## Karen's Perpetual Calendar Page

#### Contents

This is a single piece of paper that will serve as a calendar for any month provided you know the day of the week that year started on. It's not too hard to memorize the day the current year started on; you can remember one day of the week for a whole year. Calculating nearby years is fairly easy.

I got this idea from a photo in the book "The Story of Time" by Kristen Lippencott. The photo showed an engraving of a simple chart, made in 1635. The chart looked something like this one:

 Jan Oct Apr Jul Sep Dec Jun Feb Mar Nov Aug May 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
##### Figure 1

There was little explanation, beyond the observation that January and October will start on the same day of the week in any year (which is not actually true for leap year, but I'll deal with that below), and the same for September and December, and so on. After studying the chart for a while, I came up with the following enhanced chart, which I will explain.

 Jan Oct Apr Jul Sep Dec Jun Feb Mar Nov Aug May Sun Mon Tue Wed Thu Fri Sat Mon Tue Wed Thu Fri Sat Sun Tue Wed Thu Fri Sat Sun Mon Wed Thu Fri Sat Sun Mon Tue Thu Fri Sat Sun Mon Tue Wed Fri Sat Sun Mon Tue Wed Thu Sat Sun Mon Tue Wed Thu Fri 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
##### Figure 2

To understand how this calendar works, consider the following problem. Suppose you want to find what day of the week Valentine's Day falls on in 2008 and that you know 2008 begins on a Tuesday. The day in which the year began is the key to the whole calendar, so remember it. For 2008, the key day is Tuesday. Read across the top row until you come to February, and look down that column until you come to Tuesday. It lies in the row that is indigo (dark blue). Think of the indigo row as being the headings for the calendar that appears at the bottom of the chart. Look at the 14th, and look up to the indigo row to see what day of the week it is. The 14th lies under Thursday in the indigo row, so Valentine's Day is on Thursday in 2008.

Since 2008 is a leap year, you have to adjust slightly for days after February. Suppose you want to know what day of the week Christmas falls in in 2008, and that you know 2008 begins on a Tuesday. Days after February 29th in a leap year come a day after they normally would. So for months after February, you will use the next day, Wednesday, as the key day. Read across the top row until you come to December, and look down that column until you come to Wednesday. It lies in the row that is orange. Think of the orange row as being the headings for the calendar that appears at the bottom of the chart. Look at the 25th. It lies under Thursday on the orange row, so Christmas will fall on Thursday in 2008.

The colored rows aren't really necessary. They didn't appear in the 1635 engraving. If you know what the key day is, and which column is that key day, you can count across the row from there to find the days for the other columns.

I really like the simplicity of Figure 1, and with practice it's reasonably easy to use. If you know what day the year started on, it's all the calendar you need for that year. What follows is an explanation of how to find the day the year starts on for any year.

### To find the starting day for the year:

To use either version of the chart (Figure 1 or Figure 2), you need to know what day the year began on. Figure 3, below, gives the starting day for years from 1978 to 2039, which should be enough for everyday use. If you're only using the calendar for dates within a year or two of the current one, the table in Figure 3 isn't really necessary. Suppose that you know that 2006 (the year in which I am writing this) began on a Sunday. Each year begins a day later than the preceeding year, except that after a leap year the day jumps by two instead of one. So 2006 begins on Sunday, 2007 begins on Monday, and 2008 begins on Tuesday. (Since 2008 is a leap year, 2009 does not begin on Wednesday, but on Thursday.)

Figure 4 gives the starting dates for every 4 years from 1901 to 2097, which allows you to easily find the starting date of any year in the 20th and 21st centuries. Below that I include charts that allow you to find the starting day for any year in the Common Era (AD).

You can make a sliding calendar, in which the appropriate headings appear in a window above the days of the month, by following the directions here: Sliding Perpetual Calendar. The back of this calendar will have Figure 3 and/or Figure 4 on it for easy reference.

 Sun Mon Tue Wed Thu Fri Sat 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 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039
##### Figure 3

The following table gives starting dates for the 20th and 21st centuries, but only every fourth year. To interpolate, each following year starts on the following day of the week, as explained above. Note that the previous year for each year listed is a leap year, so if you work back to the previous year, you must go back two days. For example, 1957 began on a Tuesday, but 1956 began not on Monday, but on Sunday. Note that the days of the week in the headings are not in order, due to only listing every fourth year. Note also that the headings are different for the two centuries.

