1900s = 0, 2000s = 6.
• Day codes: 1 Sun · 2 Mon · 3 Tue · 4 Wed · 5 Thu · 6 Fri · 0 Sat.

Year repeats (by year mod 4): remainder 1 → +6 yr, 2 → +11, 3 → +11, 0 / leap → +28.