[math-fun] Weeks Between Easters
Long before even one round of the 5700000-year Gregorian Easter cycle is up, we will likely no longer use that formulation, assuming even that mankind survives that long. Nevertheless, as a mathematical entity - there it is, straddling the boundary between religious necessity and astronomical reality. When two separate Easters are one year apart, they can be 50, 51, 54, or 55 weeks apart. When they are 3 years apart, there are 5 weekly possibilities. When they are 10 years apart, there are 6 weekly possibilities. When they are 125 years apart, there are 7 weekly possibilities. Can there ever be 8 weekly possibilities?
I was just thinking about this, while I was making my yearly attempt to wrap my head around the official Easter algorithm. The most coherent statement of the algorithm that I know comes from Jean Meeus's masterful _Astronomical Algorithms_. In case anyone on the list doesn't have access to a clear statement of the official algorithm, I'm going to present it here, slightly adapted. First an executive summary. To find Easter Sunday in a given year, one begins on March 21 to wait for an "ecclesiastic full moon"; the following day one begins to wait for a Sunday, which will be the desired Easter Sunday. It is clear that all the ensuing mess is due to the details of what constitutes an "ecclesiastic full moon". The ecclesiastic full moon recurs every 29 or 30 days, with the pattern of short and long months determined by a complicated rule. It would be nice to be able to tell you that this rule is essentially that there is an "ecclesiastic full moon instant" with a fixed period, and that this instant is then rounded to the nearest day in some fashion. Alas, this is not what is going on. To this day I'm not sure what is. The 19-year Metonic cycle is certainly involved: this states that every 19 solar years contains 235 lunar months, an assertion which is false by only a few hours. Anyway, here is the algorithm that I adapted from Meeus, who got it from Spencer Jones. I'm not sure who determined that it accurately reflects the official definitions. Each line is an "assignment statement" of the form KX + Y = Z, where Z is an expression of some sort, K is a constant, and X and Y are variables which thereby take on the values [Z/K] and Z mod K. Suppose the given year is y. The variable x is never needed after assignment, but is included only by the requirements of this syntax, so I reuse it freely. 1. 19x + a = y. 2. 100b + c = y. 3. 4d + e = b. 4. 25f + x = b + 8. 5. 3g + x = b - f + 1. 6. 30x + h = 19a + b - d - g + 15. 7. 4i + k = c. 8. 7x + l = 32 + 2e + 2i - h - k 9. 451m + x = a + 11h + 22l 10. 31n + p = h + l - 7m + 114 Then (saith Meeus), n is the number of the month (3 for March, 4 for April), and p + 1 is the day of the month on which Easter Sunday falls. Let us work 2009 as an example. 1. 19*105 + 14 = 2009; a = 14. 2. 100*20 + 9 = 2009; b = 20, c = 9. 3. 4*5 + 0 = 20; d = 5, e = 0. 4. 25*1 + 3 = 28; f = 1. 5. 3*6 + 2 = 20; g = 6. 6. 30*9 + 20 = 290; h = 20. 7. 4*2 + 1 = 9; i = 2, k = 1. 8. 7*2 + 1 = 15; l = 1. 9. 451*0 + 256 = 256; m = 0 10. 31*4 + 11 = 135; n = 4, p = 11. You can imagine my relief at discovering that this algorithm predicts Easter Sunday to fall on April 12. In principle you now possess all you need to answer Hans Havermann's question. In practice ... well, now. On Sat, Apr 11, 2009 at 2:03 PM, Hans Havermann <pxp@rogers.com> wrote:
Long before even one round of the 5700000-year Gregorian Easter cycle is up, we will likely no longer use that formulation, assuming even that mankind survives that long. Nevertheless, as a mathematical entity - there it is, straddling the boundary between religious necessity and astronomical reality.
When two separate Easters are one year apart, they can be 50, 51, 54, or 55 weeks apart. When they are 3 years apart, there are 5 weekly possibilities. When they are 10 years apart, there are 6 weekly possibilities. When they are 125 years apart, there are 7 weekly possibilities. Can there ever be 8 weekly possibilities?
