Fifty-two and a half minutes from three ropes

Question

You have three ropes. Each burns end to end in exactly 60 minutes, but unevenly, and none can be cut or marked. Every flame you light must be started with a match at the moment you choose. Measure exactly 52.5 minutes, starting from the moment you light the first match. Describe exactly which ends you light and when.

Accepted Answer

Light rope A at both ends, and ropes B and C at one end each, all at time zero. When A burns out (30 min), light B's other end. When B burns out (45 min), light C's other end. C burns out at 52.5 minutes. Why. - Rope A with two flames is gone at 30 minutes. - At that moment rope B, lit at one end since time zero, has 30 minutes of burn left. Lighting its second end halves that: B is gone at 45 minutes. - At that moment rope C, lit at one end since time zero, has 15 minutes of burn left. Lighting its second end halves that: C is gone at 52.5 minutes. So the sequence of halvings gives 30, then …

Fifty-two and a half minutes from three ropes

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