Day of Week

How to Calculate the Day of the Week

Links to Tables are highlighted in blue.
Footnote references are shaded in yellow. Footnotes are shown below the description.

To calculate the Day-of-Week ("DoW") for any date in the Gregorian calendar, add the values for the Month, Day, and Century/Year using the following three steps:

Step 1 - Add the DAY to the value for the MONTH from the Month Table and subtract from the result the largest multiple of 7 contained in it. Hold this number until step 3. a

Step 2 - Subtract from the last two digits of the YEAR the largest multiple of 28 contained in it, then dividebthe result by 4and round down (i.e., drop the remainder or decimal). Add the two results together and then add the value for the CENTURY from the Century Table. Subtract 1 if the year is a leap yearc AND the month is January or February.d e

Step 3 - Add the results from steps 1 and 2 and then subtract from the result the largest multiple of 7 contained in it. Using the resulting number, look up the Day of the Week in the Day of Week Table.

Using January 11, 1988, as an example, the value of the MONTH (Jan) is 0, the DAY is 11, the YEAR is 88, and the value for the CENTURY (19) is 1. 84 is the largest multiple of 28 in 88, which is a leap year because it is evenly divisible by 4. Click here to see the calculation (in part I only).

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Once you have mastered the above basic technique, you can then use the same basic formula with just one more step to find the Month, the Day, the Year and even the Century (if that need ever arises) when you know the other elements of the Date including the Day-of-Week.

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FOOTNOTES: The below footnotes are not needed to master the above technique. They are here for informaiton purposes only. .

a Some people are able to hold this number (between 0 and 6) in their heads while they do step 2. I prefer to "hold" it by subtly using the fingers on my left hand so that I make no mistakes when I return to get it in step 3.

b The reason you divide by 4 is to fund the number of leap years that have occurred since the base year.

c A year is a leap year if it is evenly divisible by 4, that is, there is no remainder when you divide it by 4. The only exception to this is for turn-of century years ('00) not evenly divisible by 400. (1600 and 2000 were leap years, whereas 1700, 1800, and 1900 were not.)

d The reason you subtract 1 here is because in a leap year, the extra day from February 29th is already added in for the full year, and so if February 29th has not occurred yet, you need to take out that extra one day.

e A table with the precomputed full year (CENTURY & YEAR) values can be found by clicking here.

Notice: If computing the year value in step two is too cumbersome for you, there are two alternative methods you might wish to consider. (These are for the year values before the century value is added and the -1 adjustment is made for January/February in leap years. Click here to look at those alternative methods.

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Computing the Year Value – an alternate method

Here is are two alternative methods people may wish to think about for computing the year value. They are not pure methods, but they work, and so if they work for you better, use one them.

		0	1	2	3	4	5	6	7	8	9
00	0	0	1	2	3	5	6	0	1	3	4
10	2	5	6	1	2	3	4	6	0	1	2
20	0	4	5	6	0	2	3	4	5	0	1
30	2	2	3	5	6	0	1	3	4	5	6
40	0	1	2	3	4	6	0	1	2	4	5
50	2	6	0	2	3	4	5	0	1	2	3
60	0	5	6	0	1	3	4	5	6	1	2
70	2	3	4	6	0	1	2	4	5	6	0
80	0	2	3	4	5	0	1	2	3	5	6
90	2	0	1	3	4	5	6	1	2	3	4

For those who wish to avoid the sometimes cumbersome computation for year value of subtracting from the last two digits of the year the highest multiple of 28 in it, that is, 28, 56 or 84, there are two other ways to get the value.

The first way is of course to memorize the above 100 cell chart of years and values, 200 numbers in all, 100 for the years, 100 for the values. If you can do that, do it, but good luck.

The second way is to memorize not three numbers (the 7X4 pivotal years of 28, 56 and 84) but four of the numbers on the above chart, the pivotal leap decade years (shown in purple, above) and their values, i.e., 20=4, 40=1, 60=5, and 80=2. These numbers can be memorized not as eight numbers but only four (forget about 00=0 because that is zero): 24, 41, 65 and 82, the tens (left) digit being the decade and the units (right) digit being the value for that pivotal decade year. Then all you need to do is to subtract the lowest pivotal decade from the year and divide that by 4 dropping the decimal, then add those two numbers together, and finally add that number to the value for the pivotal decade year. The largest number you will have to deal with is (for, say, 19, 39, 59, 79, and 99) 19/4=4+19=23 (plus the pivotal year value) and not some number like 55-28=27/4=6+27=33.

The only problem with this method is that when you divide the number by 4, it does not automatically tell you if a year is a leap year in half of the years, so you need to figure out some other way to signal to you that a year is a leap year and may require an adjustment of -1 if the month is January or February. It works, of course, for the even pivotal decade years, but not for the odd ones. With the odd ones, if the remainder is 2 (.5), it is a leap year. Otherwise, the method works just as well 100 percent of the time.

Table for Century Values

1600/2000	1700/2100	1800/2200	1900/2300
0	5	3	1

When we refer to the Century, we are referring to the cardinal representation for the century years (e.g., 1600s for the 17^th century). This table repeats itself on four-century cycles into the distant future and to some extent into the past, as shown above for one extra cycle. So what would be the value for the 1500s? And the 2400s?

