How does John Horton Conway figure out what day of the week it is?

He uses his Doomsday Algorithm, explained by Rudy Limeback.

(If that link disappears, here is the raw text of the page: Doomsday Algorithm The Doomsday algorithm gives you the day of the week for any date (and you can do it in your head). Revised 2001-06-03 with examples for 2001. February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links Use the handy navigation bar at the right to find your way around. To learn the Doomsday algorithm, start with February 28/29, and continue until you have completed Even months and Odd months. For help while you are learning, see the 2001 Calendar which shows Doomsday in each month. After you can do the Doomsday algorithm for this year, you can learn it for any year in Other Years and then any year in Other centuries. A great way to remember the Doomsday algorithm is The Hand. Finally, see Origins and Links for additional information. Have fun! -------------------------------------------------------------------------------- February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links February 28/29 To start the Doomsday algorithm for any year, you first need to know the Doomsday for that year. Doomsday is February 28/29, or, in other words, Doomsday is always the last day of February. In normal years, Doomsday is February 28, and in leap years, Doomsday is February 29. In 2001, the last day of February is Wednesday the 28th. Once you know Doomsday, it's pretty easy to get the day of the week for any day in February. It's done by adding and subtracting, using multiples of 7, and you should be comfortable doing this in your head -- or else the rest of the algorithm will give you trouble! Luckily, most people, through practice or whatever, are good at mentally picturing a month if they have something to anchor it on, and for February it's the 28th in normal years, the 29th in leap years. Example: what is this year's Valentine's Day, February 14th? Answer: Doomsday 2001 is Wednesday the 28th of February. So one week earlier, the 21st is also a Wednesday. Another week earlier is Wednesday the 14th. So Valentine's Day 2001 was a Wednesday. Example: what is this year's Groundhog day, February 2nd? Answer: Doomsday 2001 was Wednesday the 28th... Wednesday the 21st... Wednesday the 14th... Wednesday the 7th. Now you have to get from the 7th to the 2nd. You could back up five days from Wednesday the 7th to reach Friday the 2nd, but that's pretty awkward. It may be easier to go forward two days from Wednesday the 7th to Friday the 9th, then back a week to Friday the 2nd; this illustrates how "5 off" is the same as "2 on" if all you want is the day of the week. So Groundhog Day, February 2nd 2001, is a Friday. -------------------------------------------------------------------------------- February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links Even Months Now we move on to the other months of the year, beginning with the even months. These are the 4th, 6th, 8th, 10th, and 12th, i.e. April, June, August, October, and December. For even months, the Nth of the month is a Doomsday (i.e. the same day of the week as the last day in February). This is a weird and wonderful coincidence, and it's so easy to remember: April 4th is a Doomsday June 6th is a Doomsday August 8th is a Doomsday October 10th is a Doomsday December 12th is a Doomsday Neat, eh? Now all you have to do is work your way around any even month based on its Doomsday. Example: what is this year's Christmas Day, December 25th? Answer: Doomsday 2001 is a Wednesday. December (even) 12th is a Wednesday, so two weeks later, December 26th is a Wednesday, so Christmas 2001 is "1 off", i.e. Tuesday December 25th. Easy! In fact, after you do the Doomsday algorithm often enough, you just start remembering things like Christmas is always the day before Doomsday. Example: what is this year's Canadian Thanksgiving Day, the second Monday in October? Answer: Doomsday 2001 is Wednesday. October (even) 10th is a Wednesday. Go "2 off" to Monday October 8th. This has to be the second Monday of October, since the 1st would be the first. So the Canadian Thanksgiving in 2001 is on Monday October 8th. -------------------------------------------------------------------------------- February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links Odd Months Now the odd months -- the 1st, 3rd, 5th, 7th, 9th, and 11th, i.e. January, March, May, July, September, and November. Skip January and March for a moment. Consider the following mnemonic phrase: I work from 9-5 at the 7-11 which means: for the 9th