# Finding the Day of the Week

When I was growing up way back in the '90s, I used to watch a show called "Figure It Out." It was a sort of game show that would feature a kid with a special talent, and the celebrity panelists would have to, well, figure out what it was. On one episode, there was a boy who could tell you the day of the week that a given date fell on. I remember being absolutely amazed. I had no idea how he did it.

It turns out that it's quite easy to determine the day of the week from a given date. In fact, you can do it in your head with a little mental math and a few simple memorizations. Read on for a quick background on the algorithm. And don't worry: we'll get to the formula soon enough.

## Overview of the Doomsday Algorithm

The method outlined here is called the Doomsday Algorithm. It's not the only way to find the day of the week of a given date, but it lends itself easily to memorization. Invented by mathematician John Horton Conway (born 1937), the algorithm takes advantage of the fact that in any given year, there are certain easy-to-remember dates that all happen to fall on the same day of the week. Conway referred to this special day of the week as the Doomsday. There are other methods which, with some more difficult memorization and twisting of numbers, will lead you directly to your answer. However, by finding a year's Doomsday first, you can easily find the weekday of any date in that year.

Besides the formula itself, all you need to memorize is a single Doomsday date for each month. Luckily, they're quite easy, and that's why the algorithm was based around them. In any given year, 4/4, 6/6, 8/8, 10/10, and 12/12 fall on the same day of the week -- the Doomsday. Conveniently, the last day of February (the 28th in a regular year or the 29th if it's a leap year) also falls on this day. It happens that 5/9, 9/5, 7/11, and 11/7 fall on the Doomsday as well. For those familiar with the 7-Eleven convenience store chain, a handy mnemonic for these months is, "I work from 9 to 5 at the 7-11." Or you could simply remember that these are the odd-numbered months after March.

What about the remaining months, January and March? Like February, their Doomsday dates hinge on the leap day as well. If it's a common year (a non-leap year), then January 3 is the Doomsday; if it is a leap year, then January 4 is the Doomsday. (You might remember that for three years, it's the January 3rd, and every fourth year, it's January 4th.) As for March, the last day of February is usually considered March's Doomsday date as "March 0." (Keep in mind that a leap year is a year that is divisible by four: a year when we have the Summer Olympics and the US Presidential election. The exception to the rule is that years divisible by 100 but not 400 are common years: thus, 1700, 1800, and 1900 were not leap years, but 1600 and 2000 were.)

So, to review, these are the only Doomsday dates you need to remember:

Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Date | 3/4 | 28/29 | 0 | 4 | 9 | 6 | 11 | 8 | 5 | 10 | 7 | 12 |

There's one more little thing you need to know. Each century has an "anchor day" that you will need to memorize as follows. Luckily, there are only four that you need to remember because the anchor days repeat every four centuries.

Centuries | Anchor day |

...1700s, 2100s, 2500s... | Sunday |

...1800s, 2200s, 2600s... | Friday |

...1900s, 2300s, 2700s... | Wednesday (remember: "We-in-dis-day") |

...2000s, 2400s, 2800s... | Tuesday (remember: "Y-Tue-K") |

The two you will probably need to know the most are those for the 1900s (1900-1999) and the 2000s (2000-2099). You can remember that the anchor day for the 1900s is "We-in-dis-day" because most of us were born during that century. Likewise, you can remember that the anchor day for the 2000s is Tuesday with the mnemonic "Y-Tue-K" (remember Y2K?).

## The Math

Yes, you do need to use a little math in order to do this. Here's the formula, broken down step-by-step. In this example, we'll find which day of the week July 13, 1989 fell on.

- Divide the last two digits of the year by 12. 89 divided by 12 is 7, with 5 left over.
- Find how many 4's go into the remainder evenly. 5 divided by 4 is 1, with a remainder that we ignore.
- Add these three numbers. 7 + 5 + 1 = 13.
- Divide the sum by 7 and take the remainder. 13 divided by 7 is 1, with a remainder of 6.
- Add the remainder to the century's anchor day to find the year's Doomsday. Wendesday + 6 = Tuesday.
- Use the month's Doomsday to find the day you need. July's Doomsday is 7/11, a Tuesday, so July 13 is a Thursday.

