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

by : $201$

 * Football Results

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.