CALCULATING THE DAY OF THE WEEK FROM THE DATE – part I
This is an ingenious method for determining, for example, on what day of the week your date of birth falls. The method was originally described by John Horton Conway and further elucidated by William Jeffries http://quasar.as.utexas.edu/BillInfo/doomsday.html and Rudy Limeback http://yak.net/fqa/238.html. Either of these sites may be accessed for further elucidation on this subject. The Jeffries site is especially useful because it contains a calculator at the bottom of the page to solve for all twentieth century dates.
The method is very intimidating appearing at first, but can be understood and mastered by most interested readers if the needed effort is expended. A few things are of value if noted before getting into the method.
1. Skipping ahead six days from any weekday is the same as going back one day. Thus, six days after Tuesday is Monday. Skipping ahead five days is like going back two days and so on.
2. Likewise, skipping ahead any multiple of 7 days doesn’t change the weekday. Thus, 21 days or 35 days after Friday is still Friday. And 17 days later is the same weekday as 3 days later (written 17à3)
3. In century years not divisible by 400 such as 1700, 1800 and 1900, the 00 year IS NOT a leap year whereas in century years divisible by 400 such as 1600 and 2000, the 00 year IS a leap year. For a fuller treatment of this subject and why this is so, see the Fascinoma chapter on leap years.
4. Apart from century years not divisible by 400, all years ending with a number that is exactly divisible by 4 is a leap year. Thus 1704, 1624 and 1972 are all leap years because 04, 24 and 72 are evenly divisible by 4.
5. There are 365 days in a regular year, or 52 weeks and 1 day. This makes the same calendar day one year
later fall on the next weekday, except when a February 29^{th} has occurred in between. In this case, the same date in the next calendar year will be two weekdays later. Thus, because September 5th, 1982 is a Sunday, September 5th, 1983 will be a Monday (there is no February 29^{th} in 1983). But when a February 29^{th} occurs in the year, 366 days will elapse between identical dates and this causes the following year’s date to fall two weekdays later. Thus, September 5^{th} in 1984 was a Wednesday, not a Tuesday. Thus, since June 6^{th}, 1931 is a Saturday, and since 1932 is a leap year (because 32 is evenly divisible by 4), June 6^{th}, 1932 is a Monday, two days more than Saturday.
All of this is leading up to how we find a given day of the week for a given year if we know another date and weekday. It will turn out that for any date in the 1900’s, if we know what day of the week that date fell in 1900, we can add one day for every twelve years, one day for every extra year remaining beyond that multiple of twelve, and one day for each leap year in that remainder. Thus, if October 10^{th}, 1900 is a Wednesday, what is October 10^{th}, 1939?
1939 is 39 years later than 1900, and this is three dozen years with a remainder of three years. For each dozen years, we advance a day beyond the Wednesday of 1900. In 1936, October 10^{th} was a Saturday because Saturday is three days after Wednesday. And In 1939, October 10^{th} was a Tuesday because we add a day for each extra year.
If we had been calculating October 10^{th}, 1941, we would have had 3 dozen plus five years more, one a leap year. We would advance a day for each dozen years, a day for each remaining year, and a day for the single leap year in the remainder of 5. This is a total of 9 days added onto the Wednesday of 1900’s October 10^{th} making October 10^{th} of 1941 a Friday (remember, 9 days later is the same as 2 days later, and Wed + 2 = Fri).
We say that for 19xx, xx has Q dozens with a remainder R, and that remainder R contains S leap years, or 1 for every four years in R. The maximum Q is 8 since 9 or more dozen years takes us into the 2000s, and our method so far is only good for the 1900’s. The maximum R is 11 since that is the maximum remainder possible when dividing by 12. And the maximum S is 2 since 11, the maximum R, has only two complete sets of 4 years. When R is 47, S is 1. When R is 3 or less, s is 0. When R is 811, S is 2.
Exercises: (a), (b), (c) and (d). What are Q, R, S and Q+R+S for 1927, 1953, 1968 and 1983? Given that July 11th, 1900 is a Wednesday, what July 11^{th} in 1927, 1953, 1968 and 1983?
Answers: (a) Q=2 because there are 2 dozens in 27, and R = 3 because there are 3 remaining in 27 when 24 are taken out. S = 0 whenever R is less than 4. Thus Q+R+S = 2+3+0 = 5; July 11, 1927 is the same as Wed + 5 = Monday (b) 53 = 4(12) + 5, therefore, Q=4, R=5 and S=1, Q+R+S=10. But 13 days later is the same weekday as 4 days later, that is, 10à3, and July 11, 1953 is Wed + 3 = Saturday. (c) For 1968, use 68. This is 5 dozens with a remainder of 8 which is two leap years. So, Q=5, R=8, S=2 and Q+R+S=15à1. Wed+1=Thursday for July 11, 1968. (d) For 83, Q=6, R=11 and S=2 so Q+R+S=19à5. Wed + 5 = Monday for July 11^{th}, 1983.
