Daylight Saving Time Calculations

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Mensa Daily Puzzlers

For over 15 years Mensa Page-A-Day calendars have provided several puzzles a year for my programming pleasure.  Coding "solvers" is most fun, but many programs also allow user solving, convenient for "fill in the blanks" type.  Below are Amazon  links to the two most recent years.

Mensa 365 Puzzlers  Calendar 2017

Mensa 365 Puzzlers Calendar 2018

(Hint: If you can wait, current year calendars are usually on sale in January.)


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Problem Description

This program  investigates how Daylight Saving Time start and end dates can be calculated using simple equations without resorting to high level date-time routines.

Background & Techniques

Daylight Saving Time is not one of mankind's greatest inventions, but it does provide an excuse to study the
mathematics of associating day of the week with the day of the month.

Delphi and most other high level languages incorporate date-time procedures to simplify the calculation, but
there are thousands of devices with embedded processors which may not have this option but still need to
accurately track the current local time. I recently researched the issue for a manufacturer of such a device who
had found formulas but wanted to understand what was behind the "magic" so they can adapt to various
localized DST definitions or future definition changes.

Most of the world (Asia, Africa, and South America except Brazil) avoid DST altogether by not defining it. USA
(except for Arizona) and the Euro zone have formal definitions. The current USA definition took effect in 2007
and the European definition in 1998. The definitions below include equations found at which also includes attributions for the developers of the

bulletUSA prior to 2007 began DST on the 1st Sunday in April (Day of month = 1 + (2+6*Y-Y/4) mod 7) and ended it on the last Sunday of October (Day of month = 31- (1 + 5*Y/4) mod 7).
bulletUSA starting in 2007, extended DST to begin on the 2nd Sunday in March (Day of month = 14 - (1+Y*5/4)
mod 7
) and end on the 1st Sunday in November (Day of month = 7 - (1 + 5*Y/4) mod 7).
bulletFor Europe, DST begins on the last Sunday in March (Day of month = 31 - (4+ 5*Y/4 ) mod 7) and ends on the last Sunday in October (Day of month = 31 - (1 + 5*Y/4) mod 7).

Notice that we have six equations with the last five being very similar in structure.  Y in the equations is the 4 digit year. For consistency,  I have rearranged the  terms  from the way they appear on the webexhibits site.  All of the  "/4" operations represent  Integer division, i.e. any fractions are dropped so the term is always a whole number.   "Mod 7" terms are modulus operations which return the remainder (0 to 6) when the first operand is divided by 7.  That explains the initial 1+ in the first equation since we need a number from 1 to 7 for the day of month for the first Sunday.  The other five equations have initial terms of  7-, 14-, or 31- representing the 1st, 2nd,  or last Sunday of a month so their Mod 7 term must return the number of days from that latest date back to a Sunday.

If we look at the Mod 7 terms, notice that the 6Y-y/4  in Equation 1 says that we are going to add 6 days for every year, except  we'll subtract one day for each leap year.  On first look this seems counterintuitive since a normal year is 52 weeks + 1 day.  But it makes sense  if you think in terms of what happens to a particular day of the week from year to year.  It's true that the date moves forward by one day per year, but that also means that the day of week moves backward  one day or forward 6 days, whichever is more convenient per year. 

 In the strange world of modulo arithmetic, it turns out that (-Y mod 7) is the equivalent to (6Y mod 7) so we could replace 6Y with -Y and  (6Y-Y/4) with (-Y-Y/4 = -5Y/4), the same term that appears in the other five equations. The only problem is that whatever day that represents might apply to any 7 day increment before or after that date.  Enter the "magic" number added to -5Y/4 that makes the equation apply to a Sunday.    These are already provided by the authors for the five equations, but let us find one for the 1st equation if we want to put it into our new "standard" form.  For the 1st Sunday in April we'll one known value and solve Day of Month = 7 - (X + 5Y/4) mod 7  for X.   Any year should do so we'll try 2012 where April 1 is a Sunday so we need to solve:

bullet  1 = 7 - (X + 5*2012 / 4) mod 7
bullet Rearranging:  (X+ 5 * 2012 / 4) mod 7 = 7 - 1 = 6
bullet The Mod operator is distributive, so X mod 7 +  2515 mod 7 = 6
bullet X mod 7 + 2  = 6 ==>  X mod 7 = 4  ==>  X = 4

So our revised equation for the first Sunday in April is:    Day of Month = 7 - (4 + 5Y/4) mod 7.  Checking for 2006 and 2013 we get 4 and  7; both correct so  4 seems to be the correct "magic" number   

 Note that none of the equations take into account 100 year and 400 year Leap Year exception rules so they only provide valid results from 1900 through 2099.  That range is probably sufficient for DST calculations.

The program implements these equations and also a version using the high level date-time routines of Delphi.
The results are compared so that for a range of input years you can display all results or only the "errors"
(useful for debugging).  Results agree within the 1900 - 2099 date range discussed above.

Running/Exploring the Program 

bulletDownload  executable
bulletDownload source 
bulletDownload current library file  One time download of (DFFLibV15) is required to recompile DSTCalc.

Suggestions for Further Explorations

Adding the 100 and 400 year exceptions would allow  more generalized calculations for any given occurrence of any day of week in any month and year.  


Original:  April 20,2012

Modified:  May 15, 2018

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