## Teaser 2942: What do points make?

**From The Sunday Times, 10th February 2019** [link]

In the Premier League table a team’s points are usually roughly equal to their goals scored (Burnley were an interesting exception in 2017-18). That was exactly the case in our football league after the four teams had played each other once, with 3 points for a win and 1 for a draw.

A ended up with the most points, followed by B, C and D in that order. Fifteen goals had been scored in total, and all the games had different scores. The best game finished 5-0, and the game BvD had fewer than three goals.

What were the results of B’s three games (in the order BvA, BvC, BvD)?

[teaser2942]

## Jim Randell 11:38 am

on8 February 2019 Permalink |I think this puzzle could have been worded more clearly.

I took the first part to mean we are looking for situations where each team has

exactlythe same number of points as the number of goals scored by that team. And that the “best game finished 5-0” means that that game had the most goals scored altogether (i.e. the other games had no more than 4 goals scored in total). I took the fact that all matches had different scores to mean that you couldn’t have (for example) one game with a score of 2-0 and another with a score of 0-2.This program uses the [[

`Football()`

]] helper class from theenigma.pylibrary. It looks for possible games for a team that can give the same number of points as the number of goals scored by that team, and then chooses outcomes for the four teams that satisfy the remaining conditions of the puzzle. It runs in 340ms.Run:[ @repl.it ]Solution:The scores in B’s games are: B vs A = 1-2; B vs C = 2-2; B vs D = 1-0.It turns out that the fact that there is a 5-0 game means that there are only 10 goals to distribute between the remaining 5 matches, and there are no further solutions if games with more than 4 goals scored are considered, so it is enough to know that there is a 5-0 score.

If we allow “mirror” scores (e.g. one game with a score of 2-0 and another with a score of 0-2), then there are no further solutions either. (Although if we allow repeated scores then there are additional solutions).

If we relax the conditions that the number of points is

exactlythe same as the number of goals scored then there are multiple solutions, even if they must be within 1 of each other.LikeLike