This would be a good place to take a break and come back to this material tomorrow.
DOOMSDAY
To recap, we can calculate the weekday of any date in the 1900’s if we know the weekday for that date in 1900 and add one day for every dozen years since 1900, one for each remaining year, and one for each complete set of 4 in that remainder. We can know the weekday for that date if we know the weekday for one day in each of the 12 months of that year.
JAN FEB MAR APR
Su M T W Th F Sa Su M T W Th F Sa Su M T W Th F Sa Su M T W Th F Sa
1 2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 1 2 3
3 4 5 6 7 8 9 7 8 9 10 11 12 13 7 8 9 10 11 12 13 4 5 6 7 8 9 10
10 11 12 13 14 15 16 14 15 16 17 18 19 20 14 15 16 17 18 19 20 11 12 13 14 15 16 17
17 18 19 20 21 22 23 21 22 23 24 25 26 27 21 22 23 24 25 26 27 18 19 20 21 22 23 24
24 25 26 27 28 29 30 28 28 29 30 31 25 26 27 28 29 30
31
MAY JUN JUL AUG
Su M T W Th F Sa Su M T W Th F Sa Su M T W Th F Sa Su M T W Th F Sa
1 1 2 3 4 5 1 2 3 1 2 3 4 5 6 7
2 3 4 5 6 7 8 6 7 8 9 10 11 12 4 5 6 7 8 9 10 8 9 10 11 12 13 14
9 10 11 12 13 14 15 13 14 15 16 17 18 19 11 12 13 14 15 16 17 15 16 17 18 19 20 21
16 17 18 19 20 21 22 20 21 22 23 24 25 26 18 19 20 21 22 23 24 22 23 24 25 26 27 28
23 24 25 26 27 28 29 27 28 29 30 25 26 27 28 29 30 31 29 30 31
SEP OCT NOV DEC
Su M T W Th F Sa Su M T W Th F Sa Su M T W Th F Sa Su M T W Th F Sa
1 2 3 4 1 2 1 2 3 4 5 6 1 2 3 4
5 6 7 8 9 10 11 3 4 5 6 7 8 9 7 8 9 10 11 12 13 5 6 7 8 9 10 11
12 13 14 15 16 17 18 10 11 12 13 14 15 16 14 15 16 17 18 19 20 1213 14 15 16 17 18
19 20 21 22 23 24 25 17 18 19 20 21 22 23 21 22 23 24 25 26 27 19 20 21 22 23 24 25
26 27 28 29 30 24 25 26 27 28 29 30 28 29 30 26 27 28 29 30 31
31
It turns out that several easily remembered dates in every calendar year all fall on the same weekday, but that that weekday that they all fall on will be different for each year. Conway, the inventor of this method, and both Jeffries and Limeback cited above, all refer to this “magical day” of any given calendar year as the Doomsday. Consider a nonleap year like the one above. This calendar could apply to any of the following 20^{th} century years: 1909, 1915, 1926, 1937, 1943, 1954, 1965, 1971, 1983, 1993 or 1999.
Notice that one day in each month is printed in bold, and that this is a Sunday in each case. For the time being, let us refer to this day, Sunday, as a Doomsday for this year. You will see why later.
Notice also that there is a Feb 0 and a Mar 0, dates that don’t normally appear on a calendar. Of course, these refer to the days before Feb 1 and Mar 1 that in a nonleap year are Jan 31 and Feb 28. Now, for the time being, forget the 1^{st} three months of the year, January through March and focus only on April through December.
Look first at the even numbered months, April, June, August, October and December. Do you notice anything interesting about the bold date, the Doomsday, in these months?
In the 4^{th} month, it is the 4^{th} that is bold (Doomsday). Likewise for 6/6, 8/8, 10/10 and 12/12. These, of course correspond to April 4^{th}, June 6^{th}, August 8^{th}, October 10^{th} and December 12^{th}, and they are all Sundays in this calendar. It turns out that if April 4^{th} was a Thursday as it was in 1901, 1907, 1918, 1929, 1935, 1957 and several more years of the 20^{th} century, that 6/6, 8/8, 10/10 and 12/12 would all also fall on a Thursday. This is what is special about a Doomsday. It specifies the weekday for all of these dates in every year.
