Mirror Clock: The Long Wait
Read the puzzle
A 24-hour flip clock shows 15:51, which reads the same forwards and backwards (a mirror time, like 12:21). A note warns that 16:61 is not a real time. How long until the clock next shows a mirror time?
How long until the clock shows the next mirror time?
Minutes only go up to 59.
4 hours 11 minutes
16:61, 17:71, 18:81 and 19:91 are not real times. The next is 20:02, 4 h 11 min after 15:51.
How to solve it
A mirror time has the form AB:BA, so the minutes are the hour digits reversed. Those minutes must also be a real time, from 00 to 59.
Step 1: test each hour after 15. Hour 16 would need 16:61, hour 17 would need 17:71, hour 18 would need 18:81 and hour 19 would need 19:91. None of 61, 71, 81 or 91 is a valid minute, so all four are skipped.
Step 2: hour 20 reverses to 02, so 20:02 is valid. That is the next mirror time.
Step 3: count the gap. From 15:51 to 19:51 is 4 hours, and from 19:51 to 20:02 is 9 + 2 = 11 minutes. The total is 4 hours 11 minutes, which is 4 × 60 + 11 = 251 minutes.
The usual wrong answer is 70 minutes, aiming for 16:61 by the same pattern that took 13:31 to 14:41. The pattern breaks as soon as the reversed hour is too big to be a minute.
Quick check: 15:51 plus 251 minutes is 20:02.
