# Did you solve it? World Cup arithmetic

The solutions to today’s puzzles

On my puzzle blog earlier today I set you three football table challenges:

1) England, Tunisia, Belgium and Panama make up Group G in the 2018 World Cup. Imagine that once they have all played each other the table looks like this.

If you know only that England beat Tunisia 3-0, can you deduce the score of every other match in the group?

(For non football fans: the F column is “goals for”, meaning how many goals the team scored in total, the A column is “goals against”, meaning how many goals the team conceded in total, and the P column is “points”, when 3 points are awarded for a win, 1 for a draw and 0 for a loss.)

EXTRA CLARIFICATION: Some readers pointed out that Belgium should be above Tunisia in the table, because goal difference is the tie-break when two teams are equal on points. This observation is correct, but does not affect the puzzle. The reason that the error crept in is that I updated the puzzle from one written in 1921 when there must have been a different criterion for the tie-break. Thanks to everyone who spotted the mistake.

Solution

We know by looking at the points that England won all three games, Tunisia and Belgium both won, drew and lost a game, and Panama lost all three games.

Let’s consider Tunisia, which lost to England 3-0. Tunisia must have beaten Panama (since everyone did) and therefore drawn with Belgium, scoring 2 goals and conceding 0 across these two games. The only way this combination is possible is if they won 2-0, and drew 0-0, since if they scored a goal in the draw with Belgium, they would have conceded another goal, which we know they didn’t since they only conceded 3 in total. So far we have:

• England 3 0 Tunisia
• Tunisia 2 0 Panama
• Tunisia 0 0 Belgium

And the games left are

• England vs Panama
• Panama vs Belgium
• England vs Belgium

England conceded only one goal. Panama scored only one goal. Let’s assume that the goal scored by Panama was conceded by England. In that case, Belgium did not score against England. So Belgium must have scored 3 goals against Panama. We know that Tunisia scored 2 against Panama. But that means that England can only have scored a single goal against Panama, since Panama only conceded 6. Yet our assumption was that Panama scored a goal against England, meaning this match would have been a draw, which we know is incorrect. We conclude that our assumption was wrong, and that the goal scored by Panama was against Belgium, and the goal conceded by England was also scored by Belgium. So, of Belgium’s three goals, one was against England. The other two must have been against Panama, since they won that game despite conceding a goal. Thus Belgium beat Panama 2-1. Panama conceded 6 in total, so England beat Panama 2-0, and beat Belgium 2-1.

2. This GCHQ puzzle:

I’m only going to give you a sketch of how to solve this one. First, you need to know that the goals for and the goals column must always be equal. Since Slovakia has a goal difference of zero, the goals they scored are equal to the goals they conceded. Which means that the goals the other three teams scored are equal to the goals the other three teams conceded. Thus Wales and Russia conceded one goal each. Next, looking at the order of the teams, you can work out how many of the games each team must have won, drawn and lost, and once you have that you can work out the exact scores. For example England are top but they cannot have won all three games, since you cannot win all three games and have a negative goal difference. Here’s the full table.

3. I asked if it was possible to design a football table puzzle in which only a single number is given and from which we can deduce every single result between the teams. (Assume that all matches have been played).

I think the only possible answer is if the team that is in first place has a 0 in the “goals for” slot. Since from this we can deduce that every match must have ended 0-0. This means all teams are on 3 points. If this were a World Cup group, there would be a tie-break to determine who comes top, which under current rules are be Fair Play points. It is the null solution...but still interesting that we can deduce all the results from a single number. Does anyone know if there has ever been a four-team group in any major competition with only 0-0 results?

I hope you enjoyed today’s puzzles. I’ll be back in two weeks.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of popular maths, including the puzzle books Can You Solve My Problems? and Puzzle Ninja. I also co-write the children’s book series Football School.

Alex Bellos

The Guardian

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