About Calendar
Calendar questions look like tests of memory, but they are really only division by seven. The weekday of any date can be worked out from any other date whose weekday is known, because weekdays repeat every seven days and every whole week can be discarded.
What you need to understand
- A week has seven days, so after removing whole weeks only the remainder matters.
- An ordinary year has 365 days, which is 52 weeks and 1 odd day, so the next year begins one weekday later.
- A leap year has 366 days, which is 52 weeks and 2 odd days, so the next year begins two weekdays later.
- February has 29 days in a leap year. A year is a leap year if it is divisible by 4, and a century year only if it is divisible by 400.
- Two dates a whole number of weeks apart always fall on the same weekday.
- The number of days between two dates is their difference, so there is no need to count every day in between.
Formulas to remember
How to work through these questions
- Fix a reference date whose weekday you know, and count the days between it and the date asked about.
- Take that count modulo 7 and move that many weekdays forward or backward.
- For a question about two dates in the same month, simply subtract the smaller date from the larger.
- For a question asking what day it will be after n days, divide n by 7 and keep only the remainder.
- Check the answer is at most six places from the starting weekday, because a shift of seven is no shift at all.
Mistakes that cost marks
- Treating a century year as a leap year when it is not divisible by 400, as with 1900.
- Counting both ends when working out the number of days between two dates.
- Using 30 days for every month.
- Dividing by 7 and then using the quotient instead of the remainder.
- Moving the count backwards when the question asks for a later date.
Worked example
If the 5th of a month is a Tuesday, what day of the week is the 27th of the same month?
- The number of days between the 5th and the 27th is 27 - 5 = 22.
- Dividing 22 by 7 gives 3 whole weeks and a remainder of 1.
- So the weekday moves on by one place from Tuesday.
- The 27th of the month is a Wednesday.
Answer: Wednesday
Practice questions with answers
A few Calendar questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Find the day of the week on 20 May 1970.
-
A
Tuesday
-
B
Friday
-
C
Wednesday
-
D
Thursday
Answer: Option C — with explanation
Work from a date whose weekday is known. 1 January 2000 was a Saturday.
From 1 January 2000 to 20 May 1970 there is a span of 10818 days, and 10818 divided by 7 leaves a remainder of 3.
Counting 3 place(s) on from Saturday gives Wednesday.
Common mistakes
- shifted the count by two days: Friday
- counted one day too few: Tuesday
- counted one day too many: Thursday
Question 2
Today is Friday. Find the day of the week 105 days from today.
-
A
Sunday
-
B
Thursday
-
C
Saturday
-
D
Friday
Answer: Option D — with explanation
Dividing by 7 leaves only the remainder to worry about: 105 = 7 x 15 + 0.
So the day moves on by 0 place(s) from Friday.
The day is Friday.
Common mistakes
- counted one day too many: Saturday
- counted one day too few: Thursday
- counted two days too many: Sunday
Question 3
In a certain month the 1th day is a Thursday. Which day of the week is the 11th day?
-
A
Sunday
-
B
Monday
-
C
Saturday
-
D
Wednesday
Answer: Option A — with explanation
Between the 1th and the 11th there are 10 days.
10 = 7 x 1 + 3, so the day moves on by 3 place(s).
Starting from Thursday, the 11th is a Sunday.
Common mistakes
- counted three days too many: Wednesday
- counted one day too many: Monday
- counted one day too few: Saturday
Question 4
Which day of the week fell on 10 March 1975?
-
A
Monday
-
B
Tuesday
-
C
Sunday
-
D
Wednesday
Answer: Option A — with explanation
Work from a date whose weekday is known. 1 January 2000 was a Saturday.
From 1 January 2000 to 10 March 1975 there is a span of 9063 days, and 9063 divided by 7 leaves a remainder of 5.
Counting 5 place(s) on from Saturday gives Monday.
Common mistakes
- counted one day too few: Sunday
- shifted the count by two days: Wednesday
- counted one day too many: Tuesday
Question 5
Today is Wednesday. Find the day of the week 101 days from today.
-
A
Saturday
-
B
Friday
-
C
Sunday
-
D
Monday
Answer: Option A — with explanation
Dividing by 7 leaves only the remainder to worry about: 101 = 7 x 14 + 3.
So the day moves on by 3 place(s) from Wednesday.
The day is Saturday.
Common mistakes
- counted one day too many: Sunday
- counted two days too many: Monday
- counted one day too few: Friday
Frequently asked questions
How do I know whether a year is a leap year?
Divide by 4. If there is no remainder it is a leap year, unless it is a century year, in which case it must be divisible by 400. So 1900 was not a leap year, but 2000 was.
Why does only the remainder after dividing by 7 matter?
Because weekdays repeat in cycles of seven. Every whole week brings you back to the same weekday, so whole weeks contribute nothing to the answer.
What is an odd day?
It is the number of days left over after removing whole weeks. A year of 365 days is 52 weeks plus 1 odd day, and a leap year has 2 odd days.
How many odd days are there in 100 years?
Five. There are 24 leap years and 76 ordinary years, giving 24 x 2 + 76 x 1 = 124 odd days, and 124 divided by 7 leaves a remainder of 5.
