This diary will be incredibly geeky and no doubt of limited interest, but people in the small group that is interested will probably enjoy it. With the second round of group play just completed, this seems just the evening to post this dense slab of whimsy.
I’m a “high holiday” soccer fan. While I’ll watch a match if it’s on, I tend not to get that excited about soccer -- other than the World Cup, which I always enjoy. (If that attitude is pathetic by the standards of soccer fandom, my guess is it probably also places me in the upper quintile of Americans, so I’m comfortable with it.)
Aside from watching the sport played at its highest level, one thing that the rudimentary math geek in me loves in watching the three rounds of group play unwind. The various possibilities regarding which two teams advance from each group contain a mathematical beauty that captures me every four years. And every World Cup I’d wonder about just what all of the various possibilities were. Then, one year, I figured it out. Below, I share the dope on all the possible end-states of group play. If that’s your cup of tea, drink up!
This will not be a diary about the possible resolutions of this year's play, except where different groups may serve as examples. Instead, it's about how group qualification works in any World Cup.
Part 1: The mathematical problem
The opening rounds of Group Play in the World Cup involve a round-robin in which each team within a four-team group plays every other team one time. Matches can end in a win, which garners a team three points; a draw, for which they get one point; or a loss, yielding no points. Each team will thus play three matches in group play. The top two teams in each group advance to the Round of 16 (aka "octofinals," though I don't hear the word used regarding soccer), which begins a series of sudden-death "knockout rounds." A series of tiebreakers will determine which squad will advance in the event that two or more teams are tied for the second qualifying spot, but those aren't the focus on this diary. (Suffice it to say, goal differential and offense are rewarded.)
As can be deduced from the above, each set of six games within a given group can yield a maximum of eighteen points (where there are no draws) down to a minimum of twelve points (where all six games are drawn.) Each draw subtracts a point from the total. This means that, by the end of group play, groups end up with wildly divergent sets of points. Right now, for example, one can see Groups B, E, and H with a relatively straightforward outcome: one team with 6 points (two victories), two with 3 points (a win and a loss), and one with none. Tie-ridden Groups C and F, by contrast, are packed together with one team with 4 points, two with 2, and one with 1. How many different outcomes, at the end of the three rounds, can there be?
I don't know if anyone besides me is interested in how many different sets of numbers will describe a Group's outcome by the end of three rounds; even after some checking in multiple years I've never seen the answer anywhere. But, if you're interested, the answer is right below.
Part 2: How many different results can a Group have after Group Play?
The answer is 40. There are 40 different possible numerical outcomes. Here, I wrote them out.
40 POSSIBLE POINT DISTRIBUTION PATTERNS FOR EACH GROUP
(TEAMS RANKED WITHIN PATTERN BY HIGH TO LOW SCORE)
[# OF TIMES PATTERN APPEARS AMONG RESULTS IS IN BRACKETS]
#01: 9 6 3 0 [24] CUMUL.
#02: 9 6 1 1 [12] 36
#03: 9 4 3 1 [24] 60
#04: 9 4 2 1 [24] 84
#05: 9 4 4 0 [12] 96
#06: 9 3 3 3 [8] 104
#07: 9 2 2 2 [4] 108
#08: 7 7 3 0 [12] 120
#09: 7 7 1 1 [6] 126
#10: 7 6 4 0 [24] 150
#11: 7 6 3 1 [24] 174
#12: 7 6 2 1 [24] 198
#13: 7 5 4 0 [24] 222
#14: 7 5 3 1 [24] 246
#15: 7 5 2 1 [24] 270
#16: 7 4 4 1 [36] 306
#17: 7 4 3 3 [24] 330
#18: 7 4 3 2 [24] 354
#19: 7 4 3 1 [24] 378
#20: 7 4 2 2 [24] 402
#21: 7 3 2 2 [12] 414
#22: 6 6 6 0 [8] 422
#23: 6 6 4 1 [24] 446
#24: 6 6 3 3 [24] 470
#25: 6 5 4 1 [24] 494
#26: 6 5 2 2 [12] 506
#27: 6 4 4 3 [36] 542
#28: 6 4 4 2 [24] 566
#29: 5 5 5 0 [4] 570
#30: 5 5 4 1 [24] 594
#31: 5 5 3 2 [12] 606
#32: 5 5 3 1 [12] 618
#33: 5 5 2 2 [12] 630
#34: 5 4 4 3 [24] 654
#35: 5 4 4 2 [24] 678
#36: 5 4 3 2 [24] 702
#37: 5 3 3 2 [12] 714
#38: 4 4 4 4 [6] 720
#39: 4 4 4 3 [8] 728
#40: 3 3 3 3 [1] 729
The number in brackets reflects the various possible ways that the pattern can occur by different teams winning. For example, pattern #40 can appear only one way: each team must tie each of the others. Pattern #29 can come about in four ways: each of teams A, B, and C beats team D, then they tie each game among themselves. Many patterns can be derives in 12 or 24 ways; two patterns, #16 (7 4 4 1) and #27 (6 4 4 3) can come about by 36 different combinations of wins and losses among the six group games. If, for example (but note that the results at the link will change over time), the France-South Africa and Mexico-Uruguay matches in Group A both end in victories for either side, the result will be (7 4 4 1). (France is still unlikely to advance, based on goal differential.) If Cameroon beats Holland in Group E while Japan and Denmark tie, we'll see a (6 4 4 3) result.
