Henry Ernest Dudeney/Modern Puzzles/201 - Football Results/Solution

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Modern Puzzles by Henry Ernest Dudeney: $201$

Football Results
The following table appeared in a newspaper:

$\qquad \begin {array} {r | cccc |cc|c} & \text {Played} & \text {Won} & \text {Lost} & \text {Drawn} & \text {Goals For} & \text {Against} & \text {Points} \\ \hline \text{Scotland} & 3 & 3 & 0 & 0 & 7 & 1 & 6 \\ \hline \text{England} & 3 & 1 & 1 & 1 & 2 & 3 & 3 \\ \hline \text{Wales} & 3 & 1 & 1 & 1 & 3 & 3 & 3 \\ \hline \text{Ireland} & 3 & 0 & 3 & 0 & 1 & 6 & 0 \\ \hline \end{array}$

It is known that Scotland had beaten England $3 - 0$.
It is now possible to find the scores in the other five matches from the table.
How many goals were won, drawn, or lost by each side in every match?


Solution

The final scores:

Scotland $3$, England $0$
Scotland $2$, Wales $1$
Scotland $2$, Ireland $0$
England $2$, Ireland $0$
England $0$, Wales $0$
Wales $2$, Ireland $1$


Proof

We are given that England conceded $3$ goals to Scotland.

Thus must have been the only $3$ goals they conceded.

Therefore the draw with Wales (which was the only drawn match) must have been goalless.

England's only other match was against Ireland, which they won by $2$ goals against $0$.

Wales must have lost against Scotland and won against Ireland.

The goal conceded by Scotland was pushed in by either Wales or Ireland.


Suppose Wales conceded $1$ against Scotland.

Then Ireland must have scored $2$ against Wales, which could not happen.

So Wales did not concede $1$ against Scotland


Suppose Wales conceded $3$ against Scotland.

Then Ireland's goal must have been against Scotland.

But then Scotland only put one past Ireland.

But that would mean the Scotland-Ireland match was a $1-1$ draw, which it could not have been.


The only other possibility is that Wales conceded $2$ against Scotland.

Then Wales also conceded $1$ against Ireland.

Hence Ireland scored no goals against Scotland.

Then Wales must have scored the only goal against Scotland.

This means they must have scored $2$ against Ireland.

Hence Scotland also scored $2$ against Ireland, and the table is complete.

$\blacksquare$


Sources

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