Soccer Mania

Number Theory Level 2

The Clay's Bar Premier Soccer League has 20 teams, each of whom plays every other team twice, so every team plays 38 games. If the match ends in a win, the winning team gets 3 points and the losing team 0 in the standings. A tie gives both teams 1 point each. The league winner is the team with the highest total points.

In 2016, Lester City won the league with 77 points, edging out Mean City and Totty Hipsters who had 75 points each (defending champion Cheesy finished in 12 $^\text{th}$ place with a mere 51 points).

In the course of winning the league, did Lester City go undefeated?

Note : Ties do not count as defeats.

Yes No There is insufficient information

3 solutions

Denton Young
Jan 31, 2016

Assume that Lester City had no losses (went undefeated.) Then every game for them ended in either a win (3 points) or a draw (1 point.) 3 and 1 are both odd numbers. So they had a total number of points that consisted of 38 odd numbers added together, which means their total number of accumulated points must be even.

But they finished with 77 points. 77 is odd. Therefore, they must have had at least one loss. (In fact, they must have had an odd number of losses.)

Moderator note:

Nice observation of parity that simplifies the problem.

Are we sure that such a system of wins/draws/losses can be set up? It seems reasonable, and that we have enough degrees of freedom to do so.

In response to Challenge Master, you can see my solution.

Rishik Jain - 5 years, 3 months ago

Rishik Jain
Feb 22, 2016

Assuming that the team did not lose any match, let $x$ be number of wins and $y$ be the number of draws.

$x+y=38 \\ 3x+y=77$ which on solving gives us $x=19.5$ which is not possible since $x,y \in \mathbb Z$ . Hence there can be no such scenario.

On the other hand if there is 1 loss, we have

$x+y=37 \\ 3x+y=77$ which on solving give us $x=20,y=17$ . Hence there is one scenario possible where the team loses 1 match and still manages to score 77 points.

$\therefore$ The team must lose at least 1 match to score exactly 77 points.

Henny Lim
Feb 26, 2016

The solution can be found without comprehending facts about all other teams. Undefeated season may end with at least 38 points and at most 114 points, and unsurprisingly any possible combination of wins and losses number will always yield an even number.

Conclusion: Regardless a loss will award negative points or number of team is changed into odd number instead of even, undefeated season of soccer will always end up with even-numbered points.

Soccer Mania

3 solutions

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