The Solution:
From January 1 of the first year of a 400 year cycle to January 1 of the first year of the following 400 year cycle, there is some net change in the day of the week, which will be some number from 0 to 6. If the number is from 1 to 6, it will take 7 cycles of 400 years before January 1 is the same day of the week as the original. However, if the offset is 0, meaning that the day of the week is the same for January 1 of the next 400 year cycle, then we are already there. The total cycle length is not 2800 years, but 400 years. This is in fact the case as I will show below. Since 400 is not divisible by 7, it is not possible for any day of the week to come up exactly the 1/7 of the time for Christmas, or any other day. Christmas will be on a Friday 58 times out of 400. 400/7 = 58 4/7.
To show that the next 400 year cycle starts on the same day of the week as the previous one, we could add the total number of days in the 400 years and check that it is divisible by 7. A much easier calculation can be made. 365 days is 52 weeks plus one day, so that the first of the year will move up one day of the week for the year after a regular year. Similarly, the fist of the year will move up two week days after a leap year of 366 days. Now all we have to do is add the number of regular years to twice the number of leap years, divide by 7, and check the remainder.
There are ¼ of 400 =100 years divisible by 4. Of these, the years 100, 200 and 300 are not leap years. This gives 100 – 3 =97 leap years and 400 – 97 = 303 regular years. We calculate 303 + 2*97 = 303 + 194 = 497 = 7*71, and so is divisible by 7, meaning that each 400 year cycle starts on the same day, specifically Monday. You might find it easier to remember that the start of the year for each year divisible by 400 starts on Sunday.
This also means that the 400th anniversary of any day falls on the same day of the week as the original. July 11, 1611 must have fallen on a Monday. We can’t go too far back, because the Gregorian calendar, which contains the changes for leap years divisible by 100 and 400, was created in the early 16th century.