Enigmatic Code
Programming Enigma Puzzles
Enigma 1438: Soccer stats
Posted by on 22 April 2013
From New Scientist #2599, 14th April 2007
There are five teams in our local league, and each team plays each of the others once a season. Below is the league table at the end of last season.
Unfortunately three of those numerical entries are incorrect. What were the scores in Arnsley’s four games?
[enigma1438]
Enigmatic Code 
A bit of analysis tells us in which part of the table the errors must lie, and this Python program does the rest. It’s a bit messy, but it runs in 71ms.
# consider the sums of the columns: # won = 8, drawn = 7, lost = 5, for = 7, against = 7 # # each won game for one team is lost for another team, so # it should be that "won" = "lost", but they are different # hence there is at least one error in the "won" and "lost" # columns. # # also each drawn game is drawn for two teams, so "drawn" should # be a multiple of 2, and it isn't so there is at least one # error in the "drawn" column. # # each goal scored for a team is scored against another team # so these columns should be equal - and they are, so if there # was an error in one column there would have to be an error # in the other column to bring the values back into line, hence # there cannot be exactly one error in the "for" and "against" # columns (which is the maximum possibility), so these columns # must contain no errors. # # hence all three errors are in the "won", "lost", "drawn" columns. from collections import namedtuple from itertools import product from enigma import irange, printf # the 10 matches are: # AvB AvC AvD AvE BvC BvD BvE CvD CvE DvE # each is either won, lost or drawn ('d') Table = namedtuple('Table', 'w l d') # generate the "won", "drawn", "lost" entries in the table def table(t, games): w, l, d = 0, 0, 0 for g in games: if g == 'd': d += 1 elif g == t: w += 1 else: l += 1 return Table(w, d, l) # generate possible scores for team <t> in games <games> # with total goals for <f> and total goals against <a> def scores(t, games, f, a): if len(games) == 0: if f == a == 0: yield [] else: g = games[0] if g == 'd': for n in irange(0, min(f, a)): for s in scores(t, games[1:], f - n, a - n): yield [(n, n)] + s elif g == t: for n in irange(1, f): for m in irange(0, min(n - 1, a)): for s in scores(t, games[1:], f - n, a - m): yield [(n, m)] + s else: for n in irange(0, f): for m in irange(n + 1, a): for s in scores(t, games[1:], f - n, a - m): yield [(n, m)] + s # make the table for A for (AB, AC, AD, AE) in product('ABd', 'ACd', 'ADd', 'AEd'): A = table('A', (AB, AC, AD, AE)) # count the discrepancies dA = sum(1 for (p, q) in zip(A, (2, 1, 1)) if p != q) # make the table for B for (BC, BD, BE) in product('BCd', 'BDd', 'BEd'): B = table('B', (AB, BC, BD, BE)) # count the discrepancies dB = sum(1 for (p, q) in zip(B, (3, 1, 0)) if p != q) if dA + dB > 3: continue # table for C for (CD, CE) in product('CDd', 'CEd'): C = table('C', (AC, BC, CD, CE)) dC = sum(1 for (p, q) in zip(C, (1, 3, 0)) if p != q) if dA + dB + dC > 3: continue # table for D and E for DE in 'DEd': D = table('D', (AD, BD, CD, DE)) E = table('E', (AE, BE, CE, DE)) dD = sum(1 for (p, q) in zip(D, (1, 2, 1)) if p != q) dE = sum(1 for (p, q) in zip(E, (1, 0, 3)) if p != q) d = dA + dB + dC + dD + dE if d != 3: continue # generate scores that match up to the "for" against "columns" for (sAB, sAC, sAD, sAE) in scores('A', [AB, AC, AD, AE], 2, 2): for (sBC, sBD, sBE) in scores('B', [BC, BD, BE], 2 - sAB[1], 1 - sAB[0]): for (sCD, sCE) in scores('C', [CD, CE], 1 - sAC[1] - sBC[1], 0 - sAC[0] - sBC[0]): for (sDE,) in scores('D', [DE], 1 - sAD[1] - sBD[1] - sCD[1], 1 - sAD[0] - sBD[0] - sCD[0]): if (sAE[1] + sBE[1] + sCE[1] + sDE[1], sAE[0] + sBE[0] + sCE[0] + sDE[0]) != (1, 3): continue printf("AvB: {sAB}, AvC: {sAC}, AvD: {sAD}, AvE: {sAE}") printf("BvC: {sBC}, BvD: {sBD}, BvE: {sBE}") printf("CvD: {sCD}, CvE: {sCE}") printf("DvE: {sDE}") print('')Solution: The scores in Arnsley’s four games were – Arnsley vs. Boldham: 0 – 2; Arnsley vs. Cleeds: 0 – 0; Arnsley vs. Drochdale: 1 – 0; Arnsley vs. Erby: 1 – 0.
The incorrect values are the “won”, “drawn” and “lost” values for Boldham.