### An Easter SQL

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Since Easter is nearly upon us, I got to thinking once again about how challenging it is to calculate the day that Easter Sunday falls on.  Since Calendar tables and their step-children Holiday tables are so interesting I thought I might write a sequel to my earlier blog link in this sentence.

There are at least three holidays (not necessarily business holidays) that are related to the calculation of Easter Sunday.  These are:

• Good Friday is the Christian holiday that is the Friday just prior to Easter Sunday.
• Ash Wednesday is another Christian holiday (the first day of Lent) that falls precisely 46 days prior to Easter Sunday.
• Mardi Gras (a.k.a. Fat Tuesday) is a lesser known holiday celebrated by the more robust partyers among us.  I have a special affinity for it as I lived in New Orleans for a few years.

We will show you how to calculate them all and use that method to populate a Holidays lookup table so hopefully you’ll never have to do it again.  Happy Easter!

## A Calendar and a Holiday Table

Discussions of Calendar tables in T-SQL are ubiquitous (here’s one by Todd Fifield), differing only in the columns actually stored in them.   We’ll draw upon the one we created in our blog to show you how easy it is here.

```-- Create a Calendar table
SELECT [Date]=ISNULL([Date], 0), [Year]=ISNULL([Year], 0), [YrNN]=ISNULL([YrNN], 0)
,[YYYYMM]=ISNULL([YYYYMM], 0), [BuddhaYr], [Month], [Day]=ISNULL([Day], 0)
,[WkDNo], [WkDName], [WkDName2], [WkDName3], [JulDay], [JulWk]
,[WkNo], [Qtr], [Last], [LdOfMo], [LDtOfMo]
INTO dbo.Calendar
-- This function can be obtained here: GenerateCalendar FUNCTION
FROM dbo.GenerateCalendar('1990-01-01', 65536);

-- Change column types to be NOT NULL so we can index them
--
-- Note: not needed thanks to the neat trick with ISNULL above
--ALTER TABLE dbo.Calendar ALTER COLUMN [Date] DATETIME NOT NULL;
--ALTER TABLE dbo.Calendar ALTER COLUMN [Year] INT NOT NULL;
--ALTER TABLE dbo.Calendar ALTER COLUMN [Month] INT NOT NULL;
--ALTER TABLE dbo.Calendar ALTER COLUMN [YYYYMM] INT NOT NULL;
--ALTER TABLE dbo.Calendar ALTER COLUMN [Day] INT NOT NULL;
GO

-- Build some representative INDEXes
ALTER TABLE dbo.Calendar

ALTER TABLE dbo.Calendar
ADD CONSTRAINT Cal_ix1 UNIQUE NONCLUSTERED([Year], [Month], [Day]);

ALTER TABLE dbo.Calendar
ADD CONSTRAINT Cal_ix2 UNIQUE NONCLUSTERED([YYYYMM], [Day]);

SELECT TOP 10 [Date], [YYYYMM], [WkDName2], [WkNo], [Last]
FROM dbo.Calendar;

-- Now create a Holidays table, from the same blog
CREATE TABLE dbo.Holidays
(
FromDate    DATETIME PRIMARY KEY
,ToDate     DATETIME
,Holiday    VARCHAR(100)
);
```

We’ve created a Holidays table that we’ll be using to further our sequel to Easter.  There are also a few other things in the above Data Definition Language (DDL) that are worthy of mention.

• The GenerateCalendar FUNCTION is an in-line, Table Valued Function (iTVF) that appears in my earlier blog (and also in the resources file attached at the end of this article), which can be used in lieu of a Calendar table should you prefer to do so.  Once again, my thanks go out to SQL MVP Jeff Moden for his help optimizing the performance of the calculations done in that FUNCTION.
• We’ve used a neat trick with ISNULL to set the NULL attribute of the column in the table we selected INTO (when Calendar gets CREATEd) to NOT NULL, so that we don’t need to run the subsequent ALTER statements on those columns.  I included those ALTER statements for your information.  Thanks again to Jeff Moden for showing me that trick (with honorable mention to SQL MCM Wayne Sheffield (MCM=Microsoft Certified Master) who came along just a few days later with the same information).

