Can a 4 and a 6 minute glass ever measure 7 minutes?

Question

You have a 4-minute and a 6-minute hourglass. You may flip either glass at any moment, but the only moments you can actually recognise are those when a glass runs empty; between those moments you have no way of knowing what time it is. Can you ever mark out an interval of exactly 7 minutes? Give a clear argument either way.

Accepted Answer

No, it is impossible. The invariant. Start at time zero. The only moments you can perceive, and therefore the only moments at which you can flip a glass, are moments when a glass runs out. Suppose every event so far has happened at an even number of minutes. A glass flipped at an even moment runs out either a full 4 or 6 minutes later, which is even plus even, so even; or, if it was a partly-run glass, it runs out after the amount that had previously run, which is the difference of two earlier event times and therefore also even. So the next event is again at an even time. Since the first eve…

Can a 4 and a 6 minute glass ever measure 7 minutes?

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