Sunday Times Teaser 2801 – End of Season

by Victor Bryant

Published: 29 May 2016 (link)

In this football league each team plays each of the others once (with three points for a win and one for a draw). The end-of-season table has teams in decreasing order of points (ties being determined by goals scored). Here are some of the entries from three rows of the table, but with digits consistently replaced by letters.

2801

How many teams are in the league, and what was the score when Villa played Wanderers?

2 Comments Leave one →

Brian Gladman permalink

This program shows that all the letters other than Y are fully determined by the constraints set by the league table rows given. The possible values for Y are 4, 5 and 6.


from itertools import permutations

header    = '|   Team    | Pts | Win | Draw | Loss | For | Agnst |'
sep     = '\n+-----------+-----+-----+------+------+-----+-------+'
united    = '|  United   |     |  {}  |  {}   |      |     |       |'
villa     = '|  Villa    |  {}  |  {}  |  {}   |  {}   |  {}  |  {}    |'
wanderers = '| Wanderers |     |     |  {}   |  {}   |  {}  |  {}    |'

for A, E, I, M, R, S, T, Y in permutations(range(10), 8):
  
  # Villa's points must be correct
  if T != 3 * E + A:
    continue
  
  # United's points cannot be less than Villa's  
  if 3 * M + Y < T:
    continue
  
  # since Villa and Wanderers are equal on points, Villa
  # must score more goals than Wanderers
  if not E > M:
    continue
  
  # Villa and Wanderers play the same number of matches 
  if T + I != E + A + S:
    continue
  
  # since Wanderers have no wins and 'for' and 'against'
  # goals in draws are equal, their 'against' goal excess
  # means that they have at least one loss; moreover lost
  # games each require at least one excess 'against' goal 
  if not E - M >= I > 0:
    continue
  
  # Villa must have at least as many goals against them 
  # as their number of losses
  if not R >= S:
    continue
  
  # output the rows in the league table
  print(sep)
  print(header, sep)
  print(united.format(M, Y) , sep)
  print(villa.format(T, E, A, S, E, R), sep)
  print(wanderers.format(T, I, M, E),  sep)

from itertools import permutations

header = '| Team | Pts | Win | Draw | Loss | For | Agnst |'

sep = '\n+-----------+-----+-----+------+------+-----+-------+'

united = '| United | | {} | {} | | | |'

villa = '| Villa | {} | {} | {} | {} | {} | {} |'

wanderers = '| Wanderers | | | {} | {} | {} | {} |'

for A, E, I, M, R, S, T, Y in permutations(range(10), 8):

# Villa's points must be correct

if T != 3 * E + A:

continue

# United's points cannot be less than Villa's

if 3 * M + Y < T:

continue

# since Villa and Wanderers are equal on points, Villa

# must score more goals than Wanderers

if not E > M:

continue

# Villa and Wanderers play the same number of matches

if T + I != E + A + S:

continue

# since Wanderers have no wins and 'for' and 'against'

# goals in draws are equal, their 'against' goal excess

# means that they have at least one loss; moreover lost

# games each require at least one excess 'against' goal

if not E - M >= I > 0:

continue

# Villa must have at least as many goals against them

# as their number of losses

if not R >= S:

continue

# output the rows in the league table

print(sep)

print(header, sep)

print(united.format(M, Y) , sep)

print(villa.format(T, E, A, S, E, R), sep)

print(wanderers.format(T, I, M, E), sep)

The resulting values show that there are eleven teams in the league.

We are told that Villa play Wanderers; and, since Villa have no draws and Wanderers have no wins, Villa must Win against Wanderers. The goals scored by Wanderers in draws must each be matched by a goal scored against them and, since this leaves just one more ‘against’ goal, this must occur in the game that they lost. The score was hence Villa 1, Wanderers 0.

John Crabtree permalink

Or, as Villa score E goals in the E games which they win, they must win those games 1-0.

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Sunday Times Teaser 2801 – End of Season

by Victor Bryant

Published: 29 May 2016 (link)

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