 Tue Sun Fri Wed Mon Sat Thu 1901 1905 1909 1913 1917 1921 1925 1929 1933 1937 1941 1945 1949 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997
 Mon Sat Thu Tue Sun Fri Wed 2001 2005 2009 2013 2017 2021 2025 2029 2033 2037 2041 2045 2049 2053 2057 2061 2065 2069 2073 2077 2081 2085 2089 2093 2097

### A small wallet-sized calendar for the 20th and 21st centuries

Here is a small, minimal version of Figure1 and Figure4 together that you can print up and fold in half and put in your wallet. To print just the chart, hold your mouse over the "Jan", hold down the left button, and drag it down to the 93 in the lower right corner. This will highlight the chart. Then select "File", "Print" from your browser menu. When it asks what to print, say "selection". To make the chart small, I left out the century headings in Figure 4. The first 25 numbers are for the 1900s, and the last 24 numbers are for the 2000s. I left out 2097, but it is the first number in the next row.

Update: If you paste this into Word, you can make it as large or small as you want by dragging the lower right corner. For three years I have had one on my office bulletin board; it replaces the calendars I used to have to buy for my office. With practice it's not hard to use. I also have a smaller one taped into my checkbook; it's a lot easier to read than the tiny calendars that come with the check registers. I've found that the following is the one version of all the calendars on this page that I actually use all the time.

 Jan Oct Apr Jul Sep Dec Jun Feb Mar Nov Aug May 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
 Tue Sun Fri Wed Mon Sat Thu 01 05 09 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 01 05 09 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93

### Making the calendar truly perpetual: adding centuries

It occurred to me that if I knew which day a century started on, I could construct a table similar to Figure 4 for any century, which would make this calendar truly perpetual.

According to the Wikipedia article on the Julian calendar, it is not clear how the sequence of leap years happened before year 1. The early historians didn't specify which years were leap years. The Julian Calendar was instituted in 45 BCE, and for the first several years every third year was a leap year instead of every fourth. No one knows for sure whether the first year of the Julian calendar was a leap year, or the third or fourth year. Apparently this was fixed just a few years before the Common Era, in about 8 BCE. Because of the confusion, my calendar starts in 1 CE (AD).

In 1582 Pope Gregory XIII adjusted the calendar. Different countries adopted the new calendar at different times. Rather than trying to deal with the details, I have given overlapping dates in the two systems in the two charts for the 17th and 18th centuries.

I have made two tables showing the starting days for each century below, one for Julian calendar, from the year 1 to 1701 , and one for the Gregorian calendar, from the year 1601 to 3501. To extend the Gregorian chart, note the pattern: each group of four starts with a century whose first two digits is a multiple of four.

To use these charts, first find the starting day for the century. Then look at the year chart. Find that row that starts with the day that century began with. Find the year in that century, and read up to the appropriate row to find the day that year started. I have only listed every four years, so you will have to interpolate for the years between, remembering to add a day for each successive year.

Once you know the starting day for the year, use the chart in Figure 2 to find the calendar for the month you want. For convenience, I will put all four charts below:

### Julian century starting days:

 Sat Fri Thu Wed Tue Mon Sun 1 101 201 301 401 501 601 701 801 901 1001 1101 1201 1301 1401 1501 1601 1701

### Gregorian century starting days:

 Mon Sat Thu Tue 1601 1701 1801 1901 2001 2101 2201 2301 2401 2501 2601 2701 2801 2901 3001 3101 3201 3301 3401 3501

### Year starting days:

 Mon Sat Thu Tue Sun Fri Wed Sat Thu Tue Sun Fri Wed Mon Thu Tue Sun Fri Wed Mon Sat Tue Sun Fri Wed Mon Sat Thu Sun Fri Wed Mon Sat Thu Tue Fri Wed Mon Sat Thu Tue Sun Wed Mon Sat Thu Tue Sun Fri 01 05 09 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

### Months:

 Jan Oct Apr Jul Sep Dec Jun Feb Mar Nov Aug May Sun Mon Tue Wed Thu Fri Sat Mon Tue Wed Thu Fri Sat Sun Tue Wed Thu Fri Sat Sun Mon Wed Thu Fri Sat Sun Mon Tue Thu Fri Sat Sun Mon Tue Wed Fri Sat Sun Mon Tue Wed Thu Sat Sun Mon Tue Wed Thu Fri 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Note: remember that if the year is a leap year, you will have to go to the next day for months after February. In the Julian calendar, all years that are multiples of four are leap years. In the Gregorian calendar, all years that are multiples of four are leap years, except for years that are multiples of 100. Those years must be multiples of 400 to be leap years.

