Sunday Times Teaser Teaser 2637 – Powerful Team

by Danny Roth

The players in a local football team have the numbers 1 to 22 with eleven playing and eleven in reserve in any match.

In one match, the players on the field were such that there was no sequence of three consecutive numbers among them. The same was true of the reserves. The sum of the fielded player’s numbers was also a perfect square.

All these facts remained true after a single substitution at half time but the perfect square was then higher.

What were the numbers (in increasing order) of the players in the first half of the match?

4 Comments Leave one →

brian gladman permalink

Here is my solution:


# place a player in the team or as a reserve
def place(team, resv):
  lt, lr = len(team), len(resv)
  # we are done - the team and the reserves each have 11 players
  if lt == lr == 11:
    # do the teams numbers add to form a perfect square?
    t = sum(team)
    if round(t ** 0.5) ** 2 != t:
      return
    # if so we have a possible solution
    yield tuple(team)
  else:
    # the next number to place
    nxt = lt + lr + 1
    # this number can be in the team or in the reserves provided
    # that it does not form three consecutive values with the two
    # previous values in these sequences  
    if lt < 11 and (lt < 2 or nxt != team[-2] + 2):
      yield from place(team + [nxt], resv)
    if lr < 11 and (lr < 2 or nxt != resv[-2] + 2):
      yield from place(team, resv + [nxt])

for sol in place([], []):
  team = set(sol)
  # select a substitute from the reserves for the second ha
  for sub in set(range(1, 23)) - team:
    # select the player that leaves the team
    for pl in team:
      # form the second half team
      new_team = (team - set((pl,))) | set((sub,))
      # and check that the numbers sum to a larger perfect square
      sq = sum(new_team)
      if round(sq ** 0.5) ** 2 == sq and sq > sum(team):
        print('Players in the first half have numbers', sol)

# place a player in the team or as a reserve

def place(team, resv):

lt, lr = len(team), len(resv)

# we are done - the team and the reserves each have 11 players

if lt == lr == 11:

# do the teams numbers add to form a perfect square?

t = sum(team)

if round(t ** 0.5) ** 2 != t:

return

# if so we have a possible solution

yield tuple(team)

else:

# the next number to place

nxt = lt + lr + 1

# this number can be in the team or in the reserves provided

# that it does not form three consecutive values with the two

# previous values in these sequences

if lt < 11 and (lt < 2 or nxt != team[-2] + 2):

yield from place(team + [nxt], resv)

if lr < 11 and (lr < 2 or nxt != resv[-2] + 2):

yield from place(team, resv + [nxt])

for sol in place([], []):

team = set(sol)

# select a substitute from the reserves for the second ha

for sub in set(range(1, 23)) - team:

# select the player that leaves the team

for pl in team:

# form the second half team

new_team = (team - set((pl,))) | set((sub,))

# and check that the numbers sum to a larger perfect square

sq = sum(new_team)

if round(sq ** 0.5) ** 2 == sq and sq > sum(team):

print('Players in the first half have numbers', sol)

BRG permalink

Here is a slightly shorter solution:


# place a player in the team or as a reserve
def place(team=[], resv=[]):
  lt, lr = len(team), len(resv)
  # we are done - the team and the reserves each have 11 players
  if lt == lr == 11:
    # do the teams numbers add to form a perfect square?
    t = sum(team)
    if (r := round(t ** 0.5)) ** 2 == t:
      # if so we have a possible solution
      yield r, tuple(team), tuple(resv)
  else:
    # the next number to place
    nxt = lt + lr + 1
    # this number can be in the team or in the reserves provided
    # that it does not form three consecutive values with the two
    # previous values in these sequences  
    if lt < 11 and (lt < 2 or nxt != team[-2] + 2):
      yield from place(team + [nxt], resv)
    if lr < 11 and (lr < 2 or nxt != resv[-2] + 2):
      yield from place(team, resv + [nxt])

# consider the possible square roots, teams and reserves  
for sr, team, resv in place():
  # pick the number to be substituted
  for p in team:
    # consider higher square roots (all roots are in 10 .. 12) 
    for nsr in range(sr + 1, sr + 4):
      # find the number of the substitute using sr^2 - p == nsr^2 - s
      if (s := nsr ** 2 - sr ** 2 + p) in resv:
        print(f'Players in the first half have numbers {team} ({p} <- {s})')

# place a player in the team or as a reserve

def place(team=[], resv=[]):

lt, lr = len(team), len(resv)

# we are done - the team and the reserves each have 11 players

if lt == lr == 11:

# do the teams numbers add to form a perfect square?

t = sum(team)

if (r := round(t ** 0.5)) ** 2 == t:

# if so we have a possible solution

yield r, tuple(team), tuple(resv)

else:

# the next number to place

nxt = lt + lr + 1

# this number can be in the team or in the reserves provided

# that it does not form three consecutive values with the two

# previous values in these sequences

if lt < 11 and (lt < 2 or nxt != team[-2] + 2):

yield from place(team + [nxt], resv)

if lr < 11 and (lr < 2 or nxt != resv[-2] + 2):

yield from place(team, resv + [nxt])

# consider the possible square roots, teams and reserves

for sr, team, resv in place():

# pick the number to be substituted

for p in team:

# consider higher square roots (all roots are in 10 .. 12)

for nsr in range(sr + 1, sr + 4):

# find the number of the substitute using sr^2 - p == nsr^2 - s

if (s := nsr ** 2 - sr ** 2 + p) in resv:

print(f'Players in the first half have numbers {team} ({p} <- {s})')

ahmet cetinbudaklar permalink

Here is my analytical solution:

1,2,4,5,7,8,10,11,13,14,16=91 giving us the minimum value of the first half team as 100 and taking into account of the fact that the maximum diffrerence in the sum of the teams for both halfs cannot be larger than 21(22-1) which gives us the sum of the second half half team as 121.The minimum sum of the first half team numbers can be found by trial and error.:

Reply
- brian gladman permalink
  
  Hi Ahmet,
  
  Thank you for your contribution.
  
  You may not be aware that this site is really intended for discussion of programmed solutions using Python.
  
  Of course, analysis as a part of the process of producing a program is fine but purely analytic solutions are more likely to be appreciated by users of this site:
  
  https://sites.google.com/site/sundaytimesteasers/
  
  which I run for people who prefer not to use programs.
  
  Reply

Sunday Times Teaser Teaser 2637 – Powerful Team

by Danny Roth

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