Since Easter Sunday is so diabolically difficult to calculate, many reputable sources have created look-up tables such as this one.  Drawing upon that source, we can create the dates for Easter Sunday from 1990 through 2020 in our Holidays table using the following script.

```WITH EasterDates (d) AS
(
SELECT '1990-04-15' UNION ALL SELECT '1991-03-31' UNION ALL SELECT '1992-04-19'
UNION ALL SELECT '1993-04-11' UNION ALL SELECT '1994-04-03'
UNION ALL SELECT '1995-04-16' UNION ALL SELECT '1996-04-07'
UNION ALL SELECT '1997-03-30' UNION ALL SELECT '1998-04-12'
UNION ALL SELECT '1999-04-04' UNION ALL SELECT '2000-04-23'
UNION ALL SELECT '2001-04-15' UNION ALL SELECT '2002-03-31'
UNION ALL SELECT '2003-04-20' UNION ALL SELECT '2004-04-11'
UNION ALL SELECT '2005-03-27' UNION ALL SELECT '2006-04-16'
UNION ALL SELECT '2007-04-08' UNION ALL SELECT '2008-03-23'
UNION ALL SELECT '2009-04-12' UNION ALL SELECT '2010-04-04'
UNION ALL SELECT '2011-04-24' UNION ALL SELECT '2012-04-08'
UNION ALL SELECT '2013-03-31' UNION ALL SELECT '2014-04-20'
UNION ALL SELECT '2015-04-05' UNION ALL SELECT '2016-03-27'
UNION ALL SELECT '2017-04-16' UNION ALL SELECT '2018-04-01'
UNION ALL SELECT '2019-04-21' UNION ALL SELECT '2020-04-12'
)
INSERT INTO dbo.Holidays (FromDate, ToDate, Holiday)
SELECT FromDate=d, ToDate=d, Holiday='Easter Sunday ' + CAST(YEAR(d) AS CHAR(4))
FROM EasterDates;
```

Before we add Good Friday, Ash Wednesday and Fat Tuesday to our Holiday table, we’d like to check that the introductory description at least results in the correct day of the week on which the holiday falls.  We can use our Calendar table for this.

```SELECT FromDate = a.FromDate+b.[days]
,ToDate     = a.FromDate+b.[days]
,b.holiday
,c.WkDName
FROM dbo.Holidays a
CROSS APPLY
(
VALUES(-2, 'Good Friday ' + CAST(YEAR(FromDate) AS CHAR(4)))
,(-46, 'Ash Wednesday ' + CAST(YEAR(FromDate) AS CHAR(4)))
,(-47, 'Mardi Gras (Fat Tuesday) ' + CAST(YEAR(FromDate) AS CHAR(4)))
) b ([days], holiday)
JOIN dbo.Calendar c ON c.[Date] = a.FromDate+b.[days]
WHERE a.Holiday LIKE 'Easter Sunday%';
```

This produces results like this (only the first two years are shown):

```FromDate                ToDate                  holiday                        WkDName
1990-04-13 00:00:00.000 1990-04-13 00:00:00.000 Good Friday 1990               Friday
1990-02-28 00:00:00.000 1990-02-28 00:00:00.000 Ash Wednesday 1990             Wednesday
1990-02-27 00:00:00.000 1990-02-27 00:00:00.000 Mardi Gras (Fat Tuesday) 1990  Tuesday
1991-03-29 00:00:00.000 1991-03-29 00:00:00.000 Good Friday 1991               Friday
1991-02-13 00:00:00.000 1991-02-13 00:00:00.000 Ash Wednesday 1991             Wednesday
1991-02-12 00:00:00.000 1991-02-12 00:00:00.000 Mardi Gras (Fat Tuesday) 1991  Tuesday
```

So now we can be confident to insert those holidays into our Holidays table.

```INSERT INTO dbo.Holidays (FromDate, ToDate, Holiday)
SELECT FromDate = a.FromDate+b.[days]
,ToDate     = a.FromDate+b.[days]
,b.holiday
FROM dbo.Holidays a
CROSS APPLY
(
VALUES(-2, 'Good Friday ' + CAST(YEAR(FromDate) AS CHAR(4)))
,(-46, 'Ash Wednesday ' + CAST(YEAR(FromDate) AS CHAR(4)))
,(-47, 'Mardi Gras (Fat Tuesday) ' + CAST(YEAR(FromDate) AS CHAR(4)))
) b ([days], holiday)
WHERE a.Holiday LIKE 'Easter Sunday%';
```

