There are fourteen different cases based on the day of the week that January 1 occurs and whether or not the year is a leap year. First we will consider the seven cases that are not leap years.
| Case: | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Jan 1 occurs on | Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
Case 1:
If January 1 occurs on a Sunday, then days of the year that are congruent to 6 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 1 | 2 | 3 | 4 | 5 | 6 | 0 |
Thus, in years such as this, Friday the 13th occurs in January and October.
Case 2:
If January 1 occurs on a Monday, then days of the year that are congruent to 5 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 |
Thus, in years such as this, Friday the 13th occurs in April and July.
Case 3:
If January 1 occurs on a Tuesday, then days of the year that are congruent to 4 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 6 | 0 | 1 | 2 | 3 | 4 | 5 |
Thus, in years such as this, Friday the 13th occurs in September and December.
Case 4:
If January 1 occurs on a Wednesday, then days of the year that are congruent to 3 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 5 | 6 | 0 | 1 | 2 | 3 | 4 |
Thus, in years such as this, Friday the 13th occurs in June.
Case 5:
If January 1 occurs on a Thursday, then days of the year that are congruent to 2 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 4 | 5 | 6 | 0 | 1 | 2 | 3 |
Thus, in years such as this, Friday the 13th occurs in February, March, and November.
Case 6:
If January 1 occurs on a Friday, then days of the year that are congruent to 1 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 3 | 4 | 5 | 6 | 0 | 1 | 2 |
Thus, in years such as this, Friday the 13th occurs in August.
Case 7:
If January 1 occurs on a Saturday, then days of the year that are congruent to 0 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 2 | 3 | 4 | 5 | 6 | 0 | 1 |
Thus, in years such as this, Friday the 13th occurs in May.
Leap Years
When we consider leap years, we have the following table for the 13th of every month:
| Date | Day of Year | Day Mod 7 |
| Jan-13 | 13 | 6 |
| Feb-13 | 44 | 2 |
| Mar-13 | 72 | 3 |
| Apr-13 | 103 | 6 |
| May-13 | 133 | 1 |
| Jun-13 | 164 | 4 |
| Jul-13 | 194 | 6 |
| Aug-13 | 225 | 2 |
| Sep-13 | 256 | 5 |
| Oct-13 | 286 | 0 |
| Nov-13 | 317 | 3 |
| Dec-13 | 347 | 5 |
We now have seven additional cases.
Case 8:
If January 1 occurs on a Sunday in a leap year, then days of the year that are congruent to 6 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 1 | 2 | 3 | 4 | 5 | 6 | 0 |
Thus, in years such as this, Friday the 13th occurs in January, April, and July.
Case 9:
If January 1 occurs on a Monday in a leap year, then days of the year that are congruent to 5 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 |
Thus, in years such as this, Friday the 13th occurs in September and December.
Case 10:
If January 1 occurs on a Tuesday in a leap year, then days of the year that are congruent to 4 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 6 | 0 | 1 | 2 | 3 | 4 | 5 |
Thus, in years such as this, Friday the 13th occurs in June.
Case 11:
If January 1 occurs on a Wednesday in a leap year, then days of the year that are congruent to 3 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 5 | 6 | 0 | 1 | 2 | 3 | 4 |
Thus, in years such as this, Friday the 13th occurs in March and November.
Case 12:
If January 1 occurs on a Thursday in a leap year, then days of the year that are congruent to 2 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 4 | 5 | 6 | 0 | 1 | 2 | 3 |
Thus, in years such as this, Friday the 13th occurs in February and August.
Case 13:
If January 1 occurs on a Friday in a leap year, then days of the year that are congruent to 1 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 3 | 4 | 5 | 6 | 0 | 1 | 2 |
Thus, in years such as this, Friday the 13th occurs in May.
Case 14:
If January 1 occurs on a Saturday in a leap year, then days of the year that are congruent to 0 mod 7 are Fridays.
| Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
| 2 | 3 | 4 | 5 | 6 | 0 | 1 |
Thus, in years such as this, Friday the 13th occurs in October.
The 14 cases can be summarized in the following table:
| Case | Leap Year? | January 1 Falls on... | Friday 13 occurs in... |
| 1 | No | Sunday | January and October |
| 2 | No | Monday | April and July |
| 3 | No | Tuesday | September and December |
| 4 | No | Wednesday | June |
| 5 | No | Thursday | February, March, and November |
| 6 | No | Friday | August |
| 7 | No | Saturday | May |
| 8 | Yes | Sunday | January, April, and July |
| 9 | Yes | Monday | September and December |
| 10 | Yes | Tuesday | June |
| 11 | Yes | Wednesday | March and November |
| 12 | Yes | Thursday | February and August |
| 13 | Yes | Friday | May |
| 14 | Yes | Saturday | October |
From the table, we can see 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.