You do not need to know the following to do the calculation of the Day-of-Week (etc.) but for those purists among our readers, here is some additional information about the Century table.

What is the meaning of the value for the Century? For centuries not evenly divisible by 400 (e.g., 1700, 1800, 1900) where there is no leap year in the turn-of-century year, the Century value means the numerical value of the day-of-week prior to the first day of the new century. For instance, the 1 (which also means Sunday) for the 1900s means that the first day of the century was the day after, a Monday or a 2. (January 1, 1900, was indeed a Monday.) For the turn-of-century years evenly divisible by 400 (e.g., 1600, 2000), the Century value means the value of the first day of the new century. For instance, the value for the 2000 century is 0, or Saturday. The value for January 1, 2000, is 0+1+0+0+0-1 equals 0, or Saturday, and it was indeed a Saturday.

How did we compute the value for the different centuries? We know that in a non-leap calendar year, the calendar moves forward one extra day of the week (365 days is 52 full weeks plus one day), but in leap years it moves forward by still one more day for the extra day in February. So in a century that starts with a leap year (e.g., 1600 and 2000), there are 75 non-leap years (1 extra day) and 25 leap years (2 extra days) for a total of 125 days or 17 full weeks (of 7 days) plus 6 days left over. So, a century starting with a non-leap year has one less day, or 5 left over. So, if we start the whole process with January 1, 1600, which actually fell on a Saturday (a value of 0), the Century value for centuries starting with a leap year means the Day-of-Week on which the Century started. Moving forward 125 days from that 1600s century, or 5 weekly calendar days forward, the value going into the 1700’s is 5 (the zero starting the prior century plus 5 equals 5, which translates to a Thursday, and so if January 1, 1700, (a non-leap year) was a Friday (a value of 6), the century value translates to the day before. The same goes for the next two centuries, both starting with non-leap years. Moving forward into the 1800s, we have the 5 going into the prior century plus another 5 days from the 1700s which equals 10, and 10 minus 7 (take out the full weeks) yields a 3 going into the 1800s. Moving forward into the 1900s, we have the 3 going into the prior century plus 5 more from the 1800s for a total of 8, and 8 minus 7 (a full week) equals 1, the value going into the 1900s. Taking the 1 going into the 1900s and adding 5 for the 1900s, we get a total of 6, which means that a Friday was the day before the first day of the new century. But since we know that 2000 is a leap year turn-of-century year (because it is divisible by 400) and that the meaning of the Century value in such years is the FIRST day of the year, the first day of the week of that new century is the day after Friday which is Saturday, and that means a 0 for that century. Got it? Again, you do not need to know this. Just apply the formula.

How far into the future does this century table (and technique in general) go? Forever, repeating the 0, 5, 3, 1 cycle every four centuries. I understand that another adjustment will have to be made around the year 20,000, but to all intents and purposes we can all safely ignore that.

How far into the past does this table go? Techically the century table goes backwards, too, repeating the 0, 5, 3, 1 cycle every four hundred years. But since Pope Gregory XIII decreed the calendar in 1582, that's a good starting point, at least for in Italy. As for other countries, the "U.S." adopted it only in 1752 when England did, which sparked the calendar riots in England when people lost 11 days of wages but still have to pay full months rent. Read more about the calendar at Wikipedia (click here). Russia adopted it on in 1917 after the Bolsheviks took over. And so on. Some web sites have detailed conversion tables from the Julian calendar to the Gregorian, but unless you need it for your job, why bother? As for dates before 1582, do it for any date you want going back to the year 0001 (and maybe before) because no one will be around who can challenge you.

I

Table for Month Values

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
0	3	3	6	1	4	6	2	5	0	3	5

- You must memorize this table. When you do, you will forget the number of the month itself (e.g., 6 for June) and know the month as its table value (4 for June). As with the other tables, one way to memorize the table is by some word as well as color associations and/or some mnemonics

You do not need to know the following to do the calculation of the Day-of-Week (etc.) but for those purists among our readers, here is some additional information.

- This is the table for every year, including leap years. The adjustment for the extra day in leap years due to 29 days in February is made elsewhere in the calculation. Were it not for this adjustment, the values for January and February for leap years would be 6 and 2 respectively.

- What is the meaning of the values in this table? The value for each month means how many days of the week removed from the day of the week of the first day of the year the first day of the week of that month is.

- How is this table derived? Given the meaning, the value of January is 0 (zero). Since we know that there are 31 days in January, that’s 3 days beyond 4 four full weeks of 28 days, so the value going into February is 3. Since there are four full weeks in February (except for leap years, but we adjust for that elsewhere), no extra days are added for March beyond the 3 we had going into February, so the value for March is 3 also. Since we know that in March there are 31 days, that’s 3 days of the week more that the monthly calendars are pushed forward, and so the 3 going into March plus the 3 from March means 6, so the value for April is 6. In April there are 30 days, 2 days beyond four full weeks. The 2 from April plus the 6 going into April yields 8, but taking out the 7 days of full weeks we do not need yields 1, the value for May. You go figure the rest.