FRIDAY THE 13TH

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?

### To start off, I looked at all possible calendars.  There are 7 days of week, and we can have a Leap Year or a Non-Leap Year, therefore, there are 14 possible calendars.  Here are all possible calendars:

 Leap Year Non-Leap Year

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

The method above can be quite time consuming, and also tedious having to look at each different calendar.  So another way available is by using the Mod function in Excel.  This allows all dates in a Leap Year to be in one spreadsheet, and very easy to move through.  See the following spreadsheets – one for Leap Years, one for Non-Leap Years.

Note: On each of the spreadsheets, the 13th of each month is highlighted in Yellow, and when the 13th falls on a Friday, it is in Red.

Now, in summary, you can determine that on the given months, you have a Friday the 13th.

Note the pattern and breakdown of 13th dates:

Non-Leap Years

 1st of January on: Months with Friday the 13th Sunday January, October Monday April, July Tuesday September, December Wednesday June Thursday February, March, November Friday August Saturday May

 Sunday Start Monday Start Tuesday Start Wednesday Start 13-Jan 12 5 13-Jan 13 6 13-Jan 14 0 13-Jan 15 1 13-Feb 43 1 13-Feb 44 2 13-Feb 45 3 13-Feb 46 4 13-Mar 71 1 13-Mar 72 2 13-Mar 73 3 13-Mar 74 4 13-Apr 102 4 13-Apr 103 5 13-Apr 104 6 13-Apr 105 0 13-May 132 6 13-May 133 0 13-May 134 1 13-May 135 2 13-Jun 163 2 13-Jun 164 3 13-Jun 165 4 13-Jun 166 5 13-Jul 193 4 13-Jul 194 5 13-Jul 195 6 13-Jul 196 0 13-Aug 224 0 13-Aug 225 1 13-Aug 226 2 13-Aug 227 3 13-Sep 255 3 13-Sep 256 4 13-Sep 257 5 13-Sep 258 6 13-Oct 285 5 13-Oct 286 6 13-Oct 287 0 13-Oct 288 1 13-Nov 316 1 13-Nov 317 2 13-Nov 318 3 13-Nov 319 4 13-Dec 346 3 13-Dec 347 4 13-Dec 348 5 13-Dec 349 6

 Thursday Start Friday Start Saturday Start 13-Jan 16 2 13-Jan 17 3 13-Jan 18 4 13-Feb 47 5 13-Feb 48 6 13-Feb 49 0 13-Mar 75 5 13-Mar 76 6 13-Mar 77 0 13-Apr 106 1 13-Apr 107 2 13-Apr 108 3 13-May 136 3 13-May 137 4 13-May 138 5 13-Jun 167 6 13-Jun 168 0 13-Jun 169 1 13-Jul 197 1 13-Jul 198 2 13-Jul 199 3 13-Aug 228 4 13-Aug 229 5 13-Aug 230 6 13-Sep 259 0 13-Sep 260 1 13-Sep 261 2 13-Oct 289 2 13-Oct 290 3 13-Oct 291 4 13-Nov 320 5 13-Nov 321 6 13-Nov 322 0 13-Dec 350 0 13-Dec 351 1 13-Dec 352 2

The Pattern for Non-Leap Years is 2-2-2-1-3-1-1.

Leap Years

Now what about for Leap Years?  See the breakdown here:

 1st of January on: Months with Friday the 13th Sunday January, April, July Monday September, December Tuesday June Wednesday March, November Thursday February, August Friday May Saturday October

 Sunday Start Monday Start Tuesday Start Wednesday Start 13-Jan 12 5 13-Jan 13 6 13-Jan 14 0 13-Jan 15 1 13-Feb 43 1 13-Feb 44 2 13-Feb 45 3 13-Feb 46 4 13-Mar 72 2 13-Mar 73 3 13-Mar 74 4 13-Mar 75 5 13-Apr 103 5 13-Apr 104 6 13-Apr 105 0 13-Apr 106 1 13-May 133 0 13-May 134 1 13-May 135 2 13-May 136 3 13-Jun 164 3 13-Jun 165 4 13-Jun 166 5 13-Jun 167 6 13-Jul 194 5 13-Jul 195 6 13-Jul 196 0 13-Jul 197 1 13-Aug 225 1 13-Aug 226 2 13-Aug 227 3 13-Aug 228 4 13-Sep 256 4 13-Sep 257 5 13-Sep 258 6 13-Sep 259 0 13-Oct 286 6 13-Oct 287 0 13-Oct 288 1 13-Oct 289 2 13-Nov 317 2 13-Nov 318 3 13-Nov 319 4 13-Nov 320 5 13-Dec 347 4 13-Dec 348 5 13-Dec 349 6 13-Dec 350 0

