Engineers' Day and Children's Day are circled. How many days lie strictly between them, not counting either day?

From 15 September to 30 September is 15 days, October adds 31, and 14 more days reach 14 November: 15 + 31 + 14 = 60 days apart. Strictly between them lie 60 − 1 = 59 days. (60 is the gap; 61 counts both days.)