### A minimal perpetual calendar for any year

It occurred to me that if I arranged the columns in the "year starting day" chart above in order of the weekdays, I wouldn't need the headings. The following chart represents every fourth year in a century, arranged so that the weekdays are in order from left to right. It's a little tricky to use, but it doesn't take much room. Look on the century chart to see what day the century starts on. That day will be the left column on the year chart. The other columns follow in order. For example, if the century starts on Tuesday, then the columns represent Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday and Monday.

### Minimal year chart

 01 13 25 09 21 05 17 29 41 53 37 49 33 45 57 69 81 65 77 61 73 85 97 93 89

What follows is a minimal list of small charts, small enough to print on one piece of paper. I will list them without headings. If you highlight all four with the mouse, and then print selection, you should get just those charts. I will list them, and then do an example of how to use them. The four in order are the Julian century chart, the Gregorian century chart, the year chart, and the month chart.

 Sat Fri Thu Wed Tue Mon Sun 1 101 201 301 401 501 601 701 801 901 1001 1101 1201 1301 1401 1501 1601 1701
 Mon Sat Thu Tue 1601 1701 1801 1901 2001 2101 2201 2301 2401 2501 2601 2701 2801 2901 3001 3101 3201 3301 3401 3501
 01 13 25 09 21 05 17 29 41 53 37 49 33 45 57 69 81 65 77 61 73 85 97 93 89
 Jan Oct Apr Jul Sep Dec Jun Feb Mar Nov Aug May 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

### An example: July 4, 1776

Find the 1700s in the Gregorian century chart (the second chart). 1701 begins on a Saturday. Look on the year chart (the third chart.) Find 73 on the right column. The left column is Saturday (since 1701 is a Saturday). Read across the columns saying "Saturday, Sunday" etc until you come to the right column, which will be Friday. So 1773 begins on Friday. That means that 1774 begins on Saturday, 1775 begins on Sunday, and 1776 begins on Monday. However, 1776 was a leap year, and July is after leap day, so use Tuesday as the key day. Now look at the month chart at the bottom. July is in the second column. The second column will be the key day, which is Tuesday. Count across from that column to the column where the 4th appears and see that it's a Thursday. July 4, 1776, was a Thursday (using the Gregorian calendar.)

#### A note about patterns:

It is easy to extend the Gregorian century chart. The first two digits of every number in the first column is a multiple of four. To remember the days, start with Tuesday on the right and work back to the left, writing down every other day: Tuesday, Thursday, Saturday, Monday. The Julian chart starts on Saturday and the weekdays go backwards as you read across from left to right. In the year chart, reading down the rows, add 28 each time. Reading across the rows, add 12 or subtract 16, whichever gives the smaller unused number.

To re-create the month chart, pick a convenient key day, like Sunday, as it is in 2006. Knowing how many days in each month, use the chart itself to figure out which day each succeeding month begins on. Count across from there to the key day and fill in the month in the proper column. See which day that month ends on, and repeat for the next month. Though this perpetual calendar is a little more difficult to use than some others I've seen (my old World Book Encyclopedia has a very nice one in the "calendar" article), mine has the advantage that it could be recreated from memory, which would be difficult with any other one. The one in my encyclopedia also only goes from 1753 to 2030. I haven't seen it online, but this one comes close.

### Karen's reformed calendar

Update: Note on the following What follows was the result of a rather fevered brainstorm back in 2006, when I spent a couple of days thinking about calendars. I didn't do any research, and in retrospect, it all seems pretty packed with hubris. I got a nice letter from a correspondent named Milton, which I will quote with his permission. He provided some history of others who have had these same ideas centuries before I did. This page really needs updating, but I don't have the time to address it now. So with Milton's permission, here are his comments:

Dear Karen,

Your ideas for a simplified solar calendar had been thought of, or even used, long ago, although nobody has ever presented them all in one package, as far as I know.

