Fusible number

Fusible numbers are a class of rational numbers arising out of the following math puzzle:


 * You are given two fuses. Each one burns for exactly one minute, but not uniformly, so one cannot predict exactly how much of the fuse will be left after a given amount of time. You are allowed to light one or more unlit ends of any fuse, but only at time \(t = 0\) or when a fuse burns out completely. How do you measure 45 seconds?

The solution is to light both ends of one fuse and one end of the other fuse at the same time. When the first fuse burns out completely, 30 seconds have passed &mdash; then light the remaining end of the other fuse. Forty-five seconds will have passed when this fuse burns out.

We call a number \(t\) fusible if it can be measured (in minutes) with a finite number of fuses. Since 45 seconds = 3/4 minutes can be measured, we say that \(3/4\) is fusible. Formally, a real number \(x\) is fusible if and only if \(x = 0\) or \(x = (a + b + 1) / 2\), with \(a\) and \(b\) fusible numbers.