"Even a stopped clock shows the right time twice a day"
Figure 1. Heuristic turns watch into compass
Many websites talk of this neat scout trick, but I haven't found any explanation for the bisection rule. We don't like things unexplained round here, and as it turns out the "proof" of this rule of thumb is pleasingly simple and easy to grasp. For example at sunrise (see figure 2), and assuming for simplicity that the day lasts 12 hours (from 6am to 6pm, it is equinox time), pointing the hour hand to the sun aligns the watch east/west, so the angle from 6 to 12 o'clock is 180°, and half the angle is 180/2 = 90 degrees, straight due south. Likewise it is easy to show that at 12 noon, the angle is 0° so the hour hand points directly south.
The semi-circle in figure 2 represents the path of sun in the heavens, from dawn to dusk; the watch is drawn sideways
Why is the sun south of us? In northern hemisphere, above the tropic of cancer, the sun always stays below the zenith, or straight above our heads, because of Earth's inclined axis of daily rotation. This is especially so during winter, when the sun stays low all day, even below the tropic of cancer. But what about the angle bisection rule? Are we sure that it holds for all times, not just for sunrise and noon?
The elder scouts are trully wise, and here's why. Half a turn round a watch, it is 6 hours, whereas the "heavenly watch" that represents the sun's path is 12 hours, so the hour hand travels twice as fast as the sun, that's why the angle must be cut in half. See the little animation below.
Say the time is 8 in the morning. If the watch is aligned east-west as in figure 2 further up, the hour hand is pointing at 10 am "sun time". If we turn our clock to point into the sun, then half the angle till 12 o'clock is due south. QED.
Now we understand why the bisection rule works, we also understand the limitations of the procedure (other than the obvious necessity of a sunny day and an old school non-digital watch, showing the correct time)