Remember the busy Bee? The one that kept flying from the bicyclist to his home and back as he approached it?

Well, at the instant when the person finally got to his house, which way was the Bee facing? (Assume that the Bee's turns are instantaneous - that it can go from facing the house to facing the cyclist in no time.)

(In reply to by Sam)

Actually, Sam is both correct and incorrect. The Bee puzzle given here is, to all intents and purposes, equivalent to the Thompson's lamp example, but that does not mean that such puzzles must necessarily fail to have an answer. The problem is that the particular description of the puzzle is incomplete, and as a result we cannot tell what would happen at the limit. Compare Sam's description of Thomson's lamp (...where he turns it on at one second, off half a second later, on a quater of a second later, and so on. Is it on or off at two seconds?), which does not allow a solution, with the following more detailed account: Imagine we have a lamp (whose bulbs, wires, etc. are idealized so that they can take the strain of infinitely many changes in current, etc.), a wire leading from the lamp, through a power source (i.e. a battery, also idealized) and connected up to a switch (also idealized) that allows current to flow (and the light to thus glow) when the switch is depressed (pressed down, not sad) but allows no current to pass through when it is not depressed. Now find a rubber ball such that, when dropped from a certain height, will hit the switch and remain on it for a half-second, then bounce into the air for a half second, then sit on the switch for a quarter second, then bounce into the air for a quarter second, then sit on the switch for an eighth of a second, then bounce into the air for an eighth of a second, etc... (We can think of the time the ball 'sits' on the switch as the time that the ball, after first contacting the switch, is compressed by the impact and then regains its shape). Clearly, the ball will have bounced (and the light will have switched on and off) infinitely many times after 2 minutes, yet it is equally obvious in this case that the ball will, after its infinitely many bounces, be sitting on the switch at the end, so the light will be on. The moral is that certain puzzles involving infinity are only insoluble because the situation has not been described in enough detail and, given a sufficiently fine-grained description of what is going on, the answer is usually straightforward.

Posted by RoyCook on 2003-10-06 13:18:56