Friday the 13th Study

## Carl Kolodzy, Spring Semester, 2003

The following project list items that can be discussed as a part of examining the pattern of occurrence of Friday the 13th.  This particular project starts with an examination of the following:

1.  Show that for any year, there must be at least one month and at most 3 months for which the 13th of the month falls on a Friday.

Explanation of Method

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

Questions that arise from this:

1.  Prove that Friday the 13th can only occur in consecutive months, when these two months are February and March.

2.  On what day of the week must January 1 occur for February and March to have Friday the 13ths?

3.  What is the next year that this will occur?

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

A review of the methods used to provide answers to these questions is provided below.  This is divided into two areas:

Deals with the first 2 questions.

Shows the patterns associated the questions 3 and 4.

### One to Three Months per Year

An Excel Spreadsheet technique can be used to show that there must be at least on month each year where the 13th of the month falls on a Friday, as well as showing that as many as three months may have the 13th fall on a Friday.  Construction of the spreadsheet is setup by establishing 2 different worksheets, one for the case where the year is a leap year (366 day year) and the other then there is no leap year (365 day year).  Construction follows the following steps:

1. Start with January 1 falling on a Sunday.  Establish 3 columns (note for leap years, include Feb, 29th and date count up to 366):
• First column contains the date, starting with January 1.

This can be accomplished by entering the date January 1st in the first cell of the column, then (by selecting the lower right corner of the cell and dragging down the column) input ever increasing dates up to and including December 31st.

• Second column contains the day of the year for the date. i.e. 0 for January 1st, 1 for January 2nd,…364 for December 31st)

Start with entering a 0 in the column next to the date column.  Continue entering increasing values (1 ... 364) for the remainder of the dates.

• Third column contains the relative day of the week ( 1 for Sunday, 2 for Monday…6 for Friday, 7 for Saturday)

This is accomplished by entering the formula:

MOD(rC,7)+1

Where r refers to the row and C the column for the day number to be converted.

When this first step is completed, you should have a spreadsheet that looks like the following:

 Calendar for Years with 365 days Start on Sunday 1-Jan 0 1 2-Jan 1 2 3-Jan 2 3 4-Jan 3 4 5-Jan 4 5 6-Jan 5 6 7-Jan 6 7 8-Jan 7 1 9-Jan 8 2 10-Jan 9 3 11-Jan 10 4 12-Jan 11 5 13-Jan 12 6 14-Jan 13 7 15-Jan 14 1 16-Jan 15 2 17-Jan 16 3 18-Jan 17 4 19-Jan 18 5 20-Jan 19 6 21-Jan 20 7 22-Jan 21 1 23-Jan 22 2 24-Jan 23 3 25-Jan 24 4 26-Jan 25 5 27-Jan 26 6 28-Jan 27 7 29-Jan 28 1 30-Jan 29 2 31-Jan 30 3

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 1-Dec 334 6 2-Dec 335 7 3-Dec 336 1 4-Dec 337 2 5-Dec 338 3 6-Dec 339 4 7-Dec 340 5 8-Dec 341 6 9-Dec 342 7 10-Dec 343 1 11-Dec 344 2 12-Dec 345 3 13-Dec 346 4 14-Dec 347 5 15-Dec 348 6 16-Dec 349 6 17-Dec 350 0 18-Dec 351 1 19-Dec 352 2 20-Dec 353 3 21-Dec 354 4 22-Dec 355 5 23-Dec 356 6 24-Dec 357 0 25-Dec 358 1 26-Dec 359 2 27-Dec 360 3 28-Dec 361 4 29-Dec 362 5 30-Dec 363 6 31-Dec 364 0

This will be the equivalent of starting January 1 on a Sunday.

1. Copy the first 3 columns over to the 4th, 5th and 6th column, but start January 1st down a row from the first.  Change the starting number for the 5th column from 0 for January 1st to 1 for January 1st.  This will be the equivalent of January 1 starting on a Monday.