Now observe the odd numbered months (ignoring January through March). The highlighted Doomsdays in these months are May 9^{th}, July 11^{th}, September 5^{th }and November 7^{th},and they are all Sundays in the example calendar shown. These correspond to 5/9, 9/5, 7/11 and 11/7. These can be easily remembered by the mnemonic, “A 9 to 5 job at the 711 (Seven11)”. Once we know what the Doomsday is for a particular year, we know all of these dates’ weekdays as well.
For March, Doomsday is March 0 (or, the last day in February, whether this is the 28^{th} or the 29^{th}) and so March 7, 14, 21 and 28 are all Doomsdays as well. For January and February, pure rote must be applied. In nonleap years, use January 3^{rd} (1/3) and February 0^{th} (2/0), For leap years, leap ahead one day each to January 4^{th} (1/4) and February 1^{st}(2/1). This completes the list of Doomsdays for each calendar month for all years.
6. Doomsday in a nonleap year will fall on 1/3 and 2/0. On a leap year it will be 1/4 and 2/1. In both, it will fall on 3/0, 4/4. 5/9, 6/6, 7/11. 8/8. 9/5, 10/10, 11/7 and 12/12.
Take a moment to see why this is true and review the material since the DOOMSDAY heading if you are not clear.
CALCULATING THE DAY OF THE WEEK FROM THE DATE – part II
If Doomsday were Tuesday in a given year, April 12^{th} would be a Wednesday because 4/4 is a Doomsday (Tuesday) by our rule above, and 4/12 is eight days later. Since 8à1, April 12^{th} is one weekday later than Tuesday.
If Doomsday were a Saturday, Halloween would also be on a Saturday because 10/10 is a Doomsday (Saturday), and 10/31 is exactly three weeks later.
Doomsday was a Monday in 1983. Christmas that year fell on a Sunday because the Doomsday for December, 12/12, is 13 days before Christmas. Adding thirteen days to Monday is like subtracting one day and arriving at Sunday.
Now we need only combine the work we have done in the two sections of this paper by citing one more rule:
1. Doomsday for 1900 was Wednesday. All other Doomsdays for all other years in the 1900’s are calculated using the Q+R+S rule and adding the result to Wednesday. Any date in that year can be translated into a weekday by identifying the Doomsday date for that month and adjusting up or down to the desired date.
Example: On what day did the U.S. bicentennial fall?
Answer: The date of the U.S. bicentennial was July 4^{th}, 1976. For 76, Q+R+S = 6+4+1 because there are six dozens in 76 with 4 remaining, and S=1 when R is 47. This totals 11à4, and Wed + 4 = Sunday. For July, Doomsday is 7/11 (remember Seven11), and July 4^{th} is exactly one week earlier. Thus the date in question fell on a Sunday.
Example: Pearl Harbor Day was December 7, 1941. What day of the week was this?
Answer: First, calculate Doomsday for 1941. There are three complete dozens in 41 with a remainder of 5. Five has one leap year (i.e., is between 4 and 7) because it has 1 complete set of four. Thus Q+R+S = 3+5+1 = 9à2. We thus add 2 to the day Wednesday, the day we use whenever we are calculating a Doomsday for the 1900’s, and we get a Doomsday of Friday. Since 12/12 is a Doomsday, and that means it is a Friday in 1941, we can subtract 5 days from December 12^{th} and arrive at Sunday. Pearl Harbor was bombed on a Sunday morning.
Example: In 1863, President Lincoln declared that the fourth Thursday in November would be Thanksgiving. In 1941, President Franklin Roosevelt declared that it would be a national holiday. What was the date of Thanksgiving in 1941?
Answer: This one will be solved from the other direction. We must work out the date of the 4^{th} Thursday in 1941. Doomsday for 1941 has been solved in the example above and is Friday. In November, 11/7 is a Friday, so November 6th is the 1^{st} Thursday of the month. The fourth Thursday falls 21 days later on November 27, 1941.
Examples to solve: (a) The Titanic sunk April 14, 1912. Prove that this was a Sunday. You might be able to do this one in your head. (b) The Challenger explosion occurred on January 28^{th}, 1986. Prove that this was a Tuesday.
OTHER CENTURIES
For the years of the century beginning in 2000, we use the same rules, except we calculate from a starting Doomsday of Tuesday for the year 2000 rather than Wednesday, the Doomsday for the year 1900.
Example: On what day will the tricentennial fall?
Answer: July 4, 2076 is calculated just as the bicentennial was, but using Tuesday for the 21^{st} century instead of Wednesday for the 20^{th} century. This leads us to Saturday in 2076 rather than Sunday as in 1976.
