Educating math by means of World Cup soccer

1. The group stage

First, we’ve got a group stage, the place the 32 qualifying groups are positioned in 8 teams 4, and in every group, the 4 groups play one another round-robin (i.e., every staff performs one another staff); every staff will get factors that means (e.g., 3 for a win, 1 for a draw), and the 2 groups with the best variety of factors in every group advance to the knockout stage. (As a result of groups may be tied on factors, this depends on a lot of tiebreaker guidelines.)

In a round-robin event, the variety of methods of populating “X performs in opposition to Y” when there are n groups is n × (n–1), as a result of you’ll be able to put n groups within the first spot and n–1 groups within the second spot (as a result of “England performs England” is not a factor). However while you do this, you are double-counting, as a result of “England performs the U.S.” is counted in a different way than “The U.S. performs England”. So we’ll simply divide by 2, and get n(n–1)/2.

Typically, that is the combinatorics idea referred to as “n select okay”, the place (n,okay) = n!/[(n–k)!k!]; right here, we’re utilizing “n select 2”, which is simply equal to n(n–1)/2.

Anyway, which means every group of 4 has 4×3/2 = 6 video games, and since there are 8 teams, that makes 8×6 = 48 video games.

2. The knockout stage

Subsequent, we’ve got a knockout stage, the place the 16 groups are decreased to 1. This can be a single-elimination event, the place the loser in any recreation is instantly eradicated. (Not like within the group stage, you’ll be able to’t have ties in particular person video games, so this requires a means of manufacturing a winner in every particular person recreation, e.g., sudden dying overtimes and penalty kicks.)

When the variety of groups is an influence of two, then it is easy to provide brackets—and it is easy to examine that with 16 groups, you get a winner with 15 video games (i.e., 8 video games within the spherical of 16, plus 4 video games within the quarter-final, plus 2 video games within the semi-final, plus 1 last recreation, and eight + 4 + 2 + 1 = 15). However you’ll be able to at all times produce a bracket with a non-power of two by utilizing some variety of byes.

So, you’ll be able to ask, what number of video games would you want typically in a single-elimination event if there are n video games, the place n is not assured to be an influence of two? You’ll be able to attempt making a pattern bracket and counting up the variety of video games, however how have you learnt that is the perfect bracket design? May you do higher? Now what if I gave you a really giant variety of groups, like 693? Are you going to check out varied brackets?

This has lengthy been one in every of my father Vladimir’s favourite logic puzzles. You’ll be able to lower this explicit Gordian knot by observing that, when you might have n video games, eliminating all the way down to 1 essentially requires eliminating n–1 members, and in a single-elimination event, taking part in 1 recreation essentially eliminates precisely 1 participant. So the variety of video games is precisely n–1. When you begin with 693 groups, you may at all times play precisely 692 video games to get a winner.

(There would possibly nonetheless be higher and worse designs of brackets: for instance, the design “A performs B, after which the winner of that recreation performs each single different staff sequentially” might be not the perfect design, as a result of then you definately’ll expect the perfect staff to play 692 video games whereas each different staff solely performs 1… and if the worst staff occurs to play on the final day whereas the perfect staff is having a nasty day, you would possibly get a perverse outcome. Higher to approximate the power-of-2-type brackets, the place each staff performs as much as roughly the log-base-2 of the variety of groups (rounding up), and no one wins except they’ve performed roughly that variety of video games (rounding down). However nonetheless, so far as the entire variety of video games is worried… the perfect bracket and the worst bracket may have precisely the identical variety of video games.)

Anyway, to get rid of 16 groups all the way down to 1, simply apply the n–1 rule, and also you get 15 video games.

3. The third-place contest

However wait a minute, we nonetheless have yet another recreation to play. FIFA occurs to have a “third-place playoff” recreation: whereas the winners of the 2 semifinal video games advance to the ultimate (and are outlined as first-place and second-place), the losers of these two semifinal video games play one further recreation (and are outlined as third-place and fourth-place).

Thus, within the 2018 World Cup (which was performed in seven completely different Russian cities), the semifinals concerned France vs. Belgium (gained by France) and Croatia vs. England (gained by Croatia). Within the last, France performed Croatia (France gained), however earlier than that recreation, Belgium performed England (Belgium gained).

Now, strictly talking, this is not actually an correct rating, as a result of how do we all know that Croatia is #2 whereas Belgium is #3? What if France and Belgium had been the highest 2 groups, whereas Belgium and England would have been ranked #9 and #10 out of the 16 groups within the knockout stage (however the brackets had been organized in such a means that the nice groups had been all on prime however the unhealthy groups had been all on the underside)? There is a sloppiness in defining the final-loser as #2 and the winner-of-semifinal-losers as #3. However hey, everybody likes rankings, even when they’re inaccurate.

4. Placing it collectively?

Placing all of it collectively, we get 48 video games within the group stage, plus 15 video games within the knockout stage, plus an additional recreation to find out third place, which makes 64.

Why not simply play round-robins, which irons out the consequences of getting unhealthy days and provides you a extra scientific estimate of who’s the higher staff? (Neither system is ideal: single-elimination places quite a lot of emphasis on not having unhealthy days, whereas round-robin includes arbitrary win-to-draw level ratios, e.g., 3:1 for FIFA and 1:0.5 for chess.)

Seems that, as a result of the round-robin rule is n(n–1)/2, the variety of round-robin video games you’d need to play will increase because the sq. of the variety of groups. With 32 groups, you’d have 32×31/2 = 496 video games. That is much more than 64!

When you’ve got T = nk groups and also you divide these groups into okay teams of n, you get kn(n–1)/2 video games, however since okay = T/n, you’ll be able to specific that as T(n–1)/2 video games. With T = 32 and n = 4, that is one other means of attending to 32×3/2 = 48 video games for the group stage. So, holding the variety of groups fixed, we’re principally linear within the number-of-teams-per-group. We may reduce the variety of video games by making n = 2, i.e., 16 video games (32 video games complete while you add within the knockout stage and third-place recreation), i.e., simply making it single-elimination all the best way again. Or we may maximize the variety of video games by making n = 32 and truly taking part in these 496 video games.

FIFA has chosen an arbitrary variety of teams after which an arbitrary place to begin the knockout stage, which supplies us 64, a pleasant compromise between 32 video games and 496.