How often does December 11 fall on Saturday over the 400 year Gregorian calendar cycle from 1600 to 2000?

Answer 1

11 December has had 56 Saturdays between 1600 and 2000 in the Gregorian calendar.

In the Gregorian calendar, centuries are leap years only if the year is divisible by 400, which means that 1700, 1800 and 1900 were not leap years.

Every year, the day of a date goes forward 1 day except in a leap year when it goes forward 2 days; this leads to cycle of 28 days that any date could have:

{Mo, Tu, We, Th}, {Sa, Su, Mo, Tu}, {Th, Fr, Sa, Su}, {Tu, We, Th, Fr}, {Su, Mo, Tu, We}, {Fr, Sa, Su, Mo}, {We, Th, Fr, Sa}

The list is in blocks of four that come between each leap year - each day of the week starts one of the blocks, and each day of the week occurs in each position in a block, one in each of fours block.

When a century is not a leap year, the sequence jumps forward 5 blocks instead of to the next block.

As a century (100 years) is divisible by 4, an exact number of blocks (100 ÷ 4 = 25) will occur within a year, and as we are interested only in Saturdays, the sequence of which blocks contain Saturday makes life easier: NYYNNYY

As 25 blocks will be used in any year, and there is a sequence of 7 blocks, they will all be used 3 times and the first 4 will be repeated at the end: 3 x 7 + 4 = 25.

If every century was a leap year, the sequence would continue across the centuries; however, with the Gregorian calendar centuries are only leap years if they are divisible by 400 - only 1 in every 4 century years is a leap year. As a result, the sequence will jump blocks at non-leap year centuries, as the next year's day for any given date is the one after the current day.

Looking at the full sequence above, it can be seen that when a non-leap year occurs, the sequence jumps an extra 4 blocks forward, making the next block 2 back from the current block.

Supposing we number the blocks 0, 1, ..., 6 with block 0 the first block used in 1600-1603. Then the sequence of blocks used in each century is:

1600s - 0123456 0123456 0123456 0123

1700s - 1234560 1234560 1234560 1234

1800s - 2345601 2345601 2345601 2345

1900s - 3456012 3456012 3456012 3456

2000s - 01234...

which means that each block of 4 centuries starting with a leap century

repeats the same sequence of the blocks, with each new century starting one block later.

One thing to note is when the block jump occurs. For Jan 1 to Feb 28 which are before the leap day (Feb 29), their blocks will run for the years 01-04, 05-08, ..., 97-00; and for those after the leap day, their blocks will run for the years 00-03, 04-07, ..., 96-99.

So now we are ready to find the number of Saturdays for 11 December for 1600-2000.

It is easier to work backwards from 2000 as the day for 11 December 2000 can easily be found.

11 December is after the leap day, so each block is used for years 00-03, 04-07, ..., 96-99. 2000 was a leap year, so no block jump occurs before it.

11 December 2000 was a Monday and is the first day of a block; using the blocks above it is in the first block, so let's number that block 0 and the rest 1-6 in order. Now, using the sequence above for the blocks used and whether they contain Saturdays, we can write out which blocks in 1600-2000 contains a Saturday. However, as each block is used 3 times there will be 3 x 4 = 12 Saturdays, plus however many Saturdays occur in the first 4 blocks of each century that are repeated; thus we need only note those 4 blocks:

1600s: NYYN - 2 Saturdays

1700s: YYNN - 2 Saturdays

1800s: YNNY - 2 Saturdays

1900s: NNYY - 2 Saturdays

An extra 8 Saturdays.

Which means there are 4 x 12 + 8 = 56 Saturdays.