(From Omer Tamuz)
A prisoner escaped from prison, and, starting from some n on the positive integer line, jumps k steps to the right at each step. An adversary tries to catch him by landing a the same place as the prisoner at the same time. The adversary chooses one place at each time step, and all choices are possible. For example: 3,10,5,2,58,9,... But the adversary knows neither k nor n.
How can he or she catch the prisoner?
What if the prisoner takes his basic calculator and uses some elementary function of time (using addition, substraction, multiplication, division, logarithms, exponentials, sine and cosine - rounded to the nearest integer) to compute where to go at time t?
What if the prisoner takes his laptop and writes a program that will, for each t, determine where he goes at time t?