SOLUTION: In a certain month (of a year that is NOT a leap year), the 15th day fell on a Wednesday. In the next month, on what day(s) of the week could the 15th fall?

Any day of the week will move forward the same number of weekdays as 
the remainder obtained when the number of days in the month is 
divided by 7:

If the month is February and has 28 days, then when we divide 28 by 7

   4
7)28
  28
   0

the remainder is 0 so it will not move forward any weekdays, so March
15th will also fall on Wednesday.

If the month has 30 days, then when we divide 30 by 7

   4
7)30
  28
   2

The remainder is 2 so it will move forward 2 days, so it will fall on 
Friday.

If the month has 31 days, then when we divide 30 by 7

   4
7)31
  28
   3

The remainder is 3 so it will move forward 3 days, so it will fall on 
Saturday.

In years which are not leap years, The dates in March are the same
as the dates in February.  This year, February 15 fell on 
Monday.  This year, being a leap year, means that since February
had 29 days, March 15 will be next Tuesday.  That's because

   4
7)29
  28
   1

the remainder is 1 and the 15th advances 1 weekday.

Edwin