Sunday Times Teaser 2528

by Nick MacKinnon

Published: 6 March 2011 (link)

Some teams, A, B, C and so on, play each other once in a football league, earning three points for a win and one for a draw. So far, each team has played two games and scored the same total number of goals. At the moment, they stand in alphabetical order, with each team having a different number of points. The result of the game BvE was 3-2, and indeed all the wins so far have been by a one-goal margin. If you knew how many teams there were in the league, then it would be possible to work out all the results so far.

What are those results?

One Comment Leave one →

Brian Gladman permalink

A solution using my football solver class:


from football import Football

max_goals_per_side = 8
fa_goals = (0, max_goals_per_side)

f = Football('ABCDEF', scoring={'w':3, 'd':1}, outcomes='ldwu', sep='')

for nt in (5, 6):
  
  d0 = {'BE':(3, 2), 'EB':(2, 3)}
  for d1 in f.games(d0, 'B'):
    bt = f.table(d1, 'B')
    if bt.p != 2 or nt == 5 and d1['BF'] != 'u':
      continue

    for d2 in f.games(d1, 'C'):
      ct = f.table(d2, 'C')
      if ct.p != 2 or ct.pts >= bt.pts or nt == 5 and d2['CF'] != 'u':
        continue

      for d3 in f.games(d2, 'A'):
        at = f.table(d3, 'A')
        if at.p != 2 or bt.pts >= at.pts or nt == 5 and d3['AF'] != 'u':
          continue

        for d4 in f.games(d3, 'D'):
          dt = f.table(d4, 'D')
          if dt.p != 2 or dt.pts >= ct.pts or nt == 5 and d4['DF'] != 'u':
            continue

          for d5 in f.games(d4, 'E'):
            et = f.table(d5, 'E')
            if et.p != 2 or et.pts >= dt.pts or nt == 5 and d5['EF'] != 'u':
              continue

            ft = f.table(d5, 'F')
            if (nt == 6 and (ft.p != 2 or ft.pts >= et.pts) 
                                           or nt == 5 and d5['EF'] != 'u'):
              continue

            for d6 in f.matches(d5, 'B', fa_goals, fa_goals):
              bt = f.table(d6, 'B')
              if any(x != y and abs(x - y) > 1 for (x, y) 
                                          in f.results(d6, 'B')[1]):
                continue
             
              for d7 in f.matches(d6, 'A', bt.gf, fa_goals):
                at = f.table(d7, 'A')
                if any(x != y and abs(x - y) > 1 for (x, y) 
                                            in f.results(d7, 'A')[1]):
                  continue

                for d8 in f.matches(d7, 'C', bt.gf, fa_goals):
                  ct = f.table(d8, 'C')
                  if any(x != y and abs(x - y) > 1 for (x, y) 
                                              in f.results(d8, 'C')[1]):
                    continue

                  for d9 in f.matches(d8, 'D', bt.gf, fa_goals):
                    dt = f.table(d9, 'D')
                    if any(x != y and abs(x - y) > 1 for (x, y) 
                                                in f.results(d9, 'D')[1]):
                      continue

                    for d10 in f.matches(d9, 'E', bt.gf, fa_goals):
                      et = f.table(d10, 'E')
                      if any(x != y and abs(x - y) > 1 for (x, y) 
                                                  in f.results(d10, 'E')[1]):
                        continue

                      for d11 in f.matches(d10, 'F', bt.gf if nt == 6 else 0, 
                                                    fa_goals if nt == 6 else 0):
                        ft = f.table(d11, 'F')
                        if nt == 6 and any(x != y and abs(x - y) > 1 
                                             for (x, y) in f.results(d1, 'F')[1]):
                          continue
                        
                        print(f"{nt} teams:")
                        print(', '.join(f"{n}:{r}" for 
                                        n, r in zip(*f.results(d11, 'A')[:2])))
                        print(', '.join(f"{n}:{r}" 
                                        for n, r in zip(*f.results(d11, 'B')[:2])))
                        print(', '.join(f"{n}:{r}" 
                                        for n, r in zip(*f.results(d11, 'C')[:2])))
                        print(', '.join(f"{n}:{r}" 
                                        for n, r in zip(*f.results(d11, 'D')[:2])))
                        print(', '.join(f"{n}:{r}" 
                                        for n, r in zip(*f.results(d11, 'E')[:2])))
                        if nt == 6:
                          print(', '.join(f"{n}:{r}" 
                                        for n, r in zip(*f.results(d11, 'F')[:2])))
                        print()

from football import Football

max_goals_per_side = 8

fa_goals = (0, max_goals_per_side)

f = Football('ABCDEF', scoring={'w':3, 'd':1}, outcomes='ldwu', sep='')

for nt in (5, 6):

d0 = {'BE':(3, 2), 'EB':(2, 3)}

for d1 in f.games(d0, 'B'):

bt = f.table(d1, 'B')

if bt.p != 2 or nt == 5 and d1['BF'] != 'u':

continue

for d2 in f.games(d1, 'C'):

ct = f.table(d2, 'C')

if ct.p != 2 or ct.pts >= bt.pts or nt == 5 and d2['CF'] != 'u':

continue

for d3 in f.games(d2, 'A'):

at = f.table(d3, 'A')

if at.p != 2 or bt.pts >= at.pts or nt == 5 and d3['AF'] != 'u':

continue

for d4 in f.games(d3, 'D'):

dt = f.table(d4, 'D')

if dt.p != 2 or dt.pts >= ct.pts or nt == 5 and d4['DF'] != 'u':

continue

for d5 in f.games(d4, 'E'):

et = f.table(d5, 'E')

if et.p != 2 or et.pts >= dt.pts or nt == 5 and d5['EF'] != 'u':

continue

ft = f.table(d5, 'F')

if (nt == 6 and (ft.p != 2 or ft.pts >= et.pts)

or nt == 5 and d5['EF'] != 'u'):

continue

for d6 in f.matches(d5, 'B', fa_goals, fa_goals):

bt = f.table(d6, 'B')