For one, the idea of a 364-day calendar repeating itself every year goes at least as far back as the Essenes. The discrepency of celebrating the Last Supper a day later in the synoptic Gospels than that of John, can only be explained by Jesus being Essene or officially welcomed to celebrate Passover as the Messiah only by them. The Pharisees would thereby have been justified in accusing Jesus of belonging to the Samaritans - also looked down on by them for adopting Gentile customs such as solar calendars, even if used in harmony with a luni-solar calendar not needing 8-day weeks: http://pesherofchrist.infinitesoulutions.com/Finding_the_pesher/Mishmarot.html

Likewise, your calendar with 28 & 35-day months goes at least as far back as Arnold Kempe around 1900, as presented in E.Zerubavel's book, The Seven Day Circle. While your leap-years follow mathematical common sense, they're impractical. During the 1980s in Mensa's Calendar SIG, I suggested for Kempe's calendar, one leap week every 6 years, and after 30 years, another leap week after just 2 years, followed by five more intervals of 6 years, for a 62-year cycle with eleven leap weeks. After 8060 years, or a hundred & thirty cycles of 62 years with a grand total of 10,010 leap days, this rule will fall behind on average by one-tenth of a day, assuming that the year's length shortens at the current rate of 0.00006 days every millenium. If so, the drift from the tropical year until A.D. 10,000 never exceeds 5 days, plus or minus. Therefore one can have one's cake & eat it too: only one exception to a leap week every 6 years, no exceptions to the exception, drift always well under a week.

In order to avoid a Friday the 13th every month, may I suggest starting each year on a Wednesday, as did the Essenes, and so honor them for implementing the 364-day calendar, as well as 6-year intervals between leap years? Kempe's idea of 28 & 35-day months can also be adopted to honor him, however, as a way of preserving Roman traditions, how about having the 35-day months be March, July, October, May - when the Ides are on the 15th day?

Have a blessed day, Milton

(End of update)

After all my calendar adventures above, I got to thinking about calendar reform. The trouble with the Gregorian calendar is that it's so complicated. This is somewhat necessary, because the solar year is not a whole number of days, nor is it a whole number of lunar months. The year is about 365 days, 5 hours and 49 minutes long. The lunar month is 29 days, 12 hours, and 44 minutes long. To further complicate matters, many religions consider the sabbath very important, and the week is 7 days long, which doesn't divide evenly into either 365 or 29. It does, however, divide evenly into 364 (52 times) and 28 (4 times).

There have been several plans proposed for a simplified calendar which would have a single page which could be used for every month. An obvious solution is to have thirteen months of 28 days each, which gives 364 days. In some proposals, the extra day is a holiday that is outside ordinary time: it isn't considered a weekday. You go from Saturday at the end of the year to the extra holiday, and then start the new year on Sunday again. The problem with that is that the 'true' sabbath is no longer matches the calendar.

Another problem with the 13-month calendar is that it doesn't divide evenly into quarters. Quarters are convenient for business purposes. It would be good to have four identical quarters. Also, we're pretty used to the month names we have. A 13-month calendar necessitates at the least introducing a new month name, and maybe starting all over with new names for all the months.

These proposals have been around for decades, so I don't expect mine to be adopted. I expect that being creatures of habit as we are, we will continue to use the Gregorian calendar for a long time to come. What follows then is just for fun.

I had several conflicting goals in coming up with a simplified calendar:

1. Every month should look the same
2. Keep the existing month names
3. Every quarter should look the same
4. Preserve the weeks: no extra days outside the week
5. Keep the dates in line with solar events such as solstices and equinoxes

I really like the simplicty of the 13-month calendar, but I decided to fudge a bit on (1) in order to obtain (2) and (3). I also decided to fudge a bit on (5) in order to obtain (4). So here's my proposal:

• Every quarter is identical. It consists of three months.
• The first month of each quarter has 35 days, and the other two have 28 days. The same calendar page will work for every month, as long as we ignore the last week on the short months.
• The regular year consists of 364 days, which means that each year we lose a day.
• At the end of every 7 years we're a week short, so insert an extra leap week at the end of June.
• At the end of every 28 years we're another week short, because of the extra quarter day in the solar year. So insert another leap week at the end of December.
• Because the "extra quarter day" in the solar year is actually 11 minutes short of being a full quarter day, this will give us an extra day every 130.9 years, or an extra week every 916.36 years. This is about 33 of the 28-year cycles. So every 33rd of the cycles, we will not insert the extra leap week in December that we normally would. (This corresponds to the correction made by the Gregorian calendar over the Julian calendar, of making days divisible by 100 not leap years unless they are also divisible by 400.)