month Doomsday is the 5th for the 5th month Doomsday is the 9th for the 7th month Doomsday is the 11th for the 11th month Doomsday is the 7th This gives you Doomsday for May, July, September, and November. Now just work your way around again within the month using Doomsday for that month. Example: what day is this year's July 4th? Answer: Doomsday 2001 is Wednesday. July (7th month) 11th is a Wednesday. So one week earlier, July 4th is also a Wednesday. In fact, after you do the Doomsday algorithm often enough, you just start remembering things like July 4th is always a Doomsday. Example: what is this year's Labour Day, the first Monday of September? Answer: Doomsday 2001 is Wednesday. September (9th month) 5th is a Wednesday. Two days off is Monday, September 3rd. This has to be the first Monday of September, because a week earlier puts you back into August. So Labour Day in 2001, the first Monday of September, is the 3rd. Now March. Doomsday, the last day of February, could also be called "March the 0th". You might have to think about that for a moment; it should make sense when you (mentally) confirm that the 7th of March is exactly one week after the last day of February (no matter whether it's the 28th or 29th). So March 7th is always a Doomsday too. Example: what day is this year's St. Patrick's Day, March 17th? Answer: Doomsday 2001 is Wednesday. March 7th is a Wednesday. March 14th is a Wednesday. Three days on is Saturday, March 17th, 2001. Finally, we have to be able to do January. The anchor for January is "usually" the last day of January, which is a Doomsday. Thus, if Doomsday is a Friday, for example, then January 31st would also be a Friday. The reason it's "usually" is because in leap years, when Doomsday is February the 29th, you have to use "January 32nd." This accounts for the extra leap day to make January come out right. Example: what was New Year's Day (January 1st) this year? Answer: Doomsday 2001 is Wednesday, and 2001 is a not leap year, so January 31st was a Wednesday. Subtract 28: January 3rd was a Wednesday. Go two off: January 1st was a Monday. New Year's Eve 2000 was a Sunday, remember? Neither do I. ;o) -------------------------------------------------------------------------------- February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links 2001 Calendar If you've worked your way through the rules but have trouble remembering them, it may help to see them in calendar form. Below is a calendar for 2001 with all the Doomsdays highlighted. Tech note: the Doomsdays in the calendar below are highlighted in bold red font. If your browser does not support coloured fonts, the bold should still work. Tech note: On a Windows computer, if it doesn't look right, select Courier New or Lucida Sans Typewriter for the Fixed-width font. Some fixed-width fonts have characters that differ in width between normal and bold face. Doomsday Calendar for 2001 1.Jan(31st) 2.Feb(28th) Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 10 7 8 9 10 11 12 13 11 12 13 14 15 16 17 14 15 16 17 18 19 20 18 19 20 21 22 23 24 21 22 23 24 25 26 27 25 26 27 28 28 29 30 31 3.Mar(7th) 4.Apr(4th) Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 1 2 3 4 5 6 7 4 5 6 7 8 9 10 8 9 10 11 12 13 14 11 12 13 14 15 16 17 15 16 17 18 19 20 21 18 19 20 21 22 23 24 22 23 24 25 26 27 28 25 26 27 28 29 30 31 29 30 5.May(9th) 6.Jun(6th) Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 1 2 6 7 8 9 10 11 12 3 4 5 6 7 8 9 13 14 15 16 17 18 19 10 11 12 13 14 15 16 20 21 22 23 24 25 26 17 18 19 20 21 22 23 27 28 29 30 31 24 25 26 27 28 29 30 7.Jul(11th) 8.Aug(8th) Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 1 2 3 4 8 9 10 11 12 13 14 5 6 7 8 9 10 11 15 16 17 18 19 20 21 12 13 14 15 16 17 18 22 23 24 25 26 27 28 19 20 21 22 23 24 25 29 30 31 26 27 28 29 30 31 9.Sep(5th) 10.Oct(10th) Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 1 2 3 4 5 6 2 3 4 5 6 7 8 7 8 9 10 11 12 13 9 10 11 12 13 14 15 14 15 16 17 18 19 20 16 17 18 19 20 21 22 21 22 23 24 25 26 27 23 24 25 26 27 28 29 28 29 30 31 30 11.Nov(7th) 12.Dec(12th) Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 1 4 5 6 7 8 9 10 2 3 4 5 6 7 8 11 12 13 14 15 16 17 9 10 11 12 13 14 15 18 19 20 21 22 23 24 16 17 18 19 20 21 22 25 26 27 28 29 30 23 24 25 26 27 28 29 30 31 Looking for other years? Here are older versions of my Doomsday Calendar. The first two are GIFs, the others are HTML tables like the one above. 