Is it possible to have such date and day combination that doesn't occur on any year ever?

Question

My friend told me a story yesterday and said "It happened on March 13th and it was Friday".

I was thinking about this and I've a question. Is it possible to have such date and day combination that it doesn't occur on any year ever?

For example, would it be possible to say that "March 13th" is never Friday (no matter which year). Or for example "Thursday" can never be on "April 3", etc.

Is there such date that doesn't occur on any year no matter what year (future or past)?

Answer 1

Every once in a while there is a seven year period with no leap years. Each of these seven years has $365$ days. Since $365\equiv1$ mod $7$, the days of the week shift by one each year. So at a minimum, in this seven year period, any date (except Feb 29) falls on each of the seven days of the week once.

Even when you consider leap years, it just means there is a shift of two days one year instead of just by one day. It may take a few more years, but again each date will fall on each day of the week. For instance, right after a leap year, the shift sequence will be $\{0,1,2,4,5,6,0,2,3,\ldots\}$ so it will only take 9 years for each date to fall at least once on each day of the week.

Answer 2

There are $7$ possible leap years (where January 1 is Sunday, Monday, ..., Saturday). The most recent examples of the $7$ leap years are 2012, 1996, 2008, 1992, 2004, 1988, 2000. Each month+day occurred on every day of the week in one of those 7 years.