Stopped Clock Paradox
Paradox
Of two clocks, one is better than the other if it shows the absolutely correct time more often than the other.
Consider:
- Clock $A$, which loses $1$ minute every day
- Clock $B$, which is stopped.
Clock $A$ is $1$ minute slow after day $1$, and $2$ minutes slow after day $2$, and so on.
There are $720$ minutes in $12$ hours.
Hence it will be $720$ days, or nearly $2$ years, before it shows the right time again.
On the other hand, clock $B$ shows the correct time every $12$ hours.
Hence clock $B$, a stopped clock, is better than clock $A$, which loses but a minute a day.
Resolution
First thing to note is that a clock that loses $1$ minute a day is a poor clock, and on an practical level not all that much better than a clock that does not run at all.
But let us modify our absolutely correct to correct within $30$ seconds, which is reasonable for an old dial clock.
One can assume that an adequate clock can be adjusted so that it will lose or gain no more than $30$ seconds over the course of a number of months.
On the old-fashioned mechanical clocks, there is usually a control on the back which one can use to minutely control the speed of a clock so that it can be made to be as accurate as your patience can achieve.
If I had a clock which was worse than that, I would throw it out and get a better one.
$\blacksquare$
Historical Note
The Stopped Clock Paradox appears in Further Nonsense of $1926$, a posthumous collection of some of Lewis Carroll's more humorous pieces.
Its original source is to be researched.
Sources
- 1926: Lewis Carroll: Further Nonsense
- 1944: Eugene P. Northrop: Riddles in Mathematics ... (previous) ... (next): Chapter Two: Paradoxes for Everyone