Electrician I.V. was called to repair a string of four lamps connected in series, one of which had burned out. It takes 10 seconds to unscrew any lamp from the string, and 10 seconds to screw it back in. The time spent on other actions is negligible. What is the minimum time the electrician can definitely find the burned-out lamp, if he has one spare lamp?
If after replacing one bulb the garland does not light up, then we replaced a working bulb.

## Solution

Suppose we did not replace some two bulbs. Then, if one of them is burnt out, we will not be able to determine which one. Therefore, to definitively identify the burnt-out bulb, we need to unscrew at least three of them (30 seconds) and screw in some others in their place (another 30 seconds).

We will show that 60 seconds will always be enough. Unscrew the first bulb and screw in a spare one in its place (20 seconds have passed). If the garland lights up, then we were lucky and it took only 20 seconds. If the garland does not light up, then the only faulty bulb is still in the garland, and we have a working bulb in our hand again. Now unscrew the second bulb and screw in the former first one in its place (40 seconds have passed in total). If we are still unlucky, then unscrew the third bulb and screw in the former second one in its place (60 seconds have passed in total). If the garland still does not light up, then the last bulb is the faulty one. The solution was also accepted for students who added another 20 seconds for replacing the last bulb.

## Answer

60 seconds.