Sunday Times Teaser 3299 – Who Wins Mostly?

by BRG on December 14, 2025

by Victor Bryant

Published Sunday December 14 2025 (link)

The teams in our local football league play each other once during the season, getting three points for a win and one for a draw. On the last day of the season each team played a game and then I worked out the league table with the teams in alphabetical order. Here is part of three rows of that league table in which digits have consistently been replaced by letters, with different letters used for different digits.

How many teams are there in the league and what number is LOST?

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BRG permalink


# From the properties of football games:
# S = 3.W + I
# T = 3.M + O
# W + H + O == W + I + N == M + O + S
#   ->  H == M + S - W
#   ->  N == H + O - I 
# L >= M and Y >= S

# consider W: S = 3.W + I -> 0 < W < 4
for W in (1, 2, 3):
  # consider M: T = 3.M + O -> 0 < M < 4
  for M in {1, 2, 3} - {W}:
    # consider I = S - 3.W
    for I in range(11 - 3 * W):
      S = 3 * W + I
      H = M + S - W
      # there are now five undetermined letters
      if len(s5 := set(range(10)) - {M, W, I, S, H}) == 5:
        # consider O = T - 3.M
        for O in range(11 - 3 * M):
          T = 3 * M + O
          N = H + O - I
          # there are now two undetermined letters
          if len(s2 := tuple(s5 - {O, T, N})) == 2:
            # assign the two remaining unused digits to L and Y
            for L, Y in (s2, s2[::-1]):
              if L >= M and Y >= S:
                LOST = 10 * (10 * (10 * L + O) + S) + T
                print(f"{W + H + O + 1} teams; {LOST = }.")

# From the properties of football games:

# S = 3.W + I

# T = 3.M + O

# W + H + O == W + I + N == M + O + S

# -> H == M + S - W

# -> N == H + O - I

# L >= M and Y >= S

# consider W: S = 3.W + I -> 0 < W < 4

for W in (1, 2, 3):

# consider M: T = 3.M + O -> 0 < M < 4

for M in {1, 2, 3} - {W}:

# consider I = S - 3.W

for I in range(11 - 3 * W):

S = 3 * W + I

H = M + S - W

# there are now five undetermined letters

if len(s5 := set(range(10)) - {M, W, I, S, H}) == 5:

# consider O = T - 3.M

for O in range(11 - 3 * M):

T = 3 * M + O

N = H + O - I

# there are now two undetermined letters

if len(s2 := tuple(s5 - {O, T, N})) == 2:

# assign the two remaining unused digits to L and Y

for L, Y in (s2, s2[::-1]):

if L >= M and Y >= S:

LOST = 10 * (10 * (10 * L + O) + S) + T

print(f"{W + H + O + 1} teams; {LOST = }.")

John Z. permalink

Mud thrown at a wall?


from itertools import permutations
# There must be an even number of teams because on the last day of the season
# each team played a game which would not be possible with an odd number of
# teams.
# The number of games each team plays in a season = nteams - 1

for W, H, O, I, N, S, M, T, L, Y in permutations(range(10)):
     # each team must play the same number of games
  if ((nteamsm1 := W + H + O) == W + I + N == M + O + S and
     # in a season the number of games played by each team is odd
     (nteamsm1 & 1) and
     # the number of points must sum correctly
     (3 * W + I == S) and
     (3 * M + O == T) and
     # if the team in the third row won more than lost then GF > GA and vv
     ((M > O and L > Y) or (M < O and L < Y))):

    print(f'| W | D | L | P | F | A |')
    print(f'-------------------------')
    print(f'| {W} | {H} | {O} |   |   |   |')
    print(f'| {W} | {I} | {N} | {S} |   |   |')
    print(f'| {M} | {O} | {S} | {T} | {L} | {Y} |')
    print()

    print('LOST:', L, O, S, T, 'Teams:', nteamsm1 + 1)

from itertools import permutations

# There must be an even number of teams because on the last day of the season

# each team played a game which would not be possible with an odd number of

# teams.

# The number of games each team plays in a season = nteams - 1

for W, H, O, I, N, S, M, T, L, Y in permutations(range(10)):

# each team must play the same number of games

if ((nteamsm1 := W + H + O) == W + I + N == M + O + S and

# in a season the number of games played by each team is odd

(nteamsm1 & 1) and

# the number of points must sum correctly

(3 * W + I == S) and

(3 * M + O == T) and

# if the team in the third row won more than lost then GF > GA and vv

((M > O and L > Y) or (M < O and L < Y))):

print(f'| W | D | L | P | F | A |')

print(f'-------------------------')

print(f'| {W} | {H} | {O} | | | |')

print(f'| {W} | {I} | {N} | {S} | | |')

print(f'| {M} | {O} | {S} | {T} | {L} | {Y} |')

print()

print('LOST:', L, O, S, T, 'Teams:', nteamsm1 + 1)

Frits permalink

@John, line 16 doesn’t always have to be the case.

Reply

John Z. permalink

Yes, lines 16 and 17 should be more rigorous in their logic so we have (Credit Brian)

Python

# Goals for must be >= wins and goals against must be >= losses (L >= M) and (Y >= S)):

1
2
3
4

# Goals for must be >= wins and goals against must be >= losses
(L >= M) and (Y >= S)):

Reply

Sunday Times Teaser 3299 – Who Wins Mostly?

by Victor Bryant

Published Sunday December 14 2025 (link)

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