1995 1996 1997 1998 1999 2000 2001 Tech note: These links will open in a second browser window, suitable for printing. -------------------------------------------------------------------------------- February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links Other Years Okay, you can do 2001. What about other years? If Doomsday is Wednesday in 2001, what was it last year, in 2000? Well, you might want to check in a calendar, but let me tell you it was a Tuesday. Doomsday advances by one day each year because 365 divided by 7 leaves 1 remainder. Doomsday advances two days each leap year, and we'll come back to more on this in a moment. Example: what day was New Year's Eve last year? Answer: Start with Doomsday for last year -- Doomsday 2000 is Tuesday. December (even) 12th is Tuesday, and so is the 26th. Five days on, December 31st, 2000, is Sunday. Or, if you're starting to get the hang of this, instead of "5 on Tuesday = Sunday" you'll immediately think "2 off Tuesday = Sunday". So New Year's Eve last Year, 2000, was Sunday. Example: what was New Year's Eve, 1999? Answer: Let's try this one by going backwards from January 1, 2000. Doomsday 2000 was Tuesday, and 2000 is a leap year (or, more correctly, it was), so we have to use January "32nd" as our Tuesday. Then January 32-28=4th was a Tuesday, so January 1, 2000, is "3 off Tuesday" -- and remember, if you find it easier to go forwards, then instead of "3 off Tuesday = Saturday" you'll think "4 on Tuesday = Saturday." Then New Year's Eve 1999 has to be the day before -- Friday. (Remember Friday December 31, 1999? the day the computers were all going to die with dreaded Y2K disease? Neither do I.) In order to do the Doomsday algorithm for any year in the 1900's (when most of us were born -- you usually want to be able to do birthdays for people), you have to memorize the fact that Doomsday for 1900 is a Wednesday. Then you do a calculation based on the number of years since 1900. First, look at the following chart of Doomsdays: Sun Mon Tue Wed Thu Fri Sat 1900 1901 1902 1903 ---- 1904 1905 1906 1907 ---- 1908 1909 1910 1911 ---- 1912 1913 1914 1915 ---- 1916 1917 1918 1919 ---- 1920 1921 1922 1923 ---- 1924 1925 1926 1927 ---- 1928 1929 1930 1931 ---- 1932 1933 1934 1935 ---- 1936 1937 1938 1939 ---- 1940 1941 1942 1943 ---- 1944 1945 1946 1947 ---- 1948 1994 1950 1951 ---- 1952 1953 1954 1955 ---- 1956 1957 1958 1959 ---- 1960 1961 1962 1963 ---- 1964 1965 1966 1967 ---- 1968 1969 1970 1971 ---- 1972 1973 1974 1975 ---- 1976 1977 1978 1979 ---- 1980 1981 1982 1983 ---- 1984 1985 1986 1987 ---- 1988 1989 1990 1991 ---- 1992 1993 1994 1995 ---- 1996 1997 1998 1999 ---- 2000 2001 2002 2003 ---- 2004 2005 2006 2007 ---- 2008 Notice that Doomsday 1900 was Wednesday. This is the anchor for all the years in the 1900's. (Notice also that 1900 is not a leap year, so Doomsday 1900 is February 28.) How do you remember 1900=Wednesday? Dr. Conway suggests the mnemonic "We-in-dis-day", indicative of the fact that most of us were born in the 1900's. Now every twelve years, Doomsday advances by one. Check for yourself. In the chart above, pick a year and look ahead twelve years -- down two rows and over one day. This leads to the following rule... For any year 19YY, use the YY part of the year and then calculate Step 1. the number of 12's in the YY part of the year Step 2. the remainder of step 1 Step 3. the number of 4's in the remainder of step 1 Feel free to throw out multiples of 7 along the way if you find this easy to do. Now add the result to 1900=Wednesday to get the Doomsday for that year. Example: what is Doomsday 1929? Answer: Step 1: 29 divided by 12 is 2 Step 2: ... remainder 5. Step 3: 5 divided by 4 is 1 Adding these up, we get 8, and we can throw out a multiple of 7 to get 1. This has to be added to 1900=Wednesday, so Doomsday for 1929 is Thursday. Example: what is Doomsday 1999? Answer: Step 1: 99 divided by 12 is 8 Step 2: ... remainder 3. Step 3: and of course 3 divided by 4 is 0 Adding these up, we get 11 i.e. 4. This has to be added to 1900=Wednesday, so Doomsday for 1999 is Sunday. Which we knew, but it's reassuring to confirm that the rule works. You should now be able to do any day in the 1900's in your head. Let's do a couple more examples... Example: what was November 27, 1982 (see Origins below)? Answer: 82 / 12 = 6 ... remainder 10 ... 10 / 4 = 2 ... 6 + 10 + 2 = 18 which is 4 days to be added to Wednesday (for 1900) ... so Doomsday 1982 was Sunday ... November (11th month) 7th is Sunday, 28th is Sunday ... November 27th 1982 was a Saturday (as all regular listeners of Q&Q know). Example: what was July 20, 1969 (the date of the first landing of humans on the Moon)? Answer: 69 / 12 = 5 ... remainder 9 ... 9 / 4 = 2 ... 