When this step is completed, you should have a spreadsheet that looks like the following:

 Calendar for Years with 365 days Start on Sunday 1-Jan 0 1 Start on Monday 2-Jan 1 2 1-Jan 1 2 3-Jan 2 3 2-Jan 2 3 4-Jan 3 4 3-Jan 3 4 5-Jan 4 5 4-Jan 4 5 6-Jan 5 6 5-Jan 5 6 7-Jan 6 7 6-Jan 6 7 8-Jan 7 1 7-Jan 7 1 9-Jan 8 2 8-Jan 8 2 10-Jan 9 3 9-Jan 9 3 11-Jan 10 4 10-Jan 10 4 12-Jan 11 5 11-Jan 11 5 13-Jan 12 6 12-Jan 12 6 14-Jan 13 7 13-Jan 13 7 15-Jan 14 1 14-Jan 14 1 16-Jan 15 2 15-Jan 15 2 17-Jan 16 3 16-Jan 16 3 18-Jan 17 4 17-Jan 17 4 19-Jan 18 5 18-Jan 18 5 20-Jan 19 6 19-Jan 19 6 21-Jan 20 7 20-Jan 20 7 22-Jan 21 1 21-Jan 21 1 23-Jan 22 2 22-Jan 22 2 24-Jan 23 3 23-Jan 23 3 25-Jan 24 4 24-Jan 24 4 26-Jan 25 5 25-Jan 25 5 27-Jan 26 6 26-Jan 26 6 28-Jan 27 7 27-Jan 27 7 29-Jan 28 1 28-Jan 28 1 30-Jan 29 2 29-Jan 29 2 31-Jan 30 3 30-Jan 30 3

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 1-Dec 334 6 30-Nov 334 6 2-Dec 335 7 1-Dec 335 7 3-Dec 336 1 2-Dec 336 1 4-Dec 337 2 3-Dec 337 2 5-Dec 338 3 4-Dec 338 3 6-Dec 339 4 5-Dec 339 4 7-Dec 340 5 6-Dec 340 5 8-Dec 341 6 7-Dec 341 6 9-Dec 342 7 8-Dec 342 7 10-Dec 343 1 9-Dec 343 1 11-Dec 344 2 10-Dec 344 2 12-Dec 345 3 11-Dec 345 3 13-Dec 346 4 12-Dec 346 4 14-Dec 347 5 13-Dec 347 5 15-Dec 348 6 14-Dec 348 6 16-Dec 349 6 15-Dec 349 6 17-Dec 350 0 16-Dec 350 0 18-Dec 351 1 17-Dec 351 1 19-Dec 352 2 18-Dec 352 2 20-Dec 353 3 19-Dec 353 3 21-Dec 354 4 20-Dec 354 4 22-Dec 355 5 21-Dec 355 5 23-Dec 356 6 22-Dec 356 6 24-Dec 357 0 23-Dec 357 0 25-Dec 358 1 24-Dec 358 1 26-Dec 359 2 25-Dec 359 2 27-Dec 360 3 26-Dec 360 3 28-Dec 361 4 27-Dec 361 4 29-Dec 362 5 28-Dec 362 5 30-Dec 363 6 29-Dec 363 6 31-Dec 364 0 30-Dec 364 0 365 31-Dec 365 1

Continue copying the 3 columns additional rows down for the year starting on Tuesday through Saturday.   Be sure to change the starting number for the 2nd column of the groups from 1 (Monday start) to 2 (for Tuesday), 3 (for Wednesday) etc.

When this step is completed, you should have a spreadsheet that looks like the following:

