Goldbug Variations
نویسنده
چکیده
Jim Propp bugs me sometimes. I’m usually glad when he does. Today, Jim’s bugs are trained to hop back and forth on the positive integers: place a bug at 1, and with each hop, a bug at i moves to either i + 1 or i − 2. Of course, it might jump off the left edge; we put two bugcatching cups at 0 and −1. Once a bug lands in a cup, we start a new bug at 1. What I haven’t mentioned is how the bugs decide whether to jump left or right. We could declare it a random walk, stepping in either direction with probability /2, but the U. of Wisconsin professor’s bugs are more orderly than that. At each location i, there is a signpost showing an arrow: it can point either Inbound, towards i − 2 and the bug cups, or Outbound, towards infinity. The bugs are somewhat contrarian, so when a bug lands at i, it first flips the arrow to point the opposite direction, and then hops that way (Fig. 1). Add an initial condition that all arrows begin pointing Outbound, and we have a deterministic system. Let bugs hop till they drop (Fig. 2). Well, what happens? Or, for those who would like a more directed question: First, show that every bug lands in a cup (as opposed to going off
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