5 + 9 + 2 = 16 which is 2 days to be added to Wednesday (for 1900) ... so Doomsday 1969 was Friday ... July (7th month) 11th is Friday, 18th is Friday ... July 20th 1969 was a Sunday -------------------------------------------------------------------------------- February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links Other Centuries In Other Years, we learned that Doomsday for 1900 was Wednesday. What is Doomsday for other centuries? Well, the 2000's should be easy, since Doomsday for this year, 2001, is Wednesday, and earlier we worked out that Doomsday for 2000 is Tuesday. Plus, there's the chart we were looking at earlier, the last few years of which look like this -- Sun Mon Tue Wed Thu Fri Sat 1999 ---- 2000 2001 2002 2003 ---- 2004 2005 2006 2007 ---- 2008 2009 2010 2011 ---- 2012 2013 2014 2015 Remember, each year Doomsday advances by one, except leap years, when it advances by two. But the further ahead you go, the harder this gets. Let's construct another chart of years, extending backwards and forwards from the previous chart, except we want it to cover a bigger range of years. Let's show only those rows with a century year: Sun Mon Tue Wed Thu Fri Sat 1497 1498 1499 1500 1501 1502 1503 1599 1600 1601 1602 1603 1700 1701 1702 1703 1704 1705 1796 1797 1798 1799 1800 1801 1897 1898 1899 1900 1901 1902 1903 1999 2000 2001 2002 2003 2100 2101 2102 2103 2104 2105 2196 2197 2198 2199 2200 2201 2297 2298 2299 2300 2301 2302 2303 2399 2400 2401 2402 2403 2500 2501 2502 2503 2504 2505 Examine this chart carefully, until you convince yourself that it is behaving exactly as you would expect for leap century years and non-leap century years. Remember the rule for leap years: a year divisible by 4 is a leap year except if it's divisible by 100, when it's not a leap year except if it's divisible by 400, when it is a leap year Each normal year advances Doomsday by one day. Each leap year advances Doomsday by two days. Now look at the century years again: Sun Mon Tue Wed Thu Fri Sat 1700 1600 1500 2100 2000 1900 1800 2500 2400 2300 2200 What's the best way to memorize century Doomsdays? I'm not sure. Here's what I use. Notice that century Doomsdays fall only on "Sun-Tue-Wed-Fri". I say "Son to wed Friday", thinking of my own son, and how pleased I would be if he were indeed getting married (well, maybe not this Friday!). Combine this with Dr. Conway's "We-in-dis-day" for 1900=Wednesday, and I can reconstruct the chart mentally. The tricky part is that the years go right to left in each row. The easy part is that if you can get just the one row (the one with 1900=Wednesday in it), the other years are the same Doomsday, plus or minus 400 years. Example: what day of the week is Canada's 300th birthday, July 1, 2167? Answer (using the same YY method as before): Step 1: 67 divided by 12 is 5 Step 2: ... remainder 7. Step 3: and 7 divided by 4 is 1 Adding these up, we get 13 i.e. 6. This has to be added to 2100=Sunday, so Doomsday for 2167 is Saturday. Now July(7) 11th is a Saturday, so July 1, 2167, is a Wednesday. Messy, isn't it? You are ready for "The hand." -------------------------------------------------------------------------------- February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links The Hand Dr. Conway now teaches the Doomsday algorithm, complete with Century adjustment, using a very simple visual aid -- your hand. _____ ____/ ___)____ <-- 1 _______) <-- 2 ________) <-- 3 ____ _______) <-- 4 \________) <-- 5 1 -- Doomsday Difference 2 -- Century Day 3 -- number of DOZENS 4 -- remainder 5 -- number of 4s in remainder The Doomsday Difference is the difference between the required date and a nearby Doomsday, recorded as so many days "on" (i.e. to be added) or "off" (subtracted) from that Doomsday. Recall a couple of the examples we've covered -- July 4th is always a Doomsday, i.e. the Doomsday Difference is 0 Christmas, December 25th, is always "1 off" Doomsday Be careful with the Doomsday Difference for dates in January and February. (Thanks to Bob Goddard for pointing this out.) In a leap year, you must subtract 1 from the Doomsday Difference for January and February dates -- Valentine's Day, February 14, is always "1 off" Doomsday in leap years because Doomsday is February 29th; in ordinary years, the Doomsday Difference for Valentine's Day is 0 Groundhog Day, February 2, is only "1 on" Doomsday in leap years because Doomsday is February 29th; in ordinary years, the Doomsday Difference for Groundhog Day is "2 on" New Year's Day, January 1, is always "3 off" Doomsday in leap years because Doomsday is the "32nd" of January; in ordinary years, the Doomsday Difference for New