On a giant tape measure sits an N-frog: that is, a frog with a special preference for the number N. The frog's location at the beginning of each of its jumps is called M. The frog moves on the tape measure according to the following rules:
Whenever M>N, then the N-frog makes an N-jump to the left and lands on the number M-N.
Whenever M<N, then the N-frog makes an N-jump to the right and lands on number M+N; during landing it also changes its preference and becomes an M-frog.
Whenever M=N then the frog is happy and stays on that number.
Where will a 851-frog that sits on 1517 be happy?
Where will an N-frog that sits on M be happy?
a) Whole frogs that start at fractional locations become fractional. A 1-frog that starts at 7/4 is happy as a 1/4 frog.
b) Rational frogs that start at irrational locations turn irratational,
hop forever, are never happy, and become closer and closer to
0-frogs. Unless they can hop faster and faster (like in zeno's
paradox) in which case after a finite amount of time and infinite
number of jumps (and a lot of computing) they do become happy
0-frogs. Both of these seem to me to defy nature. I suspect
that they die hopping, but near the end they look like they are already
dead.
c) And negative start points are a problem, so I hope your tape measure
starts at zero. If a positive frog starts on a negative location,
he tends to become negative and hop off the positive end of the tape
measure. Unless your tape measure is very long, in which case he
hops off to positive infinity.
Aberrant Frogs, a.k.a. Flys in the Ointment