Let's try another example by verifying that today, June 29, 2016, is a Wednesday.

- 16 divided by 12 is 1, with a remainder of 4.
- 4 goes into the remainder 1 time.
- 1 + 4 + 1 = 6.
- 6 divided by 7 is 0, with a remainder of 6.
- Tuesday + 6 = Monday. Therefore, 2016's Doomsday is Monday.
- Because June 6 is a Monday, then June 29, 23 days later, is a Wednesday.

Now it's your turn! Find the day of the week for the following dates. (If you need to check your answers, you can use the calculator at the bottom of the page.)

- The first Earth Day was observed on April 22, 1970.
- Napoleon was defeated at Waterloo on June 18, 1815.
- James Joyce's novel Ulysses takes place on June 16, 1904.
- The United States declared independence from Great Britain on July 4, 1776.
- A total solar eclipse will occur over the contintental United States on August 21, 2017.
- The first humans landed on the moon on July 20, 1969.
- John Horton Conway, the inventor of the Doomsday algorithm, was born on December 26, 1937 -- a Doomsday, as it turns out.
- Millard Fillmore, the 13th President of the United States, was born on January 7, 1800. (Careful: was it a leap year?)
- Gold was first discovered in California on January 24, 1849.
- The Second Continental Congress met in Lancaster, Pennsylvania, on September 27, 1777, making that city the capital of the United States for one day.
- The muntiny on the Bounty was led by Fletcher Christian, who later named his son Thursday October Christian because he was born on the second Thursday of October 1790. What date was this?
- Thursday October Christian died on April 21, 1831. What day of the week was it?
- North Korea's Juche calendar, introduced in 1997, counts the years since April 15, 1912, when "Eternal President" Kim Il-sung was born. What day was it?
- I just found an invitation to a party that I once had. The date of the party is given as Friday, July 11. The year isn't stated, but I know it was between 2001 and 2004. What year was it?
- Jorn Barger devised the word "weblog" on December 17, 1997.
- On December 13, 1795, a meteorite landed near the hamlet of Wold Newton in Yorkshire, England.
- The film Minority Report gives the date April 22, 2054, as a Tuesday. Is this correct?
- In the movie Heartbreak Ridge, a calendar shows July 21, 1983, as a Sunday. Is this right?
- In the movie Forrest Gump, the date March 22, 1982, is given as a Saturday. Is this correct?
- In the film Hairspray, a news broadcast gives the date as Friday, June 5, 1962. Could this be correct?
- In the film The Hudsucker Proxy, a newspaper dated Monday, December 19, 1958, is shown. Is this the right day of the week?
- The movie The Wedding Planner gives June 6, 2001, as a Saturday. Is this right?
- In the movie The Terminator, a character says that May 12, 1984, is a Thursday. Is he right?
- In O Brother, Where Art Thou?, a calendar shows that July 1937 begins on a Tuesday. Is it correct?
- In Remember the Titans, the first day of school is given as September 4, 1971. Could there have been school on this date?
- In My Dog Skip, a scene takes place in a school on September 27, 1942. Could this have been a school day?
- In The Thirteenth Floor, a newspaper is shown that is dated Monday, June 21, 2024.
- A pivotal scene in the film Back to the Future takes place at a high school dance on November 12, 1955. Would it have been on a Saturday, as stated?
- In American Gun, a newspaper gives the date as Thursday, March 18, 1988.
- In Superman Returns, a newspaper is dated Friday, September 29, 2006.
- In Raging Bull, a dance is said to take place on Saturday, August 6, 1941.
- The beginning of The Return of the Living Dead takes place on Friday, July 3, 1984.
- In Chances Are, a newspaper is dated Sunday, August 7, 1989.
- In Johnny Mnemonic, a date is given as Thursday, January 17, 2021.
- In Higher Learning, an event is advertised to take place on Saturday, November 15, 1995.
- In Phantom of the Paradise, a character mentions that he wanted to commit suicide on Saturday, November 19, 1953.
- In The Shootist, a newspaper gives the date as Monday, January 22, 1901.

## A History Lesson: The Julian Calendar

By now, you should be able to find the day of the week for any date just by using some mental math. Congratulations!

