FRIDAY THE 13^TH

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
^Sunday	^Sunday
^Monday	^Monday
^Tuesday	^Tuesday
^Wednesday	^Wednesday
^Thursday	^Thursday
^Friday	^Friday
^Saturday	^Saturday

On any year, there must be at least one month and at most three months for which the 13^th 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.

Non-Leap Year

Leap Year

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

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

Note the pattern and breakdown of 13^th dates:

Non-Leap Years

1^st of January on:	Months with Friday the 13^th
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:

1^st of January on:	Months with Friday the 13^th
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 1^st falls on Thursday do you have a Friday the 13^th in both February and March. Here is a summary of just February and March days…

Sunday JAN 1^st			Monday JAN 1^st			Tuesday JAN 1^st			Wednesday JAN 1^st			Thursday JAN 1^st			Friday JAN 1^st			Saturday JAN 1^st
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 13^th in February & March:

Sunday JAN 1^st			Monday JAN 1^st			Tuesday JAN 1^st			Wednesday JAN 1^st			Thursday JAN 1^st			Friday JAN 1^st			Saturday JAN 1^st
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 1^st. 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 1^st 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

Click here for the full worksheet.

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