A couple of notes on the data types used in our Calendar and Holidays tables:

• The Date column of the Calendar table was chosen to be DATETIME to retain compatibility with SQL 2005 (corresponding to the GenerateCalendar FUNCTION).  You could just as easily use a DATE datatype if you’re running SQL 2008 or later.
• The FromDate/ToDate data types are also both DATETIME.  Here it could be considered important to keep the DATETIME data type to record cases where your business offers half-day holidays.  For example, I’ve seen this for Christmas Eve and New Year’s Eve.
• You can also define a different FromDate than ToDate in the Holidays table to indicate a holiday that spans several days, like the Thailand Songkran holiday example from my blog.
```“Since SQL is a database language, we prefer to do look ups and not calculations.”
–- Joe Celko (prolific author of books on things SQL)
```

There you go Joe, we’ve honored your preference!  Now we can do lookups of not just Easter Sunday, but also Good Friday, Ash Wednesday and Mardi Gras, the latter potentially being essential to planning your next trip to New Orleans.

## The Calculation of Easter Sunday

A somewhat accurate description of the calculation of Easter Sunday is:

```“The Sunday following the full Moon which falls on or after the equinox (the Paschal full moon) will give the lawful Easter.”
–- From the Wikipedia link on Easter Sunday above
```

I say “somewhat accurate” because there’s all sorts of ifs, buts and caveats in that description, which you can read about in the Wikipedia link, however it will suffice for the time being.

The formal description of the calculation of Easter is known as Computus (no relation to Locutus of Borg), and is provided by Wikipedia at the link shown.  We shall observe this calculation in what follows.  While the formulas given are indicated to be valid for the years 1900 to 2199, since our Calendar table has been populated with years 1990-2169 we’ll limit our calculations of Easter Sunday to just those the years.  Years 1990 through 2020 will be checked against the lookup table we’ve already provided.

Of course, this calculation appears often enough in the literature and implemented in T-SQL (for example: here and here), often using methods I’d prefer to avoid due to performance.  Since our personal mantra is: “No Loops.  No Cursors.  No RBAR. Hoo-uh!” we shall endeavor to live up to that mantra today.  Think about that term “RBAR” and if you’ve never heard of it before, you’ll find it mentioned in the Conclusion to this article.

We’ll start by attempting to replicate the tabular example for calculating the Paschal full moon (PFMD) within the Gregorian calendar shown by Computus on that Wikipedia page.

Note that the algorithm described above is a step-by-step or procedural algorithm. Here is where many SQL implementations fail because they don’t try to convert it to one that is set-based. To do this, you must think of each step in the procedural algorithm as a query, and as you proceed from one step to the next, each new step is another outer layer to your set-based query.

This is easier shown that described, so here is that query for 2013-2032.

```SELECT [Year], [Y MOD 19], Addendum
,PFMD       = CASE WHEN PFMD > 31 THEN PFMD-31 ELSE PFMD END
,[Month]    = CASE WHEN PFMD > 31 THEN 'Apr' ELSE 'Mar' END
,[MonthNo]  = CASE WHEN PFMD > 31 THEN 4 ELSE 3 END
FROM
(
SELECT [Year], [Y MOD 19], Addendum
FROM
(
SELECT [Year]
,[Y MOD 19] = [Year]%19
WHEN 5 THEN 29
WHEN 16 THEN 29
WHEN 8 THEN 30
ELSE 0
END
FROM dbo.Calendar
WHERE [Month] = 1 AND [Day] = 1
) a
) a
WHERE [Year] BETWEEN 2013 AND 2032;
```

Note how the Addendum (CASE) implements the first two IFs (second and third rows) of the procedural algorithm. The results from this query for just the first 10 years are:

```Year  Y MOD 19  Addendum  PFMD  Month  MonthNo
2013  18        0         27    Mar    3
2014  0         0         14    Apr    4
2015  1         0         3     Apr    4
2016  2         0         23    Mar    3
2017  3         0         11    Apr    4
2018  4         0         31    Mar    3
2019  5         29        18    Apr    4
2020  6         0         8     Apr    4
2021  7         0         28    Mar    3
2022  8         30        16    Apr    4
```

If we take PFMD and add the first letter of the month column that follows (for example for 2013: 27M) we see exact correspondence with the row from the Computus table (graphic) labeled the “Paschal full moon date,” including for the years not shown in the results set.  So it appears that we have rather simply converted the formula for calculating PFMD from Computus to a SQL script without too much effort.

