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:

Consecutive Months with Friday the 13th   Deals with the first 2 questions.

Patterns and Cycles  Shows the patterns associated the questions 3 and 4.

RETURN TO Carl’s EMAT 6690 Page 

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):

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.

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.

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

.

.

.

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.

(EXCEL Users) Link to the Spreadsheet

  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

.

.

.

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

(EXCEL Users) Link to the Spreadsheet

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

.

.

.

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

 (EXCEL Users) Link to the Spreadsheet

  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