AHHH! I Lost the (Calculator) Game

Calculators are addicting. Well, not exactly. But one day when I finished a problem early in Chemistry, I began to punch random numbers on my calculator. Now, it so happens that when you start taking the natural logarithm of a number repeatedly, you get smaller and smaller numbers, until it becomes negative. If you push ln again, an E for error appears. To avoid this, I decided that whenever the number was negative, I would square it. I did this for a while, and found myself getting excited every time a big number appeared, and getting angry whenever I accidentally did ln to a negative number, and got an E. This turned into a game. The rules were simple, so you could play it while half-heartedly listening to a teacher lecture, and people would assume that you, being an exceptional student, were doing your own calculations.

Yesterday, Sachi and I were talking about math, and we had our calculators out, when I somehow thought to mention my silly game. "You mash your fingers over the keys," I said, "and the rules are that you push ln repeatedly until you get a negative number, in which case you square it. You get happy when you get a big number, and if you accidentally get an E, that means you lose." Now, I was sure that this would not be of very much interest to anyone, but Sachi took to it immediately. Rebecca, who was also with us, did it too. Now, the three of us were joyfully punching buttons on our calculators, and occasionally someone would shriek or cringe, and yell out, "I lost the game!" Only this game was better than the "I lost the game" game. Soon, we were pondering its mathematical properties. Sachi pointed out that it was completely deterministic--that is, once you put in the number to start with, the rest of the game is pre-determined, except when you lose. Rebecca and Sachi, who have been taking classes at MIT about chaos theory, wondered whether the calculator game was chaotic. We decided that we would research the game, but first, we just wanted to get nice big numbers to pop up. We played the game for a half hour or so, and found that the cosine of pi was an especially good number, until Rebecca had to leave. Sachi told me that she would spend hours playing the game instead of doing her homework. "Tell me your results tomorrow," I said as Sachi went home. Who knew that the game would be such a hit?

-Philip