There's a small catch, though. Your skills apply only to the Gregorian calendar, the calendar that we currently use. The Gregorian calendar was introduced by the Catholic Church in 1582. Before that, the less accurate Julian calendar was used. The only real difference between the Julian and Gregorian calendars is that they have different leap year rules (all Julian years divisible by 4 are leap years) and differ by an ever-increasing number of days. Julian and Gregorian years usually have different Doomsdays.

Finding a Doomsday in itself is relatively simple; the trick is to remember whether a given date is Julian or Gregorian. If you're working out the date that some event happened, it's important to consider where it happened. That's because not every country switched to the Gregorian calendar at the same time; in fact, only some Catholic countries adopted the new calendar right away. Great Britain and its colonies, which followed the Church of England, didn't make the switch until 1752. The last country to switch from the Julian to the Gregorian calendar was Turkey in 1926.

It's important to remember that, in the study of history, Julian dates are often not converted to their Gregorian equivalents. If a distinction between the calendars needs to be made, Julian dates are referred to as Old Style (O.S.) dates, and Gregorian dates are called New Style (N.S.) dates. Contemporaries of English thinker Isaac Newton would have said that he was born on December 25, 1642, and died on March 20, 1727. However, today we might say that he was born on January 4, 1643, and died on March 31, 1727. Though the dates are different, you'll be able to find that the corresponding days of the week are the same. Incidentally, the Julian and Gregorian calendars have the same days of the week at the same times; only the dates are different.

To find the Julian Doomsday, the first four steps are basically the same. When you add up the three numbers (the quotient of the last two digits of the year and 12, the remainder of that, and the number of fours in the remainder), subtract the century digits from the sum, and add that to Sunday. Here's an example for April 23, 1564:

- 64 / 12 = 5, remainder 4.
- 4 / 4 = 1.
- 5 + 4 + 1 = 10.
- Sunday + 10 - 15 = Sunday - 5 = Tuesday.
- April 4 is a Tuesday, so April 23, 19 days later, is a Sunday.

Julian dates are easy because you don't need to remember anchor days or the more complicated leap year rule. So, you should be able to try these:

- Verify that October 4, 1582 (Julian), and October 15, 1582 (Gregorian), occurred on consecutive days of the week.
- Verify that September 2, 1752 (Julian), and September 14, 1752 (Gregorian), occurred on consecutive days of the week.
- The Battle of Hastings occurred on October 14, 1066.
- In 1492, Columbus sailed the ocean blue. He made his first landing in the New World on October 12 of that year.
- English poet and playwright William Shakespeare and Spanish author Miguel de Cervantes both died on April 23, 1616, but not on the same day of the week. Find the day of the week that each man died.

## Advanced Techniques

This section is optional. It contains tips on finding Doomsdays more quickly if you're able to memorize more things.

### Complete List of Doomsdays

If you find it hard to remember some of the handy Doomsdays for each month, you might benefit from this complete list.

Month | Doomsdays |

January (common) | 3, 10, 17, 24, 31 |

January (leap) | 4, 11, 18, 25 |

February (common) | 7, 14, 21, 28 |

February (leap) | 1, 8, 15, 22, 29 |

March | "0", 7, 14, 21, 28 |

April | 4, 11, 18, 25 |

May | 2, 9, 16, 23, 30 |

June | 6, 13, 20, 27 |

July | 4, 11, 18, 25 |

August | 1, 8, 15, 22, 29 |

September | 5, 12, 19, 26 |

October | 3, 10, 17, 24, 31 |

November | 7, 14, 21, 28 |

December | 5, 12, 19, 26 |

Note that some other easy-to-remember Doomsdays are Valentine's Day (February 14), U.S. Independence Day (July 4), Halloween (October 31), and Boxing Day (December 26). Note, however, that February 14 is a Doomsday only in common years.

### Memorizing Years

Once you've done Doomsday calculations enough, you might start to memorize the Doomsdays for certain years. Early on in my study of the Doomsday method, I memorized that 1969's Doomsday was Friday. Why? I don't know; I just did. But it's helped me. Consider the satisfaction I got when someone "challenged" me to work out June 6, 1969, and I was able to tell them the answer immediately.