 Thursday Start Friday Start Saturday Start 13-Jan 16 2 13-Jan 17 3 13-Jan 18 4 13-Feb 47 5 13-Feb 48 6 13-Feb 49 0 13-Mar 76 6 13-Mar 77 0 13-Mar 78 1 13-Apr 107 2 13-Apr 108 3 13-Apr 109 4 13-May 137 4 13-May 138 5 13-May 139 6 13-Jun 168 0 13-Jun 169 1 13-Jun 170 2 13-Jul 198 2 13-Jul 199 3 13-Jul 200 4 13-Aug 229 5 13-Aug 230 6 13-Aug 231 0 13-Sep 260 1 13-Sep 261 2 13-Sep 262 3 13-Oct 290 3 13-Oct 291 4 13-Oct 292 5 13-Nov 321 6 13-Nov 322 0 13-Nov 323 1 13-Dec 351 1 13-Dec 352 2 13-Dec 353 3

The Pattern for Leap Years is 3 – 2 – 1 – 2 – 2 – 1 – 1.

Above you see that only when January 1st falls on Thursday do you have a Friday the 13th in both February and March.  Here is a summary of just February and March days…

 Sunday JAN 1st Monday JAN 1st Tuesday JAN 1st Wednesday JAN 1st Thursday JAN 1st Friday JAN 1st Saturday JAN 1st 13-Feb 43 2 13-Feb 44 3 13-Feb 45 4 13-Feb 46 5 13-Feb 47 6 13-Feb 48 0 13-Feb 49 1 13-Mar 71 2 13-Mar 72 3 13-Mar 73 4 13-Mar 74 5 13-Mar 75 6 13-Mar 76 0 13-Mar 77 1

Or for a Leap Year, there is no occurrence of Friday the 13th in February & March:

 Sunday JAN 1st Monday JAN 1st Tuesday JAN 1st Wednesday JAN 1st Thursday JAN 1st Friday JAN 1st Saturday JAN 1st 13-Feb 43 2 13-Feb 44 3 13-Feb 45 4 13-Feb 46 5 13-Feb 47 6 13-Feb 48 0 13-Feb 49 1 13-Mar 72 3 13-Mar 73 4 13-Mar 74 5 13-Mar 75 6 13-Mar 76 0 13-Mar 77 1 13-Mar 78 2

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?

We can answer both questions above utilizing the Excel once again.  We know that this year, 2004, was a leap year, with a Thursday January 1st.  We have leap years about every 4 years. In the Gregorian calendar, there is a leap year every year divisible by four except for years which are both divisible by 100 and not divisible by 400. Therefore, the year 2000 will be a leap year, but the years 1700, 1800, and 1900 were not.

Knowing this, we develop the following spreadsheet:

Using Excel, we’ll have 3 columns.

Column A                                 Column B                                             Column C

1-1-2000                                 “=WEEKDAY(A2)”                            =IF(WEEKDAY(A2,1)=5, "MARCH & FEBRUARY"," ")

1-1-2001                                 “=WEEKDAY(A3)”                            =IF(WEEKDAY(A2,1)=5, "MARCH & FEBRUARY"," ")

copy down to 2100                  copy down to Year 2100                      copy down to Year 2100

Since we already know that the 1st must fall on a Thursday, and only a non-leap year, we can utilize the WEEKDAY function to easily determine what years this happen.

Our spreadsheet looks like this:

 YEAR DAY 3 FRI 13TH LEAP YEAR Thursday, January 01, 2004 5 MARCH & FEBRUARY LEAP YEAR Thursday, January 01, 2009 5 MARCH & FEBRUARY Thursday, January 01, 2015 5 MARCH & FEBRUARY Thursday, January 01, 2026 5 MARCH & FEBRUARY Thursday, January 01, 2032 5 MARCH & FEBRUARY LEAP YEAR Thursday, January 01, 2037 5 MARCH & FEBRUARY Thursday, January 01, 2043 5 MARCH & FEBRUARY Thursday, January 01, 2054 5 MARCH & FEBRUARY Thursday, January 01, 2060 5 MARCH & FEBRUARY LEAP YEAR Thursday, January 01, 2065 5 MARCH & FEBRUARY Thursday, January 01, 2071 5 MARCH & FEBRUARY Thursday, January 01, 2082 5 MARCH & FEBRUARY Thursday, January 01, 2088 5 MARCH & FEBRUARY LEAP YEAR Thursday, January 01, 2093 5 MARCH & FEBRUARY Thursday, January 01, 2099 5 MARCH & FEBRUARY