 Calendar for Years with 365 days Start on Sunday 1-Jan 0 1 Start on Monday 0 0 0 0 0 2-Jan 1 2 1-Jan 1 2 Start on Tuesday 1 1 1 1 3-Jan 2 3 2-Jan 2 3 1-Jan 2 3 Start on Wednesday 2 2 2 4-Jan 3 4 3-Jan 3 4 2-Jan 3 4 1-Jan 3 4 Start on Thursday 3 3 5-Jan 4 5 4-Jan 4 5 3-Jan 4 5 2-Jan 4 5 1-Jan 4 5 Start on Friday 4 6-Jan 5 6 5-Jan 5 6 4-Jan 5 6 3-Jan 5 6 2-Jan 5 6 1-Jan 5 6 Start on Saturday 7-Jan 6 7 6-Jan 6 7 5-Jan 6 7 4-Jan 6 7 3-Jan 6 7 2-Jan 6 7 1-Jan 6 7 8-Jan 7 1 7-Jan 7 1 6-Jan 7 1 5-Jan 7 1 4-Jan 7 1 3-Jan 7 1 2-Jan 7 1 9-Jan 8 2 8-Jan 8 2 7-Jan 8 2 6-Jan 8 2 5-Jan 8 2 4-Jan 8 2 3-Jan 8 2 10-Jan 9 3 9-Jan 9 3 8-Jan 9 3 7-Jan 9 3 6-Jan 9 3 5-Jan 9 3 4-Jan 9 3 11-Jan 10 4 10-Jan 10 4 9-Jan 10 4 8-Jan 10 4 7-Jan 10 4 6-Jan 10 4 5-Jan 10 4 12-Jan 11 5 11-Jan 11 5 10-Jan 11 5 9-Jan 11 5 8-Jan 11 5 7-Jan 11 5 6-Jan 11 5 13-Jan 12 6 12-Jan 12 6 11-Jan 12 6 10-Jan 12 6 9-Jan 12 6 8-Jan 12 6 7-Jan 12 6 14-Jan 13 7 13-Jan 13 7 12-Jan 13 7 11-Jan 13 7 10-Jan 13 7 9-Jan 13 7 8-Jan 13 7 15-Jan 14 1 14-Jan 14 1 13-Jan 14 1 12-Jan 14 1 11-Jan 14 1 10-Jan 14 1 9-Jan 14 1 16-Jan 15 2 15-Jan 15 2 14-Jan 15 2 13-Jan 15 2 12-Jan 15 2 11-Jan 15 2 10-Jan 15 2 17-Jan 16 3 16-Jan 16 3 15-Jan 16 3 14-Jan 16 3 13-Jan 16 3 12-Jan 16 3 11-Jan 16 3 18-Jan 17 4 17-Jan 17 4 16-Jan 17 4 15-Jan 17 4 14-Jan 17 4 13-Jan 17 4 12-Jan 17 4 19-Jan 18 5 18-Jan 18 5 17-Jan 18 5 16-Jan 18 5 15-Jan 18 5 14-Jan 18 5 13-Jan 18 5 20-Jan 19 6 19-Jan 19 6 18-Jan 19 6 17-Jan 19 6 16-Jan 19 6 15-Jan 19 6 14-Jan 19 6 21-Jan 20 7 20-Jan 20 7 19-Jan 20 7 18-Jan 20 7 17-Jan 20 7 16-Jan 20 7 15-Jan 20 7 22-Jan 21 1 21-Jan 21 1 20-Jan 21 1 19-Jan 21 1 18-Jan 21 1 17-Jan 21 1 16-Jan 21 1 23-Jan 22 2 22-Jan 22 2 21-Jan 22 2 20-Jan 22 2 19-Jan 22 2 18-Jan 22 2 17-Jan 22 2 24-Jan 23 3 23-Jan 23 3 22-Jan 23 3 21-Jan 23 3 20-Jan 23 3 19-Jan 23 3 18-Jan 23 3 25-Jan 24 4 24-Jan 24 4 23-Jan 24 4 22-Jan 24 4 21-Jan 24 4 20-Jan 24 4 19-Jan 24 4 26-Jan 25 5 25-Jan 25 5 24-Jan 25 5 23-Jan 25 5 22-Jan 25 5 21-Jan 25 5 20-Jan 25 5 27-Jan 26 6 26-Jan 26 6 25-Jan 26 6 24-Jan 26 6 23-Jan 26 6 22-Jan 26 6 21-Jan 26 6 28-Jan 27 7 27-Jan 27 7 26-Jan 27 7 25-Jan 27 7 24-Jan 27 7 23-Jan 27 7 22-Jan 27 7 29-Jan 28 1 28-Jan 28 1 27-Jan 28 1 26-Jan 28 1 25-Jan 28 1 24-Jan 28 1 23-Jan 28 1 30-Jan 29 2 29-Jan 29 2 28-Jan 29 2 27-Jan 29 2 26-Jan 29 2 25-Jan 29 2 24-Jan 29 2 31-Jan 30 3 30-Jan 30 3 29-Jan 30 3 28-Jan 30 3 27-Jan 30 3 26-Jan 30 3 25-Jan 30 3

