Sunday Times Teaser 2765 – Prime Points
by John Owen
Published: 20 September 2015 (link)
In my fantasy football league each team plays each other once, with three points for a win and one point for a draw. Last season Aberdeen won the league, Brechin finished second, Cowdenbeath third, and so on, in alphabetical order. Remarkably each team finished with a different prime number of points. Dunfermline lost to Forfar.
In order, what were Elgin’s results against Aberdeen, Brechin, Cowdenbeath, and so on (in the form WWLDL…)?
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Brian Gladman permalink123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687from itertools import combinations, permutations, productfrom collections import defaultdictfrom number_theory import Primes, frobenius_solve# return unique permutations of an input sequencedef unique_permutations(x):seen = set()for p in permutations(x):if p not in seen:seen.update([p])yield p# compile match results that give a specified set of (win, draw, loss)# values; for the list (in res) of (win, draw, loss) values for the# teams in order of their league positions (A, B, ...) allocate their# wins, draws and losses as losses, draws and wins for the teams in# lower positions, permuting them to check all possible allocations;# repeat this until all match scores have been allocated in this way;def compile_table(res, g):if len(res) == 0:# we have a compatible set of match scores  convert the results# in g to match results for each teamr = defaultdict(str)for a in range(len(g)):for b in range(a + 1, len(g)):r[chr(ord('A') + a)] += 'LDW'[g[a][b  a  1]]r[chr(ord('A') + b)] += 'WDL'[g[a][b  a  1]]yield g, relse:# convert the current team's wins, draws and losses into losses,# draws and wins for the lower placed teams (use indexes into 'WDL')t = [2] * res[0][0] + [1] * res[0][1] + [0] * res[0][2]# permute these results for allocation to the remaining teamsfor p in unique_permutations(t):# now compile the remaining wins, draws and losses after these# results have been accounted fornew_res = tuple(tuple(x[j]  (1 if j == y else 0)for j in (0, 1, 2)) for x, y in zip(res[1:], p))# ensure that the win, draw, loss values remain validif all(x >= 0 for y in new_res for x in y):# proceed to the next team lower in the league and repeatyield from compile_table(new_res, g + [p])# for saving possible solutionspsol = []# Since teams each play (N  1) matches, the maximum points for a team is# 3N  3; so N must be such that the N'th prime is <= 3N  3, which means# that N is less than 10 with a maximum prime of 23pr = Primes().list(0, 25)# the number of teamsfor teams in range(6, 10):# the number of matchesmatches = teams * (teams  1) // 2for seq in combinations(pr, teams):# ensure that the sum of the points is valid (the minmum/maximum total# points are two/three times the number of matches)if not (2 * matches <= sum(seq) <= 3 * matches):continue# find all possible (win, draw, loss) combinations that give these numbers# of points (put higher scores first  i.e. in order of league position)for wd_l in product(*(frobenius_solve((1, 3), s) for s in reversed(seq))):# add in the losses for each team and check that they are not negativewdl_l = [(d, w, teams  1  w  d) for w, d in wd_l if w + d < teams]# check we have (win, draw, loss) values for every teamif len(wdl_l) == teams:# now compile the total wins, draws and lossesw, d, l = (sum(x[0] for x in wdl_l), sum(x[1] for x in wdl_l),sum(x[2] for x in wdl_l))# total wins must equal total losses and total draws must be even# team D must lose at least once and team F must win at least onceif w == l and d % 2 == 0 and wdl_l[3][2] and wdl_l[5][0] > 0:psol += [tuple(wdl_l)]# for each set of (win, draw, loss) sequences for the teams, look for a set of# compatible match resultsfor res in psol:first = Truefor g, r in compile_table(res, []):if first:print('WDL (A, B, ...):', *res)print('Results:')first = False# set a marker for any results where D loses to Fs = ' ***' if r['D'][4] == 'L' else ''print(' ', tuple(x + ':' + r[x] for x in sorted(r.keys())), s)