Calendars – keeping track of time
Activity 1
Calculating the day of the week
The calendar, with all its awkwardness and lack of symmetry, has provided many scientists with amusement over the years. Lewis Carroll, a mathematician with a talent for writing (his most famous work was Alice in Wonderland), once published a paper in the journal Nature describing how to determine the day of the week for any particular date in the Common Era. His method is described below. The example we've used is 26 January 1788, the date on which the First Fleet sailed into Sydney Harbour.
First, divide the date into four parts: (1) the number of centuries, (2) the number of years left over, (3) the month, and (4) the day of the month.
26 January 1788 gives (1) 17; (2) 88; (3) 1; (4) 26.
Compute the following four items, adding each, when found, to the total of the previous items. Whenever an item or total exceeds seven, divide by 7 and keep only the remainder. In the example, the number in bold is the total carried on to the next step.
- The century item: Divide by 4, subtract the remainder from 3 and then multiply by 2. (Note: for dates prior to 14 September 1752, the date at which the calendar was reformed in Great Britain, simply subtract from 18.)
17 divided by 4 = 4 with a remainder of 1.
3 minus 1 = 2.
2 × 2 = 4.
- The year item: Add together the number of dozens, the remainder, and the number of 4s in the remainder.
88 divided by 12 = 7 (the number of dozens) with a remainder of 4.
The number of 4s in the remainder = 1.
Therefore: 7 + 4 + 1 = 12.
Divide by 7 = 1 with a remainder of 5.
Add 5 to 4 [from (1) above] = 9.
Divide this by 7 gives 1 plus a remainder of 2.
- The month item: If the name of the month begins or ends with a vowel, the item is obtained by subtracting the number denoting its place in the calendar from 10 (eg, if the month is April, the value is 10 minus 4 = 6). Months ending with vowels can also be used to give the value for the following month: as before, subtract the number denoting its place in the calendar from ten but this time also add its number of days. In this way, April can be used to calculate the item for May: 10 minus 4 = 6 plus 30 = 36 (which must then be divided by 7, to give a remainder of 1, which is the value for May). The value for January is zero, for February or March, 3, and for December, 12.
The value for January is set at zero.
Add 0 to 2 [from (2) above] = 2. - The day item: This item is simply the day of the month.
The day is 26, which must be divided by 7 and the remainder kept.
Thus, 26 divided by 7 = 3 with a remainder of 5.
Add 5 to 2 [from (3) above] = 7.
Leap-year adjustment
The total arrived at must be corrected by deducting 1 (first adding 7 if the total is zero) if the date is January or February in a leap-year (remembering that every year divisible by 4 is a leap-year except in century-years, which are only leap-years if divisible by 400). The corrected number gives the day of the week, if the days of the week are numbered such that 1 = Monday and 7 = Sunday.
1788 is divisible by 4 and is therefore a leap-year.
Thus, the value becomes 7 minus 1 = 6.
Therefore, Governor Phillip planted the British flag in Sydney Harbour on a Saturday.
You might like to try this procedure. Test it first on a few recent dates to make sure you’ve got it right, then calculate the day of the week on which important events have occurred throughout history (eg, the first lunar landing on 20 July 1969, or the date of your birth).
You can check your calculations by entering the date you have chosen at the following site:
How to calculate mentally the day of the week for any date (Guy Rimmer)
Page updated January 2005.