# Reconstructing World Cup results

In the group stage of the World Cup, teams compete within eight groups of four teams each. Each group plays a round-robin tournament, in which each team plays three matches, one against each other team in the same group. This means that a total of six matches are played within a group. Points are used to rank the teams within a group, and the top two teams advance further. Three points are awarded for a win, one for a draw, and none for a loss. The scoring table for a group also includes the total number of goals scored for and against a given team in its three matches. (There are also usually columns for wins, draws, losses, and goal difference.)

Play in Group H of the 2002 FIFA World Cup completed on 14 June 2002. Japan won the group, and advanced to the second round, along with Belgium. Russia and Tunisia failed to advance. The final standings were as follows:

Team Goals for Goals against Points Qualification
Japan (host) 5 2 7 Advance to knockout stage
Belgium 6 5 5 Advance to knockout stage
Russia 4 4 3 eliminated
Tunisia 1 5 1 eliminated

What were the scores of the six individual matches?

It's interesting that the answer is unique, given that it's from a real game.

• Japan - Belgium: Draw (2:2)
• Japan - Russia: Japan Wins (1:0)
• Japan - Tunisia: Japan Wins (2:0)
• Belgium - Russia: Belgium Wins (3:2)
• Belgium - Tunisia: Draw (1:1)
• Russia - Tunisia: Russia Wins (2:0)

Let's start by finding out the outcomes (win/draw/lose).

Since the sum of points is 16,

which is 18 - 2, we know that the draw happened exactly twice,

and based on the individual points modulo 3,

the two draws were Japan-Belgium and Belgium-Tunisia.

Then we can deduce that

Japan won against Russia and Tunisia, Belgium won against Russia, and Russia won against Tunisia.

Now Belgium has a win against Russia, which must be

a win by 1 goal difference.

And Japan has two wins with the goal difference of 3, so the individual matches must be

one win by 1 diff, and the other by 2 diff, in some order.

At this point, we can start building equations:

Belgium-Russia: Belgium wins ($$x+1:x$$)
Belgium-Tunisia: Draw ($$y:y$$)
Japan-Belgium: Draw ($$5-x-y:5-x-y$$) (since Belgium earned 6 and lost 5 goals)
Japan-Russia: Japan wins ($$a+1:a$$ or $$a+2:a$$)
Japan-Tunisia: Japan wins ($$b+2:b$$ or $$b+1:b$$) (respectively)
Russia-Tunisia: Russia wins ($$c+2:c$$ or $$c+3:c$$) (respectively, since Russia's overall goal diff is 0)

Then, from the individual earned and lost goals for each team,

Japan having lost 2 goals gives $$5-x-y+a+b = 2$$, which gives $$a+b+3 = x+y$$
Russia having scored 4 goals gives $$a+x+c+2 = 4$$ or $$a+x+c+3 = 4$$
Tunisia having scored 1 goal gives $$b+y+c=1$$

Now, the following observation gives the answer:

since $$x+y\ge 3$$, only the first option can be true, with $$x,y$$ being maximum (2 and 1 respectively) and $$a,b,c$$ being minimum (all zero).

• Nice job! I picked a group where the answer was unique, and I also test-solved it to make sure I thought the solving process was pleasing. Dec 8, 2020 at 2:52

I want to present a different solution based more on logical deduction and football knowledge.

First, we know that:

- 7 points in a 4 team round robin means 2 wins, 1 draw
- 5 points means 1 win and 2 draws
- 3 points means 1 win and 2 losses or 3 draws
- 1 point means 1 draw and 2 losses

As Russia is 4-4 in goals, it's possible to have drawn three times, but since Belgium drew twice, Japan and Tunisia only once, it's impossible for Russia to have drawn three times, as there would be no 2nd draw for Belgium possible, therefore Russia must be 1 win and 2 losses

From this we can deduce the results:

Belgium must have drawn Japan and Tunisia to meet their two draws and win against Russia, therefore Japan must have won their other games.