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I neglected to mention that this algorithm works only after the official start of the Gregorian calendar, in 1583. You can apply it as a mathematical function to earlier years, but the results will not tell you when Easter was actually celebrated. On Sat, Apr 11, 2009 at 4:50 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I was just thinking about this, while I was making my yearly attempt to wrap my head around the official Easter algorithm. The most coherent statement of the algorithm that I know comes from Jean Meeus's masterful _Astronomical Algorithms_. In case anyone on the list doesn't have access to a clear statement of the official algorithm, I'm going to present it here, slightly adapted.
First an executive summary. To find Easter Sunday in a given year, one begins on March 21 to wait for an "ecclesiastic full moon"; the following day one begins to wait for a Sunday, which will be the desired Easter Sunday.
It is clear that all the ensuing mess is due to the details of what constitutes an "ecclesiastic full moon". The ecclesiastic full moon recurs every 29 or 30 days, with the pattern of short and long months determined by a complicated rule. It would be nice to be able to tell you that this rule is essentially that there is an "ecclesiastic full moon instant" with a fixed period, and that this instant is then rounded to the nearest day in some fashion. Alas, this is not what is going on. To this day I'm not sure what is. The 19-year Metonic cycle is certainly involved: this states that every 19 solar years contains 235 lunar months, an assertion which is false by only a few hours.
Anyway, here is the algorithm that I adapted from Meeus, who got it from Spencer Jones. I'm not sure who determined that it accurately reflects the official definitions. Each line is an "assignment statement" of the form KX + Y = Z, where Z is an expression of some sort, K is a constant, and X and Y are variables which thereby take on the values [Z/K] and Z mod K. Suppose the given year is y. The variable x is never needed after assignment, but is included only by the requirements of this syntax, so I reuse it freely.
1. 19x + a = y. 2. 100b + c = y. 3. 4d + e = b. 4. 25f + x = b + 8. 5. 3g + x = b - f + 1. 6. 30x + h = 19a + b - d - g + 15. 7. 4i + k = c. 8. 7x + l = 32 + 2e + 2i - h - k 9. 451m + x = a + 11h + 22l 10. 31n + p = h + l - 7m + 114
Then (saith Meeus), n is the number of the month (3 for March, 4 for April), and p + 1 is the day of the month on which Easter Sunday falls.
Let us work 2009 as an example.
1. 19*105 + 14 = 2009; a = 14. 2. 100*20 + 9 = 2009; b = 20, c = 9. 3. 4*5 + 0 = 20; d = 5, e = 0. 4. 25*1 + 3 = 28; f = 1. 5. 3*6 + 2 = 20; g = 6. 6. 30*9 + 20 = 290; h = 20. 7. 4*2 + 1 = 9; i = 2, k = 1. 8. 7*2 + 1 = 15; l = 1. 9. 451*0 + 256 = 256; m = 0 10. 31*4 + 11 = 135; n = 4, p = 11.
You can imagine my relief at discovering that this algorithm predicts Easter Sunday to fall on April 12.
In principle you now possess all you need to answer Hans Havermann's question. In practice ... well, now.
On Sat, Apr 11, 2009 at 2:03 PM, Hans Havermann <pxp@rogers.com> wrote:
Long before even one round of the 5700000-year Gregorian Easter cycle is up, we will likely no longer use that formulation, assuming even that mankind survives that long. Nevertheless, as a mathematical entity - there it is, straddling the boundary between religious necessity and astronomical reality.
When two separate Easters are one year apart, they can be 50, 51, 54, or 55 weeks apart. When they are 3 years apart, there are 5 weekly possibilities. When they are 10 years apart, there are 6 weekly possibilities. When they are 125 years apart, there are 7 weekly possibilities. Can there ever be 8 weekly possibilities?
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To focus the discussion somewhat back to my original query, I did work on this last year: http://chesswanks.com/pxp/weeksbetweeneasters.html My empirical brute-force approach was able to snapshot a couple of hundred instances out of the maximum 2850000 required for a definitive solution. I had hoped that a theoretical understanding of the Gregorian Easter algorithm might somehow shortcut thereto but perhaps brute-force is all that is possible with this particular arithmetic.
When two separate Easters are one year apart, they can be 50, 51, 54, or 55 weeks apart. When they are 3 years apart, there are 5 weekly possibilities. When they are 10 years apart, there are 6 weekly possibilities. When they are 125 years apart, there are 7 weekly possibilities. Can there ever be 8 weekly possibilities?
participants (2)
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Allan Wechsler -
Hans Havermann