Ponder This Challenge - April 2026 - The Unlabeled Clock

We are given a strange analog clock - its dial has no markings, and although it has three hands that move continuously (not in steps), they are all of the same length and are indistinguishable. Quickly it becomes obvious that the clock is far from useless, since we can usually deduce the time from the cyclic triple of relative angles between its three hands.

However, there are still many moments in time that cannot be safely deduced from the state of the clock, because there are other moments in time that result in the same relative angles.

Here are some examples of both unique and undeducible clock configurations. Note that the first angle denotes the position of the first hand encountered clockwise from the unmarked 00:00 point, the second angle denotes the space between the first and second hands, and the third angle is between the second and third hands:

• 05:25 results in $150^\circ, 12.5^\circ, 197.5^\circ$ and this angle configuration is unique to it.
• Both 00:30 and 06:30 result in $15^\circ, 165^\circ, 180^\circ$ angles and cannot be distinguished.
• Both 03:00 and 09:00 result in $0^\circ, 90^\circ, 270^\circ$. These cases can be distinguished by checking where the two co-located hands are in respect to the third one - if they are located clockwise (09:00) or counterclockwise (03:00) to it.

Your goal: With the usual conventions of 60 seconds per minute (and full rotation), 60 minutes per hour (and full rotation), and 12 hours per full rotation, determine the number of moments in time that cannot be deduced.

A bonus "*" will be given for finding the "worst" convention, in the following sense: The clock still represents $12\cdot 60 \cdot 60=43200$ seconds, but the convention is now that there are $S$ seconds in a minute, $M$ minutes in an hour, and $H$ hours per rotation, such that $S\le M\le H$ and $S\cdot M\cdot H=43200$. What choice of $H,M,S\in \mathbb{N}$ yields the largest number of moments that cannot be deduced? Give both $(H,M,S)$ and the number of moments in time that cannot be deduced.