Clock and calendar problems: angles, odd days and no memorising

Clock and calendar questions are worth doing because they are formulaic: two small ideas cover the whole topic, and neither requires memorising tables. One is a rate, the other is a remainder.

Clocks: the minute hand gains 5.5 degrees a minute

The minute hand sweeps 360° in 60 minutes, so 6° per minute. The hour hand sweeps 360° in 12 hours, so 0.5° per minute. The minute hand therefore gains 5.5° on the hour hand every minute, and that single number drives every clock question.

The clean angle formula follows directly:

angle = | 30H − 5.5M |

where H is the hour and M the minutes past it. The 30H places the hour hand (30° per hour), the 5.5M corrects for both hands moving as the minutes pass. At 3:40, that is | 30×3 − 5.5×40 | = | 90 − 220 | = 130°. If the formula gives more than 180°, subtract from 360° to get the smaller angle.

Hand events, counted for the day

Because the minute hand laps the hour hand 11 times every 12 hours:

• Overlap (0°): 22 times a day.
• Straight line, opposite (180°): 22 times a day.
• Right angle (90°): 44 times a day.

Questions like “at what time between 4 and 5 o’clock are the hands together” are solved by asking how many minutes at 5.5°/min are needed to close the gap the hour hand starts with — no formula to memorise beyond the rate.

Calendars: everything is odd days

An ordinary year is 365 = 52 weeks + 1 day, so it carries 1 odd day; a leap year carries 2. “Odd days” are simply the leftover days after whole weeks, and since the weekday cycle is 7 long, the count of odd days is the weekday shift. That is why your birthday usually lands one weekday later each year, and jumps two across a February 29.

For longer spans, the 400-year cycle repeats exactly:

• 100 years carry 5 odd days, 200 years 3, 300 years 1, and 400 years 0.

So the calendar for 1 January repeats every 400 years, a favourite one-mark fact.

Finding a weekday

Most exam questions give you an anchor and ask you to shift:

• “If 15 August is a Saturday, what day is it a year later?” An ordinary year adds 1 odd day, so it becomes Sunday; if a February 29 falls in between, add 2 and it becomes Monday.
• “What day after 100 days from a Monday?” 100 mod 7 = 2, so two weekdays on from Monday is Wednesday. Every “after N days” question is just N mod 7.

Count the days between the two dates, take the remainder on division by 7, and shift the known weekday by that many.

The leap-year rule, stated exactly

A year is a leap year if it is divisible by 4, except century years, which are leap years only if divisible by 400. So 2024 and 2000 are leap years; 1900 and 2100 are not. Getting this right is the difference between 1 and 2 odd days for a span, and therefore the whole answer.

Where the marks leak

Forgetting the 5.5, not the 6. The correction term in the angle formula is 5.5M, because the hour hand also moves as minutes pass; using 6M is the classic slip.

Taking the reflex angle. If the formula exceeds 180°, the question almost always wants the smaller angle — subtract from 360°.

Mishandling a century year. 1900 is not a leap year; only centuries divisible by 400 are.

Counting the boundary date twice. Decide whether the start date is day zero or day one and stay consistent.

Practice that transfers

Clock and calendar reward pattern recognition more than calculation, so drill a mixed set under time in a full-length mock test or the daily reasoning quiz, and keep the quick facts — the 5.5°/min rate, odd days per year, the leap rule — on your reasoning shortcuts cheat sheet for last-week revision.