This problem has several parts:
1. Show that for any year there must be at least one and at most three months for which the 13th of the month falls on a Friday.
2. Observe that in 1998, both February and March have a Friday the 13th. 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? What is the next year in which this will occur again? Is there a pattern or cycle by which you can determine which years between 2000 and 2100 that this will occur?
For a link to the statement of this problem, click here or here.
The hint for this problem is to use a spreadsheet and number the days of the year by mod 7. I made a spreadsheet with three columns: the date, the day of the year, and the day of the year mod 7. The first few rows of the spreadsheet look like this:
|
Date |
Day of Year | Day Mod 7 |
| Jan-1 | 1 | 1 |
| Jan-2 | 2 | 2 |
| Jan-3 | 3 | 3 |
| Jan-4 | 4 | 4 |
| Jan-5 | 5 | 5 |
| Jan-6 | 6 | 6 |
| Jan-7 | 7 | 0 |
| Jan-8 | 8 | 1 |
| Jan-9 | 9 | 2 |
| Jan-10 | 10 | 3 |
| Jan-11 | 11 | 4 |
| Jan-12 | 12 | 5 |
| Jan-13 | 13 | 6 |
| Jan-14 | 14 | 0 |
| Jan-15 | 15 | 1 |
| Jan-16 | 16 | 2 |
| Jan-17 | 17 | 3 |
| Jan-18 | 18 | 4 |
| Jan-19 | 19 | 5 |
| Jan-20 | 20 | 6 |
| Jan-21 | 21 | 0 |
| Jan-22 | 22 | 1 |
| Jan-23 | 23 | 2 |
| Jan-24 | 24 | 3 |
| Jan-25 | 25 | 4 |
| Jan-26 | 26 | 5 |
| Jan-27 | 27 | 6 |
| Jan-28 | 28 | 0 |
| Jan-29 | 29 | 1 |
| Jan-30 | 30 | 2 |
| Jan-31 | 31 | 3 |
| Feb-1 | 32 | 4 |
| Feb-2 | 33 | 5 |
| Feb-3 | 34 | 6 |
| Feb-4 | 35 | 0 |
| Feb-5 | 36 | 1 |
| Feb-6 | 37 | 2 |
| Feb-7 | 38 | 3 |
| Feb-8 | 39 | 4 |
| Feb-9 | 40 | 5 |
| Feb-10 | 41 | 6 |
| Feb-11 | 42 | 0 |
| Feb-12 | 43 | 1 |
| Feb-13 | 44 | 2 |
| Feb-14 | 45 | 3 |
| Feb-15 | 46 | 4 |
Now we need to look at the thirteenth of every month:
| Date | Day of Year | Day Mod 7 |
| Jan-13 | 13 | 6 |
| Feb-13 | 44 | 2 |
| Mar-13 | 72 | 2 |
| Apr-13 | 103 | 5 |
| May-13 | 133 | 0 |
| Jun-13 | 164 | 3 |
| Jul-13 | 194 | 5 |
| Aug-13 | 225 | 1 |
| Sep-13 | 256 | 4 |
| Oct-13 | 286 | 6 |
| Nov-13 | 317 | 2 |
| Dec-13 | 347 | 4 |
Click here for a demonstration of the fact that for any year, there is at least one month and at most three months for which the 13th of the month falls on Friday.
Click here for a discussion of a Friday the 13th in two consecutive months.