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Friday the 13th

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First

Show that for any year there must be at least one month and at most three months for which the 13th of the month falls on Friday.

Suggestion: Consider using a spreadsheet and numbering the days of the year by mod 7.

Second*

Observe that in 1998 both February and March have a Friday the 13th.

1. Prove that Friday the 13th can occur in two consecutive months only in February and March in a year that is not a leap year. On what day of the week must January 1 occur for February and March to have Friday the 13ths?

2. What is the next year in which this will occur again?

3. Is there a pattern or cycle by which you can determine which years between 2000 and 2100 that this will occur?

*Problem stated by Joe Hooten, Jr.