Now we’ll use our Calendar table yet again, along with a little additional magic:

• Put the pieces of the Paschal full moon date back together into a date using a formula that includes a couple of calls to the T-SQL built-in function DATEADD.
• Do a CROSS APPLY (explained in Part 1 and Part 2 by SQL MVP Paul White) to pick up the Sunday after the Paschal full moon, which is Easter Sunday.
```WITH CalculatePaschalFullMoon AS
(
-- Original query to calculate the components of the
-- Paschal Full Moon date
SELECT a.[Year], [Y MOD 19], Addendum
,PFMD       = CASE WHEN PFMD > 31 THEN PFMD-31 ELSE PFMD END
,[Month]    = CASE WHEN PFMD > 31 THEN 'Apr' ELSE 'Mar' END
,[MonthNo]  = CASE WHEN PFMD > 31 THEN 4 ELSE 3 END
-- Put the date/time pieces together to get a real DATETIME for PFMD
,[PFMD2]    =
DATEADD(day, CASE WHEN PFMD > 31 THEN PFMD-31 ELSE PFMD END-1 -- PFMD
,DATEADD(month, CASE WHEN PFMD > 31 THEN 4 ELSE 3 END-1   -- [MonthNo]
, a.[Date]))
FROM
(
SELECT [Year], [Y MOD 19], Addendum, [Date]
FROM
(
SELECT [Year]
,[Y MOD 19] = [Year]%19
WHEN 5 THEN 29
WHEN 16 THEN 29
WHEN 8 THEN 30
ELSE 0
END
,[Date]
FROM dbo.Calendar
WHERE [Month] = 1 AND [Day] = 1
) a
) a
)
SELECT a.[Year]
,PaschalFullMoon=CAST(a.PFMD2 AS DATE)
,EasterSunday
FROM CalculatePaschalFullMoon a
-- Easter Sunday follows the Paschal Full Moon date
-- so pick that up from the Calendar table
CROSS APPLY
(
SELECT TOP 1 EasterSunday=CAST([Date] AS DATE)
FROM dbo.Calendar b
WHERE b.[Date] > a.PFMD2 AND b.WkDName3 = 'Sun'
ORDER BY [Date]
) b
WHERE [Year] BETWEEN 2013 AND 2032;
```

