“A broken clock is right twice a day”

It's a common observation that an analog clock, once it ceases to function, will still manage to display the correct time on two occasions within a single day. This isn't a magical occurrence or a testament to the clock's lingering accuracy, but rather a simple consequence of how we measure time and how these particular timepieces are designed. The belief stems from the basic mechanics of a 12-hour clock face and the cyclical nature of a 24-hour day.

The mathematical truth behind this widely accepted notion is straightforward. A standard analog clock completes a full rotation of its hour hand every twelve hours. Therefore, over the course of a full twenty-four hour day, the actual time will pass through every position on the clock face exactly twice. If a clock is broken and its hands are stuck at a particular position, say 3:00, then at 3:00 AM and again at 3:00 PM, the hands of the stopped clock will accurately reflect the current time. This principle holds true regardless of where the hands are frozen (Review), as the real time will inevitably catch up to and align with that stationary position twice daily.

People commonly believe this because it's easily observable and intuitively makes sense. Almost everyone has encountered a stopped clock at some point, perhaps in a public space or even in their own home, and noticed this brief moment of accidental accuracy. It serves as a simple, relatable example of how mathematical principles govern everyday phenomena, making it a satisfying and easily verifiable "truth" that resonates with common experience.