Originally posted by Paul Dirac II
To clarify a bit more...
If we "played the game" five times and noted that the path lengths happened to be (in any order) 2, 2, 2, 3, 4, we calculate "experimentally" that the mean is 13/5.
If instead we convince ourselves that there are only five paths the piece could take (rather than the infinity of paths available in the actual knight problem ...[text shortened]... ges to the theoretical mean when the game is played enough times, such as by a computer program.
Well, I have
now run this a couple of times, and I've come to the following conclusions:
- The mode is indeed, as was obvious from the start, 2, which is also the minimum. It seems that very small numbers are also slightly more common than higher ones, but after that, I can't at a glance see if very high is predictably
less common than any merely high count, though in aggregate it does seem to hold.
- The maximum isn't in the millions; of course, in theory that's a possibility, but the highest I've seen was 1390. Tours in the high decades and low hundreds are common; high hundreds are not unheard of.
- The running average rambled all over the 140s in all experiments, with somewhat of a predilection for the higher ones, but looked like it refused to approach any single value more closely than the others. One time it veered into the low 150s but quickly dropped again; another time it stayed in the low 140s for a while but after a while got back up - and then down again.
- There is no indication in my (smallish; a couple of runs of roughly thousand tours each) experiment that the final value, if there is a fixed one, ends up being an integer. If forced to pick, I'd choose 148 and 149, with no idea which it is.