Note that I've ordered and numbered patterns of results by putting the scores in order from highest to lowest and then sorting the pattern as if it were a four digit number. You'll see those numbers come in handy in the tables below.
Part 3: How many different total outcomes of the six games can there be in a group?
The answer is 3^6 (three to the sixth power) or 729, as shown in the cumulative total in the chart above. Here's how we can derive it through a "brute strength" approach.
The first think you'll find out is that dealing with six games in which each game can have three results is a computation pain in the butt. I find it easier to separate out one team and then multiply their outcome by the results of a three-team round robin. Let's consider the squad that comes first alphabetically within the group to be "Team A," although the choice of which to designate as Team A is actually arbitrary (and inconsequential, so long as one is consistent.)
27 POSSIBILE OUTCOMES FOR TEAM A [RESULT VS. B, C, D]
9 pts [1] WWW (B+0, C+0, D+0)
7 pts [2] WWT (B+0, C+0, D+1)
7 pts [3] WTW (B+0, C+1 D+0)
7 pts [4] TWW (B+1, C+0, D+0)
6 pts [5] WWL (B+0, C+0, D+3)
6 pts [6] WLW (B+0, C+3, D+0)
6 pts [7] LWW (B+3, C+0, D+0)
5 pts [8] WTT (B+0, C+1, D+1)
5 pts [9] TWT (B+1, C+0, D+1)
5 pts [10] TTW (B+1, C+1, D+0)
4 pts [11] WTL (B+0, C+1, D+3)
4 pts [12] WLT (B+0, C+3, D+1)
4 pts [13] TWL (B+1, C+0, D+3)
4 pts [14] TLW (B+1, C+3, D+0)
4 pts [15] LWT (B+3, C+0, D+1)
4 pts [16] LTW (B+3, C+1, D+0)
3 pts [17] WLL (B+0, C+3, D+3)
3 pts [18] LWL (B+3, C+0, D+3)
3 pts [19] LLW (B+3, C+3, D+0)
3 pts [20] TTT (B+1, C+1, D+1)
2 pts [21] TTL (B+1, C+1, D+3)
2 pts [22] TLT (B+1, C+3, D+1)
2 pts [23] LTT (B+3, C+1, D+1)
1 pts [24] TLL (B+1, C+3, D+3)
1 pts [25] LTL (B+3, C+1, D+3)
1 pts [26] LLT (B+3, C+3, D+1)
0 pts [27] LLL (B+3, C+3, D+3)
This probably doesn't need to be explained, but the three letter code there is the outcome of Team A's three matches against, respectively, Teams B, C, and D. You see Team A's point total to the left; on the right you see how many points Teams B through D earn from their matches with Team A.)
Once you've figured out all of the 27 possible results for Team A, you can create a cross-product for each of the 27 possible results of round-robin play among Teams B through D. (This, not coincidentally, generates the same 729 total number of possible outcomes that we saw above.) I've drafted the ampersand into emergency service as a twenty-seventh consonant.