 Now for a quiz on the above code where the lines are bold: Why do we subtract 1 from each of the CASE statements that appear as the second argument to DATEADD? We’ll give you the answer to this in the Conclusion section of this article, so stay tuned… Here are the first ten years of the results set from the last T-SQL query shown above: ```Year PaschalFullMoon EasterSunday 2013 2013-03-27 2013-03-31 2014 2014-04-14 2014-04-20 2015 2015-04-03 2015-04-05 2016 2016-03-23 2016-03-27 2017 2017-04-11 2017-04-16 2018 2018-03-31 2018-04-01 2019 2019-04-18 2019-04-21 2020 2020-04-08 2020-04-12 2021 2021-03-28 2021-04-04 2022 2022-04-16 2022-04-17 ``` Our Computus Wikipedia page also has the inset table (at right) presenting the Western calculation of Easter Sunday, which corresponds quite nicely to our calculation of Easter Sunday. That really wasn’t so hard was it? Now comes the packaging into a tool. Since we like to keep things clean, we’re going to make this wonderful bit of SQL code into an in-line, table-valued function that we can call anytime we need to calculate Easter Sunday for a specific year. We’ll then continue on to use our tool to calculate the other holidays by using the known number of days offset, just like we did before with our Easter Sunday lookup table, and insert that result into our Holidays table for years that are missing.
```CREATE FUNCTION CalculateEasterSunday
(
@Year       INT
)
-- Calculate Easter Sunday for any given year
RETURNS TABLE WITH SCHEMABINDING
AS RETURN

WITH CalculatePaschalFullMoon AS
(
-- Original query to calculate the components of the
-- Paschal Full Moon date
SELECT a.[Year] --, [Y MOD 19], Addendum
--,PFMD       = CASE WHEN PFMD > 31 THEN PFMD-31 ELSE PFMD END
--,[Month]    = CASE WHEN PFMD > 31 THEN 'Apr' ELSE 'Mar' END
--,[MonthNo]  = CASE WHEN PFMD > 31 THEN 4 ELSE 3 END
-- Put the date/time pieces together to get a real
-- DATETIME for PFMD
,[PFMD2]    =
DATEADD(day, CASE WHEN PFMD > 31
THEN PFMD-31
ELSE PFMD
END-1           -- PFMD
WHEN PFMD > 31
THEN 4
ELSE 3
END-1           -- [MonthNo]
, a.[Date]))
FROM
(
SELECT [Year], [Y MOD 19], Addendum, [Date]
FROM
(
SELECT [Year]
,[Y MOD 19] = [Year]%19
WHEN 5 THEN 29
WHEN 16 THEN 29
WHEN 8 THEN 30
ELSE 0
END
,[Date]
FROM dbo.Calendar
WHERE [Month] = 1 AND [Day] = 1 AND [Year] = @Year
) a
) a
)
SELECT a.[Year]
,PaschalFullMoon=CAST(a.PFMD2 AS DATE)
,EasterSunday
FROM CalculatePaschalFullMoon a
-- Easter Sunday follows the Paschal Full Moon date
-- so pick that up from the Calendar table
CROSS APPLY
(
SELECT TOP 1 EasterSunday=CAST([Date] AS DATE)
FROM dbo.Calendar b
WHERE b.[Date] > a.PFMD2 AND b.WkDName3 = 'Sun'
ORDER BY [Date]
) b;
```

You can see where I commented out a few lines of code in the FUNCTION that represent intermediate results that our FUNCTION doesn’t really need to retain. Notice also how the filter on the @Year parameter was moved up into the innermost SELECT from the Calendar table, because when using the FUNCTION you’ll be selecting only one row at a time at that point. Calling the iTVF using CROSS APPLY in-lines the code to make this a non-RBAR query.

Now we can use our FUNCTION to check that the dates we’ve prior stored for Easter Sunday in our Holidays table are correct.

```SELECT HolidayFromDate  = CAST(a.FromDate AS DATE)
,b.EasterSunday
,[Check]            = CASE
WHEN a.FromDate = b.EasterSunday
THEN 'MATCHED!'
ELSE 'WRONG!'
END
FROM dbo.Holidays a
CROSS APPLY dbo.CalculateEasterSunday(YEAR(a.FromDate)) b
WHERE Holiday LIKE 'Easter Sunday%';
```

The first ten years of our results show us all MATCHED! (and so do the others that are not shown).

```HolidayFromDate  EasterSunday   Check
1990-04-15       1990-04-15     MATCHED!
1991-03-31       1991-03-31     MATCHED!
1992-04-19       1992-04-19     MATCHED!
1993-04-11       1993-04-11     MATCHED!
1994-04-03       1994-04-03     MATCHED!
1995-04-16       1995-04-16     MATCHED!
1996-04-07       1996-04-07     MATCHED!
1997-03-30       1997-03-30     MATCHED!
1998-04-12       1998-04-12     MATCHED!
1999-04-04       1999-04-04     MATCHED!
```

So we can be reasonably confident of our Easter Sunday calculation for the years indicated by Computus (1900-2199)!

