3.7 Clocks & Calendars
Key Takeaways
- The minute hand gains 5.5 degrees per minute over the hour hand, and the angle between the hands equals the absolute value of 30H minus 5.5M.
- The hands of a clock coincide 22 times and are at right angles 44 times in a 24-hour period, because they overlap 11 times every 12 hours rather than 12.
- An ordinary year contains 1 odd day and a leap year contains 2 odd days, which is the entire basis of every day-of-the-week calculation.
- A century year is a leap year only when it is divisible by 400, so 2000 was a leap year while 1900 and 2100 are not.
Part 1: Clocks
The basic geometry
A clock face is a circle of 360 degrees divided into 12 hours, so each hour mark spans 30 degrees and each minute mark spans 6 degrees.
| Hand | Full revolution | Rate |
|---|---|---|
| Minute hand | 60 minutes | 6 degrees per minute |
| Hour hand | 12 hours | 0.5 degrees per minute |
The minute hand therefore gains on the hour hand at
This single number generates the whole topic.
The angle formula
At $H$ hours and $M$ minutes, the angle between the hands is
If the result exceeds 180 degrees, subtract it from 360 to obtain the smaller angle.
Worked example. Find the angle at 3:40.
Worked example. Find the angle at 8:20.
Coincidence, opposition and right angles
Because the minute hand gains 5.5 degrees per minute, it takes
for the minute hand to gain a full lap on the hour hand. Consequently the hands coincide once every $65\tfrac{5}{11}$ minutes, not once an hour.
| Event | In 12 hours | In 24 hours |
|---|---|---|
| Hands coincide | 11 times | 22 times |
| Hands are opposite (180 degrees) | 11 times | 22 times |
| Hands at right angles | 22 times | 44 times |
The reason coincidences number 11 and not 12 in a half-day is that between 11 o'clock and 1 o'clock the hands coincide only once, at 12. The same logic removes one opposition, between 5 and 7 o'clock.
Faulty clocks
If a clock gains or loses time, work in proportion. A clock that gains 5 minutes every hour shows 65 minutes of apparent time for every 60 true minutes, so
If two clocks, one gaining and one losing, start together, the time until they differ by a given amount is that amount divided by the sum of their rates.
Worked example. A watch loses 3 minutes per day. If it is set right at noon on Monday, what does it show at noon on Thursday? Three days pass, so it loses 9 minutes and shows 11:51.
Part 2: Calendars
Odd days: the whole method
An odd day is the remainder when a number of days is divided by 7. Because the week repeats every 7 days, only this remainder affects the day of the week.
| Period | Days | Odd days |
|---|---|---|
| Ordinary year | 365 | 1 (365 = 52 weeks + 1) |
| Leap year | 366 | 2 |
| 100 years | — | 5 |
| 200 years | — | 3 |
| 300 years | — | 1 |
| 400 years | — | 0 |
Because 400 years contain zero odd days, the calendar repeats exactly every 400 years.
The leap-year rule
A year is a leap year if it is divisible by 4, except that a century year must be divisible by 400.
| Year | Leap? | Why |
|---|---|---|
| 2024 | Yes | Divisible by 4, not a century |
| 1900 | No | Century, not divisible by 400 |
| 2000 | Yes | Century, divisible by 400 |
| 2100 | No | Century, not divisible by 400 |
This exception is the single most examined point in the topic.
Month codes and day counting
Days in the months: 31 for January, March, May, July, August, October and December; 30 for April, June, September and November; and 28 or 29 for February.
To count odd days across months, take each month's days modulo 7:
| Month | Days | Odd days |
|---|---|---|
| January | 31 | 3 |
| February | 28 / 29 | 0 / 1 |
| March | 31 | 3 |
| April | 30 | 2 |
| May | 31 | 3 |
| June | 30 | 2 |
Worked example: same date, different year
If 15 August 2026 is a Saturday, what day is 15 August 2027?
From August 2026 to August 2027 the intervening February belongs to 2027, which is not a leap year, so the span is an ordinary year carrying 1 odd day. The day advances by one: Sunday.
And 15 August 2028? The span from 2027 to 2028 includes February 2028, which is a leap February, so it carries 2 odd days. Sunday plus two gives Tuesday.
Worked example: counting within a year
If 1 January of a non-leap year is a Monday, what day is 1 March?
January contributes 31 days and February 28, a total of 59 days. Then $59 \div 7$ leaves a remainder of 3. Monday plus 3 gives Thursday.
Repeating calendars
A calendar repeats when the accumulated odd days return to zero. For an ordinary year the calendar typically repeats after 6 or 11 years; for a leap year, after 28 years, provided no century-year exception intervenes.
Exam Technique
Both halves of this topic are formula-driven, so the marks go to whoever recalls the formula rather than whoever reasons hardest. Commit three things to memory: the angle formula $|30H - 5.5M|$, the coincidence count of 22 in a day, and the odd-day values of 1 for an ordinary year and 2 for a leap year. With those, most items in this bullet resolve in under thirty seconds.
The angle between the hour and minute hands of a clock at 3:40 is:
In a 24-hour period, the hands of a clock coincide:
Which of the following is NOT a leap year?
If 1 January of a non-leap year falls on a Monday, then 1 March of the same year falls on a: