frogs.py

import fractions

for num_pads in range(15):
    options = {num_pads: fractions.Fraction(1)}
    for pad in range(num_pads-1, 0, -1):
        free_pads = [p for p in range(num_pads+1) if p > pad]
        path_lengths = []
        for free_pad in free_pads:
            path_lengths.append(options[free_pad] + 1)
        options[pad] = sum(path_lengths) / len(path_lengths)
    print("{} pads: {} hops".format(num_pads, sum(options.values()) / len(options)))