The World Cup, Plus Math

Yes, I have been following the World Cup tournament. Though I cannot say I know all that much about soccer, I appreciate the "beautiful game" as much as anyone in my knowledge range. I particularly enjoy beautiful things like, oh, the delightful drone of the airhorns, the charming attempts by France's media, athletes, and managers to blame everybody else for the French team's insipid play, and the ever-admirable arguments over the eyesight, credibility, and integrity of various officials. And the goals are pretty cool too. (Though I have to say that as a one-time [very low-level] water polo goalie I appreciate some of those saves even more. Man oh man alive, the things those guys can do...]

But what attracts us Math Guys to the World Cup, of course, isn't flags, or controversy, or those funky yellow shoes the Honduras players are wearing. No, the real draw, OB-viously, is the mathematics of the tourney. There really are a lot of interesting mathematical puzzles and problems surrounding the World Cup, and on the chance that your child--or you yourself--might be interested in a few World Cup Math Queries, here's your chance.

Some background: In the first round, teams are divided into groups of 4. Each team plays each other team in its group once, for a total of 3 games. You get 3 points for a win, 0 points for a loss, and 1 point for a tie.

A warm-up or two first.

Question 1. What is the greatest number of points a team can get during the first round of the tournament?

Question 2. What is the fewest number of points a team can get during this round?

Too easy? All right. Here's a slightly harder one.

Question 3. A team can get various single-digit point totals during the first round. For instance, 2 ties will result in 2 points. However, there is one single-digit point total that a team canNOT get during this round. What is it?

Question 4. If I told you my team finished round 1 with just 1 point, you'd be able to tell me its record. (You'd also be able to tell me my team was going home.) What would its record be? That is, how many wins, how many losses, how many ties?

Question 5. There's one point total, though, that's ambiguous. That is, there are two possible won-loss-tie records that will result in that number of points. What is that total? And what are the two different records that will get you there?

I alluded a couple of questions ago to the fact that some teams GO HOME. Yes. Only the top two teams from each group survive to play another day. I think we call this Darwinism. If teams are tied for second/third place there's a series of tiebreakers to resolve who moves on. But just two teams advance. (Oh! Now I understand the message of the Australian patriotic song "Advance, Australia Fair"! It's a song urging the Socceroos to move forward in the tournament! See how neatly all this fits together?)

So. Question 6. What is the greatest number of points a team can get in the first round and FAIL TO ADVANCE? (Yes, it involves losing a tiebreaker.)

And Question 7. How about the least number of points a team can get in the first round and ADVANCE ANYWAY? (If you're reasoning by analogy from number 6, you might think again.)

Perhaps the most interesting challenge, though, is trying to reconstruct the individual results from the group standings. Surprisingly often, it can be done. Here's a current example. This is the UP TO THE MOMENT scoresheet for the four nations of Group G after each has played two games:

Country Points
Brazil 6
Portugal 4
Ivory Coast 1
N. Korea 0

Question 8: What were the four games already played, and how did they turn out?

(Spoiler alert.) Okay. If you've been paying attention you know that Brazil has 2 wins (3 + 3 = 6). Ivory Coast has a tie. North Korea has no points, so it's lost twice, and Portugal, with 4 points, must have a win (3 pts) and a tie (add 1 more). Now, follow along:

1. Brazil has only wins, but Portugal has no losses. Therefore Brazil hasn't played Portugal yet.

2. Since Brazil hasn't playd Portugal, then its two games were against North Korea and Ivory Coast.

3. Brazil won both of these games, since Brazil only has wins.

4. Portugal also has two games to be accounted for, and since the Portuguese haven't played Brazil they also must have played NK and IC.

5. Portugal has a win, which matches up nicely with North Korea's loss (the one that wasn't to Brazil);

6. and since Portugal and Ivory Coast are the only teams with ties, that game must have been a draw.

Therefore, the four games thus far:
Brazil over Ivory Coast
Brazil over North Korea
Portugal ties Ivory Coast
Portugal over North Korea
--Gimme a Q! Gimme an E! Gimme a D!

One more before I let you go. It's a bit tricky.

Question 9. WITHOUT LOOKING UP THE RESULTS, give the result AND THE FINAL SCORE of each of the four games played thus far by the teams of Group E.

Country Points Goals Scored Goals Allowed
Netherlands 6 3 0
Japan 3 1 1
Denmark 3 2 3
Cameroon 0 1 3

My friend Cheerful Charlie says the problem can't be solved, and he's (sort of) right; if you only look at the points column, you CAN'T tell who played (and beat) who. But if you look closely at the goals columns as well, you'll realize that there is a unique solution...one which, fortunately enough, happens to match reality. Good luck!

**Want answers? Want to know how you were supposed to know? Disagree with me and want me to (ahem) prove you're wrong? Fine. Send me an SASE containing your questions and comments and $1000 and... No, just email me at scurrie at-sign poughkeepsieday dot org and I'll be happy to respond. As soon as I'm done blaming France's troubles on the Slovakian equipment manager and a God-awful offsides call by a crooked linesman from Qatar, that is.**