Editing Doomsday rule (section)

===Why it works===
[[File:Doomsday Rule.svg|thumb|Doomsday rule]]
The doomsday's anchor day calculation is effectively calculating the number of days between any given date in the base year and the same date in the current year, then taking the remainder modulo 7.  When both dates come after the leap day (if any), the difference is just {{math|365''y'' + {{sfrac|''y''|4}}}} (rounded down).  But 365 equals 52 × 7 + 1, so after taking the remainder we get just

:<math>\left(y + \left\lfloor \frac{y}{4} \right\rfloor\right) \bmod 7.</math>

This gives a simpler formula if one is comfortable dividing large values of {{math|''y''}} by both 4 and 7.  For example, we can compute

:<math>\left(66 + \left\lfloor \frac{66}{4} \right\rfloor\right) \bmod 7 = (66 + 16) \bmod 7 = 82 \bmod 7 = 5</math>

which gives the same answer as in the example above.

Where 12 comes in is that the pattern of <math>\bigl(y + \bigl\lfloor \tfrac{y}{4} \bigr\rfloor \bigr) \bmod 7</math> ''almost'' repeats every 12 years.  After 12 years, we get <math>\bigl(12 + \tfrac{12}{4}\bigr) \bmod 7 = 15 \bmod 7 = 1</math>.  If we replace {{math|''y''}} by {{math|''y'' mod 12}}, we are throwing this extra day away; but adding back in <math>\bigl\lfloor \tfrac{y}{12} \bigr\rfloor</math> compensates for this error, giving the final formula.

For calculating the Gregorian anchor day of a century: three “common centuries” (each having 24 leap years) are followed by a “leap century” (having 25 leap years). A common century moves the doomsday forward by
:<math> (100 + 24) \bmod 7 = 2 + 3 = 5 </math>
days (equivalent to two days back). A leap century moves the doomsday forward by 6 days (equivalent to one day back).

So ''c'' centuries move the doomsday forward by
:<math> \left(5c + \biggl\lfloor {c \over 4} \biggr\rfloor \right) \bmod 7 </math>,
but this is equivalent to
:<math> (5 (c \bmod 4)) \bmod 7</math>.
Four centuries move the doomsday forward by
:<math> -2 - 2 - 2 - 1 = -7, \qquad -7 \equiv 0 \quad \pmod{7}</math>;
so four centuries form a cycle that leaves the doomsday unchanged (and hence the “mod 4” in the century formula).