ID specific years which use the 14 leap and non leap year formats, beginning in the year 1700 to year 2100.?

Question

It may have been difficult to understand the question as it was posed due to limits on number of characters to use.

I do remember seeing a list of years for each yearly format possible. One format would be years beginning on Sunday that are NOT leap years, another beginning on Sunday that IS a leap year, and so on for all the days of the week. Problem is, I can't find it on the internet.

Answer 1

I think I understand your question, and I know how to figure it out. I'm not sure why you picked 1700-2100, but you do have a 400-year span, and that's correct.

Here's the problem. A year can begin with any of 7 days Sun-Sat, and a year can be a leap year or not. Therefore, there are 7x2=14 different calendar arrangements.

A regular year has 365 days, which is 52 weeks and one day. So, for example, if one regular year begins on a Tuesday, the next year begins on a Wednesday.

A leap year has 366 days, which is 52 weeks and two days. So if a leap year begins on a Friday, the next year begins on Sunday.

Every year divisible by 4 (1700, 1704, 1708, etc.) is a leap year "century" years not divisible by 400. Thus, 1700, 1800, 1900, and 2100 are/were not leap years, but 2000 was. In a 400-year period, there are 97 leap years.

That's enough to get you going. This year (2006) began on a Sunday, so you could work forward and backward from there.

But here's a shortcut. Every years (including a leap year) have 1461 days, which is 208 weeks and five days. So you could make your 400-year table by listing every year instead of every single year. That cuts your work by a factor of four.

2008 will be a leap year beginning on a Tuesday ... 2006 (Sunday) plus two days. Backing up from there, 2004 began five days Tuesday ... it began on a Thursday. And instead of going five days, you can go two days. Easier, and you get the same result. Thursday is five behind Tuesday, but it's also two days ahead of Tuesday.

I think this is enough. You can make a list, all on one sheet of paper (4 columns of 25 entries each) listing every fourth year from 1700 to 2100, and, starting from 2008, enter the first day of every fourth year.

The only places you'll have to be careful are the years 1700, 1800, and 1900, because they were leap years. There, instead of adding two days (or subtracting five), you'll have to add three days (or subtract four).

Last comment. There will be a pattern in here somewhere. This whole thing is repetitive, so look for the pattern. The problem, though, is that it might take more than 400 years before it repeats.

I think that does it.