It may be convienient to know some of the Gregorian years whose Doomsdays are the same as the anchor day for their century. For example, 1945's Doomsday is Wednesday, just as 1900's was. In the Gregorian calendar, the following years of a century have the same Doomsday as the anchor day for their century: '00, '06, '17, '23, '28, '34, '45, '51, '56, '62, '73, '79, '84, '90.

Incidentally, I've discovered that intervals between these years follows a pattern: 6, 11, 6, 5, 6, 11, 6, 5, 6, 11, 6, 5, 6. Using the 1900s as an example, the pattern actually continues into the next century: ...1984, 1990, 2001, 2007, 2012, 2018... This pattern breaks after 2100, probably because it will not be a leap year. It would be interesting to investigate whether any Doomsday (not just those that are anchor days) can be found as a member of a sequence, and whether this extends to the Julian calendar as well. I think the answer is yes to both, but I don't feel like crunching the numbers. Maybe someone reading this can prove it one way or the other.

### Advance of the Doomsdays

You may have observed in the past that if a certain date falls on a Monday this year, it will be a Tuesday next year (that is, if the next year is a common year). The Doomsday advances one day each year, except in the case of leap years, when it advances two days. If you've memorized the Doomsdays for certain years, you can use this to your advantage. For example, you don't need to do the math for 1999 if you know that 2000's Doomsday is a Tuesday. Because 2000 was a leap year, 1999's Doomsday was two days before it and was therefore Sunday. Likewise, you know that 1900's Doomsday was Wednesday, so 1899's must have been Tuesday, and 1898's was Monday.

### Anchor Days for Future Centuries

If you're trying your newly found skills at a party, there's always going to be the snarky person in the bunch who asks you to find the day of the week for January 31, 6732. How are you supposed to find the anchor date for the 6700s? Recall that the anchor dates for the Gregorian calendar repeat every four centuries. (As such, the Gregorian calendar repeats every 400 years. Just as June 29, 2016, is a Wednesday, so too was June 29, 1616.) If you divide the century digits (the first two digits in four-digit years) by 4, the remainder determines the anchor day. When you 17 by 4, the remainder is 1. Recall that the anchor day for the 1700s is Sunday. When the remainder is 2, the anchor day is Friday. When it's 3, it's Wednesday, and when the remainder is 0 (as in the 2000s), the anchor day is Tuesday. Now for the 6700s: 68 is evenly divisible by 4, so the remainder of 67 divided by 4 is 3. Thus, the anchor day for the 6700s is Wednesday. Now, you should be able to find the day of the week for January 31, 6732. Here's another challenge: January 1, 10,000. (For 5-digit years, the century digits are the first three.)

## Date Calculator

This calculator uses the Doomsday method. You must have JavaScript enabled in your browser in order for the calculator to work. You can use it to check your answers (but no cheating!). Enter numeric values for the month, day, and year, then click "Find day of week." The calculator accepts all dates from the years 1 to 9999. However, Gregorian dates prior to October 15, 1582, will be entered as Julian dates instead.

Here's a more easily accessible version of this calculator, better for bookmarking and frequent use.

## More Information

According to a source who has spoken to Dr. Conway, he devised the Doomsday Algorithm after a colleage showed him a day-of-the-week algorithm by Lewis Carroll and challenged him to simplify it. Conway first published the algorithm in his article, "Tomorrow is the Day After Doomsday," Eureka 36 (1973): 28-31. It is described briefly in a book that he co-authored with Richard K. Guy and Elwyn R. Berkelamp called Winning Ways for Your Mathematical Plays, Vol. 2, London: Academic Press, 1982. Since then, the algorithm appears to have been simplified so that the "handy Doomsdays" for the months are easier to remember. Conway now teaches his algorithm using the hand as a mnemonic. He himself can perform the necessary calculations in under two seconds. His secret? Besides being a mathematician, he has a program on his computer that gives him random dates to try whenever he logs on.

These resources have proven useful in the research for this page. They might also be useful for you if you've found this article just too confusing.

- Doomsday rule on Wikipedia
- Doomsday Algorithm by Rudy Limeback
- What Is the Day of the Week, Given Any Date? by Bill Jefferys
- Julian and Civil Date Calculator by Numerical Recipies Software

Last updated January 2, 2007.

Created December 29, 2006.