Solving a Neat Puzzle: How Many Times in a 12-Hour Period Can You Not Tell the Time?
Have you ever come across a puzzle that you can't seem to solve? This blog post discusses a neat puzzle involving a clock with identical hands and how many times in a 12-hour period you can't tell the time. Be careful not to make assumptions, as the hour hand won't always point straight up!
Dave Richeson
Mathematician. John J. & Ann Curley Chair in Liberal Arts at Dickinson College. Author of Tales of Impossibility and Euler's Gem. Coffee drinker.
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I just saw a neat puzzle: If both hands of a clock are identical, how many times in a 12-hour period can you not tell the time? Be careful: you might think that 2:00 and and 12:10 are indistinguishable, but they're not since the hour hand at 12:10 won't point straight up.
— Dave Richeson (@divbyzero) March 8, 2023
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Spoiler: I'll share the answer in the tweets that follow. Don't scroll if you don't want to know the solution.
— Dave Richeson (@divbyzero) March 8, 2023
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I saw the solution described in words, but here's my topological/pictorial answer.
— Dave Richeson (@divbyzero) March 8, 2023
Every position of the hour hand can be represented by a point on a circle, and every position of the minute hand is also a point on a circle. So every time is a point on a torus, which is a
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circle cross a circle. That is, one circular direction around the tube of the torus tells where the hour hand is, and the other way around tells the location of the minute hand. So the hands of the clock running for 12 hours is a path that wraps once around the torus one
— Dave Richeson (@divbyzero) March 8, 2023
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way and 12 times around the other way. It is a torus knot! You can view this on the torus itself or on the square with sides identified.
— Dave Richeson (@divbyzero) March 8, 2023
Now suppose we have a second clock, but we've swapped the roles of the minute and hour hands. That's also a torus knot, but it wraps 12
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times around the first direction and once in the other direction.
— Dave Richeson (@divbyzero) March 8, 2023
The hands of the two clocks coincide when the two curves cross each other. This happens 143 times (noon and midnight are the same point in the figure).
That looks like the answer, but it isn't quite. This
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counts the times when the two hands are on top of each other (like at noon). We can't tell which hand is which, but we can tell the time! This happens 11 times (the points along the diagonal of the square). So the answer is 132.
— Dave Richeson (@divbyzero) March 8, 2023
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Note: I said these are torus knots. That's true, but they happen to be unknotted torus knots.
— Dave Richeson (@divbyzero) March 8, 2023
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