In the long run, this calendar is as accurate as the Gregorian calendar. In the short run, the solstices and equinoxes can be off by as much as eight days, though this is corrected every 28 years. I don't really see this as being any more problem than the fact that the moon phases don't line up with the months. Here's the calendar:

 Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

On long months, use the entire page. On short months, use only the first four rows; ignore 29-35.

Long months are January, April, July, and October. Every 7 years, June is also a long month. Every 28 years, June and December are both long months. Every 924 years, only June is a long month.

This makes every 7th year 371 days long, and every 28th year 378 days long. If this seems too much, we could stagger the 28th years so they didn't fall the same year as the 7th years. In other words, have the first leap year (with the extra week in June) come the third or fourth day of the 28-year cycle.

The other disadvantage to this calendar is that every month has a Friday the 13th in it. If this seems unpleasant, we could start the month on Monday instead of Sunday. I kind of like it, though, myself.

A big advantage is that with only one page being used over an over again, in time we would come to have it memorized without trying to. We wouldn't have to consult a calendar to know that the third Wednesday is the 18th, because it always is.

#### Another reform calendar

I can meet conditions 1 and 5 if I abandon the other ones. The Lunar/solstice calendar below has months that match what the moon is doing, and begins on the first new moon after the winter solstice. I could make a calendar using the existing weeks, but the pages end up being almost as complicated as the Gregorian calendar. On the other hand, if I abandon the 7-day week and allow one or two 8-day weeks each month, I get the following simple page for every month. Some months have 29 days, some have 30, so the fourth week of the month may have 7 or 8 days.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 (30)

This is a major overhaul, since weekdays are so ingrained in our consciousness. Therefore, in the lunar/solstice calendar that follows, I will keep our existing weekdays and not use the above chart.

### Karen's lunar/solstice calendar

This calendar has months that all start on the new moon. Some years have 12 months, some have 13 months, and the year starts on the first new moon after the winter solstice. This isn't really new at all. The Jewish calendar is pretty similar, except that it starts in the fall.

The idea behind this calendar is simple. The first lunar month starts on the first new moon after the winter solstice. The months alternate 30 and 29 days. The first of each month is on or near a new moon, the fifteenth on or near a full moon. If the 12th month ends before the solstice, insert a 13th month. If not, return to the first month. The 13th month will have either 29 or 30 days, depending on when the next new moon is. Calibrate the calendar at the begining of each lunar year, the first new moon after the winter solstice.

Unlike the reformed calendar above, this one doesn't have a nice easy calendar page for every month, unless I monkey with the weeks as in the above chart. In practice, I prefer not to do that, since the week is more ingrained in our lives than any other calendar division. On the other hand, you only have to look up at the sky to tell about which day of the month it is.

I have made a chart showing the dates for the lunar months for the years 2006-2011. It's pretty big, so I put it on a separate page here: Lunar/Solstice Calendar The chart on that page matches the day in the Lunar/solstice calendar to the day in the Gregorian calendar, including the (actual) day of the week.

By adding a row to the top of Figure 1 or Figure 2 that shows the lunar months in Roman numerals, you can figure out weekdays for the lunar calendar in the same way you used those figures for the Gregorian or Julian calendars. You have to know what day of the week the lunar year began on. (2006 began on Saturday, December 31, 2005). You can find out by looking on my Lunar/Solstice Calendar for the years 2006-2011. Or you can look up new moons on the US Naval Observatory page, which was the source I used.

Warning: If you use the Roman numerals, you will get the calendar pages for the lunar calendar, not the Gregorian calendar. The 15th day of Month II, for example, is usually not the 15th of February. The order in which the traditional months are listed also does not match the order in which the lunar months are listed, since they have different lengths. It might be better to take off the second row, but I wanted to make the chart useful for both calendars.

I X
V IV IX
VIII XIII
III XII
II VII
VI XI
Jan Oct Apr Jul Sep Dec Jun Feb Mar Nov Aug May
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
##### Figure 1
I X
V IV IX
VIII XIII
III XII
II VII
VI XI
Jan Oct Apr Jul Sep Dec Jun Feb Mar Nov Aug May
Sun Mon Tue Wed Thu Fri Sat
Mon Tue Wed Thu Fri Sat Sun
Tue Wed Thu Fri Sat Sun Mon
Wed Thu Fri Sat Sun Mon Tue
Thu Fri Sat Sun Mon Tue Wed
Fri Sat Sun Mon Tue Wed Thu
Sat Sun Mon Tue Wed Thu Fri
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31