Write each date on a calendar as a fraction MM/DD. Every month has some fractions which are equivalent in other months. For example, January 2 (1/2) is equivalent to February 4 (2/4) and September 18 (9/18).
In addition, the odd denominators of 2/1 to 2/15 are repeated as 4/2 to 4/30 (8 dates not already counted), and some of them as 6/3 to 6/27 (5 dates), 8/4 to 8/28 (4 dates), 10/5 to 10/25 (3 dates), and 12/6 to 12/30 (3 dates).
The non-multiple-of-3 denominators from 3/1 to 3/14 are repeated as 6/2 to 6/28 (10 dates), and some of them as 9/3 to 9/30 (7 dates) and 12/4 to 12/28 (5 dates).
The odd denominators 4/1 to 4/15 are repeated as 8/2 to 8/30 (8 dates) and some of them as 12/3 to 12/27 (5 dates).
The non-multiple-of-5 denominators 5/1 to 5/14 are repeated as 10/2 to 10/28 (12 dates).
The dates 6/1, 6/5, 6/7, 6/11, and 6/13 are repeated as 12/2, 12/10, 12/14, 12/22, and 12/26.
Subtracting each of these numbers from the total days in the month, you can find the total number of unique fractions for each month, and by not deleting them from the first month where they occur, the total number of unique fractions in the year (which includes a leap day).
Month Total Unique First Number Days Fractions Occurrences 1 31 17 31 2 29 7 15 3 31 11 21 4 30 7 15 5 31 13 25 6 30 5 10 7 31 27 27 8 31 16 16 9 30 20 20 10 31 13 13 11 30 28 28 12 31 11 11 Total # of fractions: 232This reveals that June has the fewest unique fractions (only 5: 6/17, 6/19, 6/23, 6/25, 6/29), November has the most (28, all but 11/11 and 11/22), and the total number of unique fractions in a leap year is 232.