FOR EACH OUTCOME LISTED ABOVE, POSSIBLE OUTCOMES FOR THE THREE GAMES AMONG B, C, & D
[A] B=6, C=3, D=0 B>C, B>D, C>D
[B] B=6, C=1, D=1 B>C, B>D, C=D
[C] B=6, C=0, D=3 B>C, B>D, C<D
[D] B=4, C=3, D=1 B>C, B=D, C>D
[E] B=4, C=1, D=2 B>C, B=D, C=D
[F] B=4, C=0, D=4 B>C, B=D, C<D
[G] B=3, C=3, D=3 B>C, B<D, C>D
[H] B=3, C=1, D=4 B>C, B<D, C=D
[I] B=3, C=0, D=6 B>C, B<D, C<D
[J] B=4, C=4, D=0 B=C, B>D, C>D
[K] B=4, C=2, D=1 B=C, B>D, C=D
[L] B=4, C=1, D=3 B=C, B>D, C<D
[M] B=2, C=4, D=1 B=C, B=D, C>D
[N] B=2, C=2, D=2 B=C, B=D, C=D
[O] B=2, C=1, D=4 B=C, B=D, C<D
[P] B=1, C=4, D=3 B=C, B<D, C>D
[Q] B=1, C=2, D=4 B=C, B<D, C=D
[R] B=1, C=1, D=6 B=C, B<D, C<D
[S] B=3, C=6, D=0 B<C, B>D, C>D
[T] B=3, C=4, D=1 B<C, B>D, C=D
[U] B=3, C=3, D=3 B<C, B>D, C<D
[V] B=1, C=6, D=1 B<C, B=D, C>D
[W] B=1, C=4, D=2 B<C, B=D, C=D
[X] B=1, C=3, D=4 B<C, B=D, C<D
[Y] B=0, C=6, D=3 B<C, B<D, C>D
[Z] B=0, C=4, D=4 B<C, B<D, C=D
[&] B=0, C=3, D=6 B<C, B<D, C<D
In this way, one can designate each possible match outcome by a code such as "3J" or "21W". Here's what one code looks like for all possible combinations involving scenario 5, in which Team A beats Team B and Team C but loses to Team D:
ALL POSSIBLE PATTERN OUTCOMES WHERE TEAM A
BEATS TEAMS B AND C BUT LOSES TO TEAM D,
ALONG WITH THE RESULTING PATTERN NUMBER
5A A=6 B=6 C=3 D=3 #24
5B A=6 B=6 C=1 D=4 #23
5C A=6 B=6 C=0 D=6 #22
5D A=6 B=4 C=3 D=4 #27
5E A=6 B=4 C=1 D=5 #25
5F A=6 B=4 C=0 D=7 #10
5G A=6 B=3 C=3 D=6 #24
5H A=6 B=3 C=1 D=7 #11
5I A=6 B=3 C=0 D=9 #01
5J A=6 B=4 C=4 D=3 #27
5K A=6 B=4 C=2 D=4 #28
5L A=6 B=4 C=1 D=6 #23
5M A=6 B=2 C=4 D=4 #28
5N A=6 B=2 C=2 D=5 #26
5O A=6 B=2 C=1 D=7 #12
5P A=6 B=1 C=4 D=6 #23
5Q A=6 B=1 C=2 D=7 #12
5R A=6 B=1 C=1 D=9 #02
5S A=6 B=3 C=6 D=3 #24
5T A=6 B=3 C=4 D=4 #27
5U A=6 B=3 C=3 D=6 #24
5V A=6 B=1 C=6 D=4 #23
5W A=6 B=1 C=4 D=5 #25
5X A=6 B=1 C=3 D=7 #11
5Y A=6 B=0 C=6 D=6 #22
5Z A=6 B=0 C=4 D=7 #10
5& A=6 B=0 C=3 D=9 #01
Now, you may at this point be asking yourself why one would do this. The answer is: don't ask. If you think it's cool, you don't need to ask; if you don't, no explanation is likely to satisfy you.
For my part, I just think it's sort of cool to be able to summarize the result of groups so easily. Taking Algeria as the "Team A" in Group C and proceeding with England, Slovenia, and the US alphabetically, I find it that during the game I may well be rooting for 24N (to make it more likely that we advance, despite my being pissed with the Slovene holding on the disallowed goal), while secretly fearing 22E. Your mileage may vary.
Part 4
Enjoy round 3 and the knockout rounds! Feel welcome to post soccer/futbol speculation/commentary!