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 1-Dec 334 6 30-Nov 334 6 29-Nov 334 6 28-Nov 334 6 27-Nov 334 6 26-Nov 334 6 25-Nov 334 6 2-Dec 335 7 1-Dec 335 7 30-Nov 335 7 29-Nov 335 7 28-Nov 335 7 27-Nov 335 7 26-Nov 335 7 3-Dec 336 1 2-Dec 336 1 1-Dec 336 1 30-Nov 336 1 29-Nov 336 1 28-Nov 336 1 27-Nov 336 1 4-Dec 337 2 3-Dec 337 2 2-Dec 337 2 1-Dec 337 2 30-Nov 337 2 29-Nov 337 2 28-Nov 337 2 5-Dec 338 3 4-Dec 338 3 3-Dec 338 3 2-Dec 338 3 1-Dec 338 3 30-Nov 338 3 29-Nov 338 3 6-Dec 339 4 5-Dec 339 4 4-Dec 339 4 3-Dec 339 4 2-Dec 339 4 1-Dec 339 4 30-Nov 339 4 7-Dec 340 5 6-Dec 340 5 5-Dec 340 5 4-Dec 340 5 3-Dec 340 5 2-Dec 340 5 1-Dec 340 5 8-Dec 341 6 7-Dec 341 6 6-Dec 341 6 5-Dec 341 6 4-Dec 341 6 3-Dec 341 6 2-Dec 341 6 9-Dec 342 7 8-Dec 342 7 7-Dec 342 7 6-Dec 342 7 5-Dec 342 7 4-Dec 342 7 3-Dec 342 7 10-Dec 343 1 9-Dec 343 1 8-Dec 343 1 7-Dec 343 1 6-Dec 343 1 5-Dec 343 1 4-Dec 343 1 11-Dec 344 2 10-Dec 344 2 9-Dec 344 2 8-Dec 344 2 7-Dec 344 2 6-Dec 344 2 5-Dec 344 2 12-Dec 345 3 11-Dec 345 3 10-Dec 345 3 9-Dec 345 3 8-Dec 345 3 7-Dec 345 3 6-Dec 345 3 13-Dec 346 4 12-Dec 346 4 11-Dec 346 4 10-Dec 346 4 9-Dec 346 4 8-Dec 346 4 7-Dec 346 4 14-Dec 347 5 13-Dec 347 5 12-Dec 347 5 11-Dec 347 5 10-Dec 347 5 9-Dec 347 5 8-Dec 347 5 15-Dec 348 6 14-Dec 348 6 13-Dec 348 6 12-Dec 348 6 11-Dec 348 6 10-Dec 348 6 9-Dec 348 6 16-Dec 349 6 15-Dec 349 6 14-Dec 349 6 13-Dec 349 6 12-Dec 349 6 11-Dec 349 6 10-Dec 349 7 17-Dec 350 0 16-Dec 350 0 15-Dec 350 0 14-Dec 350 0 13-Dec 350 0 12-Dec 350 0 11-Dec 350 1 18-Dec 351 1 17-Dec 351 1 16-Dec 351 1 15-Dec 351 1 14-Dec 351 1 13-Dec 351 1 12-Dec 351 2 19-Dec 352 2 18-Dec 352 2 17-Dec 352 2 16-Dec 352 2 15-Dec 352 2 14-Dec 352 2 13-Dec 352 3 20-Dec 353 3 19-Dec 353 3 18-Dec 353 3 17-Dec 353 3 16-Dec 353 3 15-Dec 353 3 14-Dec 353 4 21-Dec 354 4 20-Dec 354 4 19-Dec 354 4 18-Dec 354 4 17-Dec 354 4 16-Dec 354 4 15-Dec 354 5 22-Dec 355 5 21-Dec 355 5 20-Dec 355 5 19-Dec 355 5 18-Dec 355 5 17-Dec 355 5 16-Dec 355 6 23-Dec 356 6 22-Dec 356 6 21-Dec 356 6 20-Dec 356 6 19-Dec 356 6 18-Dec 356 6 17-Dec 356 7 24-Dec 357 0 23-Dec 357 0 22-Dec 357 0 21-Dec 357 0 20-Dec 357 0 19-Dec 357 0 18-Dec 357 1 25-Dec 358 1 24-Dec 358 1 23-Dec 358 1 22-Dec 358 1 21-Dec 358 1 20-Dec 358 1 19-Dec 358 2 26-Dec 359 2 25-Dec 359 2 24-Dec 359 2 23-Dec 359 2 22-Dec 359 2 21-Dec 359 2 20-Dec 359 3 27-Dec 360 3 26-Dec 360 3 25-Dec 360 3 24-Dec 360 3 23-Dec 360 3 22-Dec 360 3 21-Dec 360 4 