## Using our FUNCTION to Create Lookup Values in our Holidays Table

Now to keep Joe Celko happy (i.e., so we can do lookups instead of calculations), let’s insert the rest of our holidays, including Easter Sunday, Good Friday, Ash Wednesday and Mardi Gras, into our Holidays table, using all the years available to us in our Calendar table.

```INSERT dbo.Holidays (FromDate, ToDate, Holiday)
SELECT FromDate, ToDate, holiday
FROM
(
SELECT FromDate = DATEADD(day, c.days, b.EasterSunday)
,c.holiday
FROM dbo.Calendar a
CROSS APPLY dbo.CalculateEasterSunday(a.[Year]) b
CROSS APPLY
(
VALUES (0, 'Easter Sunday ' + CAST(a.[Year] AS CHAR(4)))
,(-2, 'Good Friday ' + CAST(a.[Year] AS CHAR(4)))
,(-46, 'Ash Wednesday ' + CAST(a.[Year] AS CHAR(4)))
,(-47, 'Mardi Gras (Fat Tuesday) ' + CAST(a.[Year] AS CHAR(4)))
) c ([days], holiday)
WHERE a.[Month] = 1 AND a.[Day] = 1
) a
WHERE NOT EXISTS
(
SELECT 1
FROM dbo.Holidays x
WHERE x.FromDate = a.FromDate AND x.ToDate = a.ToDate
);
```

This added 596 holidays to our table, and I’d guess we’re pretty well set for our lifetimes!

Notice how, in the above query we had to use DATEADD instead of simple arithmetic as we did before when using the DATETIME data type.  This is because simple arithmetic doesn’t work with the DATE data type returned by our CalculateEasterSunday function.

## Conclusion and What we Learned

While Joe likes lookups, I find calculations in T-SQL to be quite a bit of fun.  In this article (as in my blog on Calendar tables), we got to do a little of both.

If you went ahead and read my blog on Calendar tables in conjunction with this article, I am pleased because you probably learned a lot about how useful they can be, particularly in conjunction with a Holidays table. In this article (in the FUNCTION) we used our Calendar table to find the next Sunday occurring after our calculated date of the Paschal full moon.

Besides learning about the trivialities of the relationship between Easter, Good Friday, Ash Wednesday, and planning for your next visit to New Orleans for Mardi Gras, we’d like to think that you may have learned a few other things as well:

• You can use simple arithmetic on a DATETIME data type to add or subtract days, however you must use DATEADD to do the same thing with a DATE data type.
• You now have efficient tools in your toolbox to generate a calendar table (GenerateCalendar) and CalculateEasterSunday, whenever you have a need to use them.  The latter function is also included in the resources file attached at the end of this article.
• You learned how to CROSS APPLY VALUES to do an UNPIVOT operation, even though I didn’t specifically mention that was what we were doing when we used it (where we created rows with dates for Good Friday, Ash Wednesday and Mardi Gras).  If you haven’t seen that method before you may want to look into that link as well. That is something that will only work in SQL 2008 (or later), but there is an equivalent construct that will work in SQL 2005.
• We learned how to use ISNULL on a SELECT/INTO operation, to set the attribute of the column created in the target table to NOT NULL.
• You learned how to think your way through a rather complex, procedural calculation and convert it into set-based code, by wrapping each successive step in another layer of query.
• You learned my personal SQL mantra, in case you hadn’t seen that before.

In case you’ve been wondering what RBAR stands for, it means “Row By Agonizing Row” and it is a term coined by Jeff Moden which I fortuitously picked up on before I got to know him.

Oh yes, then there’s the quiz I mentioned. There are two cases:

• If we’re starting with 01 Jan of a year as our date, and the Paschal full moon month is three (meaning it should fall in Mar), if we add three months to 01 Jan it puts us in Apr. So we subtract one month to get us back to where we need to be.
• If we’re starting with 01 Mar of a year as our date, and the Paschal full moon day within that month is (say) 14 (meaning it should fall on 14 Mar), if we add 14 days to 01 Mar it puts us in 15 Mar. So we subtract one day to get us back to where we need to be.

After all this, I still have no clue what a Paschal full moon is but what the heck, at least I know I can calculate when it is!  That’s all for today folks.  I hope you found this article useful and just a bit of fun.

Resources for An Easter SQL.doc

The Resources above are stored in a Microsoft Word (TM) document. There are two sections:

```-- Create GenerateCalendar, Calendar and Holiday Tables and Populate.sql
... page break ...
-- Create CalculateEasterSunday and Add Holidays.sql
```