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Twosday Tuesday: Calindrome of the Century

Happy Twosday! In honor of this rare occasion, we shall discuss the Doomsday Algorithm.

We can employ this algorithm to calculate the day of the week for any given date in our minds. And that’s the best part- your friends might think that you have superhuman computing abilities when you’ve really just memorized a series of numbers and calculations.

Credit for the development of this algorithm goes to British mathematician John Conway, (whose name you will undoubtedly be seeing in future posts)

Why does he call this technique the doomsday algorithm? The crux of Conway’s method is that, in any given year, there exist certain dates which always fall on the same day regardless of the year in discussion. Conway refers to these days as doomsdays.

The following dates are doomsdays:

the last day of February (29 February for a leap year, 28 February in a non-leap year)

March 7

April 4

May 9

June 6

July 11

August 8

September 5

October 10

November 7

December 12

We shall first denote each day of the week by numbers from 0 to 6, starting from Sunday:

Sunday 0

Monday 1

Tuesday 2

Wednesday 3

Thursday 4

Friday 5

Saturday 6

Notice how Tuesday is Twosday in the table too!

Calendars exactly repeat every 400 years. Let us designate “anchor” days to each century.

1800 - 1899 Friday

1900 - 1999 Wednesday

2000 - 2099 Tuesday

2100 - 2199 Sunday

Conway assumes that these anchor days also reappear every 400 years. From 1400-1499 the anchor day is Friday, for years 1500 to 1599 it is Wednesday and so on.

Note that these rules apply to the Gregorian calendar only, not the Julian calendar. Did you know that Britain lost 11 days in September 1752 when it switched from the Julian to the Gregorian calendar? Jeez, imagine waking up and looking at the calendar to see that 11 days have passed! That’s a scary thought.

Now let’s figure out what day of the week July 7 1967 fell on:

  • What is the quotient when the number formed by the last two digits of the year is divided by 12?

The last two digits are 67. 67/12 gives a quotient of 5.

  • What is the remainder when the number formed by the last two digits of the year is divided by 12?

The last two digits are 67. 67/12 leaves a remainder of 7.

  • Divide this result by 4. What is the integral quotient?

7/4 leaves a remainder of 3 and a quotient of 1.

  • What is the anchor day of the century’s number?

The anchor day for the twentieth century is Wednesday. Its number is 3.

  • Add all these answers up.


  • What is the remainder when this result is divided by 7?

16/7 leaves a remainder of 2.

2 corresponds to a Tuesday. The doomsday of 1967 was therefore a Tuesday.

Now it’s quite simple to compute the day the given date fell on. From the doomsday table we can see that July 11 was a doomsday. July 11 1967 therefore fell on a Tuesday. This means that July 7 1967 fell on a Friday.

And we’re done! Go ahead, throw a Twosday party and watch your friends be amazed.

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