A boy comes to meet his friend in the morning at 8:30 AM, he again comes at…

Concept: In a repeating-visit or repeating-event puzzle, the successive gaps between the known times often follow their own numeric pattern (a constant difference, a constant ratio, or a second layer of differences). Find that gap pattern, extend it one more step, and add the resulting gap to the last known time to get the next occurrence.

1. List the four known visit times: 8:30 AM, 9:15 AM, 10:15 AM, 11:30 AM.

2. Find the gap between each consecutive pair: 9:15 AM minus 8:30 AM equals 45 minutes; 10:15 AM minus 9:15 AM equals 60 minutes; 11:30 AM minus 10:15 AM equals 75 minutes.

3. Find the pattern in these gaps: 60 minus 45 equals 15 minutes, and 75 minus 60 equals 15 minutes, so each gap is 15 minutes longer than the one before it.

4. Extend the pattern one more step: the next gap equals 75 plus 15, which is 90 minutes.

5. Add 90 minutes (1 hour 30 minutes) to the last known time, 11:30 AM: 11:30 AM plus 1 hour 30 minutes equals 1:00 PM.

Cross-check: the gaps 45, 60, 75, ... form an arithmetic progression with first term 45 and common difference 15, so the nth gap equals 45 plus (n minus 1) times 15. The fourth gap (from the fourth visit to the next one) is 45 plus 3 times 15, which is 90 minutes -- independently confirming that the boy comes again at 1:00 PM.