A reader asks:
What’s the biggest lead a football team can have in the table after $n$ games?
In a typical football league, teams get three points for a win, one for a draw, and none for getting beat. After, for example, one game, if one team wins and all of the other games are draws, the winners will have three points, while everyone except the team they beat will have one point — the winners will be two points ahead.
There’s not a whole lot more to it — after two games, the biggest possible lead is four points (one team wins both of its games to get six points, and all of the others are draws, leaving everyone else with at most two points). As long as the winning team hasn’t played all of the teams, the biggest lead after $n$ games is $2n$ points.
But what if they’ve played everyone?
In a four-team group, it’s possible to have a seven-point lead after three games, rather than just six: if you beat all three of the other teams, you’ll have nine points; if they all draw with each other, they each have two points. Assuming you always win and everyone else always draws, once you’ve played everyone once, you’ll have $3n$ points, and the best of the rest will have $n-1$ points - they’ll have drawn every game except for the one they lost to you - giving you a margin of $3n - (n-1) = 2n+1$.
In general, if you’ve played everyone at least $m$ times, your biggest possible margin is $2n + m$. So, when Dunfermline beat the other nine teams in Scottish League One four times, and they all draw with each other, they’ll have a lead of $2 \times 36 + 4 = 76$ points.
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