## Brainteaser 1567: How many teams?

**From The Sunday Times, 20th September 1992** [link]

The teams in our local football league each play each of the others once in a season, earning three points for a win and one point for a draw. After the last Saturday of the season (when all the teams had played one game that day) I worked out the league table (with the teams in alphabetical order). I then consistently replaced digits in the table by letters, using different letters for different digits.

Here is part of that table, showing some of the entries in three of the rows:

What is the number of teams in the league? And what is the number TEAMS?

This puzzle was selected for the book *Brainteasers* (2002, edited by Victor Bryant). The wording here is taken from the book and was only slightly changed from the original puzzle.

[teaser1567]

## Jim Randell 8:11 am

on30 April 2019 Permalink |If there are

nteams in the league, then at the end of the season each team has played all the other(n – 1)teams, and this total number of matches must be represented in the sum of the “won”, “drawn”, “lost” columns in the table.If each team played one match on the final Saturday, then there must be an even number of teams. And if

nis even, then(n – 1)is odd.We are given the points in the third row:

And we are given the “goals against” for two teams. Each “lost” game involves conceding at least one goal, so:

(and the values cannot be equal).

Together these give us enough information to assign values to the letters.

We can solve these alphametic expressions using the [[

`SubstitutedExpression()`

]] solver from theenigma.pylibrary.The following run file executes in 216ms.

Run:[ @repl.it ]Solution:There are 12 teams in the league. TEAMS = 16459.The first row has “won”, “drawn”, “lost” values of 0, 3, 8, but we don’t know which order.

The second row has 5 “won”, 4 “drawn”, 2 “lost”, and 7 “goals against”.

The third row has 1 “won”, 6 “drawn”, 4 “lost” and 5 “goals against”, giving 9 points.

There are 12 teams, so there are 66 matches played in total.

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