Calendar: Formulas, Shortcuts and Solved Examples

What do calendar questions ask?

They ask for the day of the week on a given date. Or they ask for the dates on which a weekday falls. Or they ask which years share a calendar.

All of them use one idea: odd days.

What is an odd day?

A week has 7 days. After you remove complete weeks, the days left over are odd days.

Give each day a number: Sunday 0, Monday 1, Tuesday 2, up to Saturday 6. Odd days then tell you how many steps to move along the week.

How many odd days are in a year?

An ordinary year has 365 days, which is 1 odd day. A leap year has 366 days, which is 2 odd days.

Why it works: 365 = 52 × 7 + 1. And 366 = 52 × 7 + 2. The extra days are the odd days.

Which years are leap years?

A year divisible by 4 is a leap year. A century year must be divisible by 400.

So 2000 is a leap year. 1900 and 2100 are not.

Why it works: a true year is about 365.2425 days. The century rule removes three leap days in every 400 years to keep the calendar in step with the sun.

How many odd days are in 100, 200, 300 and 400 years?

100 years = 5 odd days. 200 years = 3. 300 years = 1. 400 years = 0.

Why it works: 100 years have 76 ordinary and 24 leap years. That is 76 + 48 = 124 odd days. And 124 = 17 × 7 + 5.

Every 400 years the calendar repeats exactly. So 400, 800 and 1200 years all have 0 odd days.

How do you find the day for a date?

Total odd days = odd days of centuries + odd days of remaining years + odd days of the current year. Divide the total by 7 and keep the remainder.

Why it works: only the remainder after full weeks changes the day. So you add the remainders, not the days.

When do two years share the same calendar?

The two years must both be ordinary or both be leap. And 1 January must fall on the same weekday.

Why it works: the first condition makes February line up. The second makes every date start on the same day.

How do you list the dates for a weekday in a month?

Find the first date, then add 7 again and again.

Why it works: seven days is one full week, so the weekday repeats.

Easy: 1 January 2057 was a Monday. What day is 1 January 2058?

Step 1: 2057 is an ordinary year. It has 1 odd day.

Step 2: Monday is 1. Add 1 odd day to get 2.

Step 3: code 2 is Tuesday.

Medium, step by step: 28 July 1986 was a Monday. What day was 28 July 1985?

Step 1: the gap is one year. It runs from July 1985 to July 1986.

Step 2: there is no 29 February in that gap, since 1986 is not a leap year. So the gap is 365 days, which is 1 odd day.

Step 3: go back 1 day from Monday. The answer is Sunday.

Exam level, step by step: what day was 15 August 1947?

Step 1: take 1600 years as 0 odd days, and 300 years as 1 odd day.

Step 2: the years 1901 to 1946 are 46 years. They hold 11 leap years. That gives 46 + 11 = 57 odd days, and 57 leaves a remainder of 1.

Step 3: days from 1 January to 15 August 1947 are 212 + 15 = 227. That leaves a remainder of 3.

Step 4: the total is 1 + 1 + 3 = 5. Code 5 is Friday.

Exam level, step by step: is the calendar of 1975 the same as that of 1986?

Step 1: 1975 is ordinary. 1986 is also ordinary. So the types match.

Step 2: the years from 1975 to 1985 are 11 years. They include three leap years: 1976, 1980 and 1984.

Step 3: odd days = 11 + 3 = 14, which is 0 after full weeks.

Step 4: so 1 January falls on the same day. Yes, the calendars are the same.

Which traps cost marks?

Treating 1900 or 2100 as leap years. Check the century rule first.

Counting a leap day that is not inside the gap. Only count 29 February if it falls between the two dates.

Counting the starting date twice. Count the days after the start.

Mixing up the day codes. Sunday is 0, not 7.

Which shortcuts save time?

For the same date one year later, move 1 day ahead, or 2 days if a 29 February lies in between.

For the same date one year earlier, move back by the same amount.

For a repeat calendar after an ordinary year, the usual gaps are 6, 11 or 5, 6, 11 years.

Use the first date and add 7. You never need to count the whole month.

What patterns come up in exams?

Four patterns repeat. The day for a given date, the dates of a weekday, the same-calendar year, and a leap year test.

For the leap test, check the century rule first. For the rest, count the odd days.

Write the steps in a column. It prevents the most common slips.