28-Dec 361 4 27-Dec 361 4 26-Dec 361 4 25-Dec 361 4 24-Dec 361 4 23-Dec 361 4 22-Dec 361 5 29-Dec 362 5 28-Dec 362 5 27-Dec 362 5 26-Dec 362 5 25-Dec 362 5 24-Dec 362 5 23-Dec 362 6 30-Dec 363 6 29-Dec 363 6 28-Dec 363 6 27-Dec 363 6 26-Dec 363 6 25-Dec 363 6 24-Dec 363 7 31-Dec 364 0 30-Dec 364 0 29-Dec 364 0 28-Dec 364 0 27-Dec 364 0 26-Dec 364 0 25-Dec 364 1 365 31-Dec 365 1 30-Dec 365 1 29-Dec 365 1 28-Dec 365 1 27-Dec 365 1 26-Dec 365 2 366 366 31-Dec 366 2 30-Dec 366 2 29-Dec 366 2 28-Dec 366 2 27-Dec 366 3 367 367 367 31-Dec 367 3 30-Dec 367 3 29-Dec 367 3 28-Dec 367 4 368 368 368 368 31-Dec 368 4 30-Dec 368 4 29-Dec 368 5 369 369 369 369 369 31-Dec 369 5 30-Dec 369 6 370 370 370 370 370 370 31-Dec 370 7
1. We now can look for all instances where the 13th of the month falls on a Friday.  This will occur when day number is equal to 6 and the date is a 13th.  This cells should then be highlighted.  (use cell formatting to change the color or the pattern).
2. We can then go through and hide unnecessary rows – those without any Friday the 13th indications. Select all rows that want to be hidden and use the Format, Column, Hide function to hide the rows.
3. Finally, we can hide the column showing the day of the year, and just note the date column and the day of the week column.  This results in the following:
 Calendar for Years with 365 days Start on Sunday 1-Jan 1 Start on Monday 2-Jan 2 1-Jan 2 Start on Tuesday 3-Jan 3 2-Jan 3 1-Jan 3 Start on Wednesday 4-Jan 4 3-Jan 4 2-Jan 4 1-Jan 4 Start on Thursday 5-Jan 5 4-Jan 5 3-Jan 5 2-Jan 5 1-Jan 5 Start on Friday 6-Jan 6 5-Jan 6 4-Jan 6 3-Jan 6 2-Jan 6 1-Jan 6 Start on Saturday 7-Jan 7 6-Jan 7 5-Jan 7 4-Jan 7 3-Jan 7 2-Jan 7 1-Jan 7 8-Jan 1 7-Jan 1 6-Jan 1 5-Jan 1 4-Jan 1 3-Jan 1 2-Jan 1 9-Jan 2 8-Jan 2 7-Jan 2 6-Jan 2 5-Jan 2 4-Jan 2 3-Jan 2 10-Jan 3 9-Jan 3 8-Jan 3 7-Jan 3 6-Jan 3 5-Jan 3 4-Jan 3 11-Jan 4 10-Jan 4 9-Jan 4 8-Jan 4 7-Jan 4 6-Jan 4 5-Jan 4 12-Jan 5 11-Jan 5 10-Jan 5 9-Jan 5 8-Jan 5 7-Jan 5 6-Jan 5 13-Jan 6 12-Jan 6 11-Jan 6 10-Jan 6 9-Jan 6 8-Jan 6 7-Jan 6 17-Feb 6 16-Feb 6 15-Feb 6 14-Feb 6 13-Feb 6 12-Feb 6 11-Feb 6 17-Mar 6 16-Mar 6 15-Mar 6 14-Mar 6 13-Mar 6 12-Mar 6 11-Mar 6 14-Apr 6 13-Apr 6 12-Apr 6 11-Apr 6 10-Apr 6 9-Apr 6 8-Apr 6 19-May 6 18-May 6 17-May 6 16-May 6 15-May 6 14-May 6 13-May 6 16-Jun 6 15-Jun 6 14-Jun 6 13-Jun 6 12-Jun 6 11-Jun 6 10-Jun 6 14-Jul 6 13-Jul 6 12-Jul 6 11-Jul 6 10-Jul 6 9-Jul 6 8-Jul 6 18-Aug 6 17-Aug 6 16-Aug