Year's Day is "2 off" The Century day for the 1900's is Wednesday ("We-in-dis-day"). Here, in his own description, is how Dr. Conway would calculate the day of the week for Pearl Harbor Day, December 7th, 1941. The various numbers to be attached to the hand are (reading from the thumb): "2 on" (for Dec 7) "Wednesday" (for 1900) "3 dozen" (getting us to 1936) "5 remainder" (number of years after 1936) "and 1" (since one of those 5 years was a leap year). Don't start adding these up until you've formed them all, and then proceed as far as possible by cancelling first 14s, then 7s. To make sure we haven't forgotten them, let's say: " 2, Wed, 3, 5, 1 " (touching the appropriate digits as we do so), and then cancel that 2+5=7 (and folding down the thumb and ring finger) to get "Wed, 3 and 1 " = Wed + 4 = Sun. I also advise use of my mnemonic names for weekdays, namely NUNday, ONEday, TWOSday, tWEBLESday, FOURSday, FIVEday, SIXurday, SE'ENday which can be pronounced so that they both sound like numbers and weekdays, and so help you do the addition, for example " TREBLES, 3 and 1 = SEVENday " (Sunday) in the above case. The nice part about Dr. Conway's Hand is that you do the calculations in the same order you usually say the date -- month/day, century, year. For example, for August 4, 1997, you do August 4, then 19, then 97. Example: what day is August 4, 1997? Answer: _____ ____/ ___)____ <-- 4 off (Aug 4) _______) <-- Wed (for 1900) ________) <-- 8 DOZENS ____ _______) <-- remainder 1 \________) <-- and 0 which is "4 off, tWEBLESday, 8, 1" or -4+3+8+1 which is 1, so August 4, 1997 is a Monday. Finally, one last warning: Watch out for Gregorian versus Julian dates. The Doomsday algorithm described up to this point covers only Gregorian dates. Example: what day was September 14, 1752? Answer: _____ ____/ ___)____ <-- 2 on (Sep 14) _______) <-- Sun (for 1700) ________) <-- 4 DOZENS ____ _______) <-- remainder 4 \________) <-- and 1 which is "2, Sun, 4, 4, 1" and you can throw out the 2, a 4 and the 1 to get 4 on Sunday, so September 14, 1752 was a Thursday. That was a trick question, sort of. September 14, 1752 was the first day of the Gregorian calendar in England and its colonies. (The Gregorian calendar was originally adopted in parts of Europe in 1583). So September 1752 actually looked like this: Sun Mon Tue Wed Thu Fri Sat 1 2 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Neat, eh? -------------------------------------------------------------------------------- February 28/29 Even Months Odd Months 2001 Calendar Other Years Other Centuries The Hand Origins Links Origins The Doomsday algorithm was created by John Horton Conway, an eminent mathematician, perhaps best known as the inventor of the Game of Life (see Link below). I first heard about the Doomsday algorithm on November 27, 1982, on a CBC Radio program called Quirks and Quarks. Dr. Conway was interviewed by Jay Ingram, who's now a co-host of the television program @discovery.ca. In those days Quirks and Quarks occasionally made typed transcripts available, and I sent away for one. Dr. Conway had just published a book that year (co-authored by Elwyn R. Berlekamp and Richard K. Guy) called Winning Ways For Your Mathematical Plays, Volume 2: Games in Particular, Academic Press, London, 1982, ISBN 01-12-091102-7. The Doomsday algorithm is on pages 795-797, and the rest of the book is mainly about games, with substantial emphasis on their mathematical underpinnings. In the original version of the Doomsday algorithm, the odd months were a bit harder to remember than "I work from 9-5 at the 7-11." You had to remember if the odd month was a long month or a short month. The 3rd, 5th, and 7th months are "long" because March, May, and July have 31 days, while the 9th and 11th months are "short" because September and November have only 30 days. You could remember "30 days hath September... and November" (but be careful because this old rhyme includes April and June which are even months). Anyway, for long odd months, Doomsday is the (N+4)th, while for short odd months, Doomsday is the (N-4)th. The mnemonic was long=add, short=subtract. Thus March (3rd month, long) 3+4=7th is Doomsday May (5th month, long) 5+4=9th is Doomsday July (7th month, long) 7+4=11th is Doomsday September (9th month, short) 9-4=5th is Doomsday November (11th month, short) 11-4=7th is Doomsday I'd agree that it's easier to remember "I work from 9-5 at the 7-11" together with "March 0th=7th". ------------ Copyright � 2000 Rudy Limeback http://www.interlog.com/~r937/ this page: doomsday.html last update: 2001-06-03 ------------- )