Henry Ernest Dudeney/Modern Puzzles/201 - Football Results/Solution
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
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $201$. -- Football Results
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $452$. Football Results