Example: On what day did 9/11 occur?
Answer: September 11, 2001 uses Q+R+S = 0+1+0 = 1. Added to Tuesday, the starting doomsday for 20002009, we get a Doomsday of Wednesday for 2001. Because 9/5 is a Doomsday (Wednesday), 6 days later was a Tuesday.
The Doomsday for any century beginning with Friday, October 15, 1582 can be determined according to the formula
8. For years of the form 15xx, 19xx, 23xx, 27xx, …,Wednesday is the century doomsday for the century.
For years of the form 16xx, 20xx, 24xx, 28xx, …, use Tuesday.
For years of the form 17xx, 21xx, 25xx, 29xx, …, use Sunday.
For years of the form 18xx, 22xx, 26xx, 30xx, …, use Friday.
This date was the first day that the new Gregorian calendar was in use in Catholic countries such as Italy, Spain and France. This date immediately followed Thursday, October 4, 1582, the last day of the old Julian calendar. October 514 did not exist in 1582 in Catholic countries. Protestant nations made the change much later. Britain and her possessions did not do so until nearly two centuries later. In 1752, Wednesday September 2, 1752 on the Julian calendar was immediately followed by Thursday, September 14, 1752 of the new Gregorian calendar, September 313 being thrown out as a correction. So for Anglo purposes, rule 8 above should not have the 15xx or 16xx entries since rule 8 applies only to the Gregorian calendar.
Example: Calculate the day of the first February 29^{th} of the third millennium.
Answer: People argue as to whether the millennium begins in 3000 AD or 3001 AD, neither is a leap year since xx00 years are only leap years in the years exactly divisible by 400 such as 2800 AD and 3200 AD. The first leap year of the third millennium will occur February 29^{th}, 3004. For 04, Q+R+S = 0+1+0 = 1. Using Friday as the century day for the century beginning 3000AD, we find that Doomsday that year is a Saturday. February 29^{th} (2/29) is the same as 3/0 (March 0^{th}), a Doomsday. Hence February 29^{th}, 3004 falls on a Saturday.
Example: Calculate the date of the signing of the Declaration of Independence
Answer: Because 1776 falls after 1752, we may use the Gregorian rules to which we are accustomed. For the 1700’s, the century Doomsday is Sunday. 76 corresponds to 6+4+1 = 11à4, Sunday + 4 = Thursday, the Doomsday for the year 1776. As before, July 4^{th} is the same day as 7/11, which is the Doomsday Thursday, the day of the week when the first signatures appeared on the Declaration.
9. To calculate century Doomsdays in the Julian period (prior to October 4^{th}, 1582 in Catholic Europe or September 2, 1752 in the British Empire), subtract the century number (i.e., 13 for the 1300’s, 7 for the 700’s, etc.) from Sunday to get the century Doomsday for that century and proceed as before.
Although the Julian calendar had been in effect since 45 B.C., it did not have regular fouryear leap years until about 4 A.D., and thus, this method is good back to approximately this 1 A.D but breaks down for the previous year, 1 B.C. (there was no year zero between them) which should have been but was not a leap year.
Example: King John of England issued in the era of limited monarchy when he signed the Magna Carta on June 15, 1215 and for the first time in history, gave a small portion of the power of the British throne to the nobles. What day of the week was this?
Answer: For Julian calendar dates, we subtract the century numbers (12xx) from Sunday. Sunday minus 12 is the same as Sunday + 2 = Tuesday, the century Doomsday for 1200’s. Next, calculate Q+R+S for 1215 using 15: 1+3+0 = 4. Tuesday + 4 = Saturday, the Doomsday for 1215. The memorized Doomsday for June is 6/6, a Saturday, so June 15^{th}, 9 days later, is Monday.
Example: Verify that September 2, 1752, the last day of usage of the Julian calendar in the British Empire was a Wednesday, and that the following day, September 14, 1752 was a Thursday on the Gregorian calendar.
Answer: The Julian century Doomsday for the 1700’s was Sunday minus 17 = Sunday + 4 = Thursday. The Q+R+S for 1752 uses 52: 4+4+1 = 9 making the Julian doomsday for 1752 a Saturday. 9/5 would have been a Saturday had it existed, so 9/2 (September 2^{nd}) was a Wednesday. The Gregorian century Doomsday for the 1700’s from rule 8 is Sunday. Q+R+S for 52 is still 9 making the Gregorian Doomsday for 1752 a Tuesday (Sun + 9 = Sun + 2 = Tue). Again, 9/5 did not exist, but would have been a Tuesday making 9/14 (September 14^{th}), 9 days later, a Thursday as required.
