Sun Bin's, the descendant of Sun Tzu, generalized solution for how General Tianji should race his horses against the King of Qi.

tianji_horse_race_problem.py

"""
References:
 + Shu, J. (2012). On Generalized Tian Ji's Horse Racing Strategy.
   Interdisciplinary Science Reviews, 37, 187 - 193.
 + Leng, M., & Parlar, M. (2006). Game-theoretic analysis of an ancient Chinese
   horse race problem. Comput. Oper. Res., 33, 2033-2055.
"""

def get_horses(n):
  for horse in range(1, n+1):
    yield horse

def get_strategy(n):
  # g(x), h(x), and f(x) definitions
  g = lambda x: x - 1
  h = lambda x, n: n * (0 ** (x - 1))
  f = lambda x, n: g(x) + h(x, n)

  # use generator for horses
  for horse in get_horses(n):
    # calculate optimal horse pairs
    yield (horse, f(horse, n))

def print_choice(n):
  # loop over choice pairs
  for matches in get_strategy(n):
    # pretty print matches
    print(f'race: {matches[0]:{len(str(n))}} -> {matches[1]}')


# executable
if __name__ == '__main__':
  # get argparsing lib
  import argparse

  # describe
  parser = argparse.ArgumentParser(
    description='Calculate optimal horse racing strategy.'
  )

  # add arguments
  parser.add_argument('horses', metavar='n', type=int,
                      help='number of horses to race')

  # now get args
  args = parser.parse_args()

  # finally execute
  print_choice(args.horses)