The Two Fuses Problem

Here's a classic lateral thinking puzzle. You have two lengths of fuse and some matches. Each fuse takes exactly a minute to burn from end to end, however there is no guarantee that it burns at a constant rate along its length. For example, it may burn really quickly along the first 10 cm, but the next portion might burn more slowly; the only guarantee from the manufacturer is that the whole length takes a minute.

Your task is to use these fuses to time out exactly three quarters of a minute. Solution below.

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Anything that involves stopping the fuse halfway along its length isn't going to work because it won't necessarily take 30 seconds. However there is trick to get around this. Start by setting alight to both ends of one fuse and one end of the other. When the two flames meet on the first fuse you will know that 30 seconds have elapsed even though they may not not have met at the middle of the fuse.

When that happens set alight to the second end of the second fuse. It will have already burned through 30 seconds worth of length, so the two flames both have a further 15 seconds to travel. Overall this process takes 45 seconds. With higher numbers of fuses you can do more binary fractions of minutes. In fact n fuses lets you do binary fractions up to a denominator of 2^n.