- Japan beats Russia
- Japan beats Tunisia
- Japan draws Belgium
- Belgium beats Russia
- Belgium draws Tunisia
- Russia beats Tunisia

Now we can start to look at the goals, starting with Belgium:

- Tunisia has 1 goal scored, Belgium - Tunisia must be 0-0 or 1-1
- Japan has two goals against, so Japan - Belgium must be 0-0, 1-1 or 2-2
- Since Belgium is +1 Goal difference, they must have won against Russia with one goal difference, with Russia having another loss, they must have scored at maximum 3 goals against them, so the result is 1-0, 2-1 or 3-2
-> This leaves 3-2 vs Russia, 2-2 vs Japan and 1-1 vs Tunisia as the only possible conclusion to meet Belgiums 6-5 scoreline.

Next we can look at the remaining three teams:

- Russia is now 2-1 goals with 1 win and 1 loss remaining, this means they must win 2-0 and lose 0-1
- Japan wins both games with 3-0 total goals, so one game must be 1-0 and the other 2-0
- Tunisia loses both remaining games with a 0-4 goalscore, so they lose both games 0-2 or 0-1 and 0-3. We can eliminate the second possibility right away.
-> This leads us to the only possible solution of Japan - Russia 1-0, Japan - Tunisia 2-0 and Russia - Tunisia 2-0

Final results:

- Japan - Belgium 2-2
- Japan - Russia 1-0
- Japan - Tunisia 2-0
- Belgium - Russia 3-2
- Belgium - Tunisia 1-1
- Russia - Tunisia 2-0

• Nice job, and welcome to puzzling.SE! Dec 8, 2020 at 12:48

Since each team played 3 games, we can work out their Win-Loss-Draw record without too much difficulty from their number of points

Japan: The only way 3 games can contribute 7 points is from 2 wins and a draw Belgium: 5 points must be one win and two draws Tunisia: 1 draw and 2 losses.

The only one that is a priori ambiguous is Russia; however

Since Belgium's two draws must have been against different teams, there are insufficient opposing teams with draws in their record for Russia to have drawn three games. So Russia won once and lost twice.

We can now work out the results (if not the scores) of each tie:

Belgium have only one game which did not result in a draw. As Russia did not draw any games, Belgium - Russia cannot have been a draw, so it must be a win for Belgium. Furthermore, Belgium drew against Japan and Tunisia, and so 1) Japan won all their non-Belgium games and 2) Tunisia lost all their non-Belgium games, i.e. Japan beat Russia, Japan beat Tunisia, and Russia beat Tunisia.

Draws impose the strictest condition on number of goals: the goals scored by opposing teams must be equal. So let us consider draws first.

In fact, since Belgium has the most draws, let's consider Belgium. Their two draws will have resulted in the same number of goals for as goals against. So the game they won (– against Russia), they won by 1 goal. Since Japan only conceded 2 goals total, Japan - Belgium can be 0-0, 1-1 or 2-2. Since Tunisia scored 1 goal total, Belgium - Tunisia can be 0-0 or 1-1. This accounts for at most 3 of Belgium's goals scored. So the score of Belgium-Russia must have been 3-2 or higher. This gives at least 3 goals against for Russia; however, Russia also lost to Japan, so must have conceded in that game; so Belgium's score could not have been any higher and so the result was Belgium 3-2 Russia; we must then have the highest possible scores in the two draws we considered, i.e. Japan 2-2 Belgium and Belgium 1-1 Tunisia; and we must have the score Japan 1-0 Russia.

Russia's remaining game is

versus Tunisia, and they have 2 more goals to score; Tunisia scored their lone goal in the draw against Belgium so this game ended Russia 2-0 Tunisia.

The final match is

Japan versus Tunisia; we have accounted for 3 of Japan's goals scored so this game was Japan 2-0 Tunisia

• Nice job, and welcome to puzzling.SE! I like how the reasoning avoids case analysis (I chose this group with that in mind). Dec 8, 2020 at 13:11