# Rotational Symmetry on Digital Clock

In the NPR Sunday Puzzle recently, the "weekly challenge" asked about what times on a digital clock (not using military i.e. 00-24:00 time) would read the same if the clock was turned upside down, giving the example 6:29 (the colon is to be ignored).

58 times

I did not get that answer. My reasoning was

In a time with three digits, a middle digit of 0, 1, 2, or 5, it will rotate onto itself. Thus we have 1(0125)1, 2(0125)2, 5(0125)5, 6(0125)9, 8(0125)8, and 9(0125)6 as 24 symmetric times. With 4 digits, the first two digits determine what the last two must be. So, 10:01, 11:11, and 12:21 are 3 more symmetric times. That's 27, but the puzzle specifies both AM and PM should be counted, so my final answer is $$27\times2=54$$ times.

Which symmetric times am I missing?

• Should we use 0:00-0:59 to represent 12:00-12:59? And are you sure this is 12-hour clock? Thanks. Nov 7, 2023 at 1:43
• The statement of the puzzle in the link says "we are not accepting military time", so I think there is no hour 0 and it should be assumed it's a 12 hour clock. Nov 7, 2023 at 1:49
• You've made a minor but important error in transcribing the puzzle - it asks for the maximum number of times per day the clock can be read the same upside down, not how many times there are that read the same upside down (it's possible to see a clock show the exact same time twice on certain days). Nov 7, 2023 at 13:48

I agree with the supplied answer of

58 times

The answer calculated in the question is off by

4

So, I'm forced to pick between the person asking this question and Will Shortz, who has proven to me, at least, to be very specific in his puzzles. This makes me inclined to trust Mr. Shortz until proven otherwise. Therefore, I ask myself, why might the answers be different, and does the time the question was asked have any bearing? This leads me to the possibility that

Daylight saving time is part of the answer.

Sure enough, if I consider this, I will find

The extra hour in the day at the end of daylight saving time adds 4 to the presumed answer, giving the supplied answer of 58.

• It is relevant to specify rot13(gur ubhe gung qnlyvtug fnivat gvzr ercrngf, juvpu vf bar nz gb gjb nz), and also rot13(abirzore gur svsgu, gur fhaqnl gur nafjre vf eryrnfrq, vf rknpgyl gur qngr bs gur gvzr punatr). Nov 7, 2023 at 13:26
• Nice! Wish I'd thought of that :) Nov 7, 2023 at 14:03
• The Will Shortz bit is real. I really did consider the possibility that the posted answer was wrong. Had it not been from him, I would likely have decided it was a typo and given up. Nov 7, 2023 at 16:12
• Is Will on this site? Nov 9, 2023 at 11:27
• I have never seen him mention it and there is no obvious user, but I'd be very surprised if he wasn't at least lurking. Maybe he's you??? Nov 10, 2023 at 14:38