Lets see if I can figure this out. Lets look at non-leap years and see what is the maximum of 13th Friday’s that we can have. What are the numbers of these days? The first one is 13the day of the year, the second one is 13+31=44th day of the year. Next one is 44+28=72th day. If you do the calculation you get the following numbers for the 13th of different months:

13, 44, 72, 103, 133, 164, 194, 225, 256, 286, 317, 347

Now if two month have their 13th day in Friday the number of these two days should differ by a multiple of 7. So its enough to find the remainder of these number when divided by 7 And here is the remainders:

6, 2, 2, 5, 0, 3, 5, 1, 4, 6, 2, 4

As you see there are three 2’s; two 4’s, two 5’s, two 6’s, one zero, one 1, and one 3 . So the maximum for non-leap years is 3 and it could happen in Feb, March, and November.

For leap years the number of days would change to the following numbers:

13, 44, 73, 104, 134, 165, 195, 226, 257, 287, 318, 348

And the remainders when divided by 7 would be as follows:

6, 2, 3, 6, 1, 4, 6, 2, 5, 0, 3, 5

There is three 6’s, two 2’s, two 3’s, two 5’s, one zero, one 1, and one 4. Which means for leap years the maximum is 3 and it could happen in January, April and July.

You can check that these maximums do happen.