NHL Brain Trinket

What I’m going to tell you—I warn you—is of no consequence whatever. And it won’t even be of interest to you unless you’re an NHL hockey fan. And, worse, it’s going to test your algebra skills. The only thing I can say in favour of saying it at all is that you just won’t ever read anything else about hockey like what I’m going to tell you right now. It just doesn’t happen. This is the unicorn of hockey writing. Right here, right now.

It’s so messed up to be telling you this at all that I have to give you something to get you to read it. What I’m going to give you is like something you’d get out of a bubble-gum machine. But maybe the best such thing you’d get. Cuz it’s an idea. It’s from the bubble gum machine of the mind. I figured it out myself. I haven’t seen it mentioned anywhere else. I’m going to give you a little brain trinket. It’s a formula. The formula tells you how many points a perfectly average NHL team should have after they’ve played N games. That’s it. That’s all this is about. The only use it has is to be able to tell if a team is above or below average. Think of this as a peculiarly Canadian gift. It’s a way to think a little bit more clearly about something that is barely worth thinking about at all. But, you know, in Canada, we think about hockey. It’s more pleasant than thinking about the mess we’re in.

This used to be dead simple before the introduction of the so-called “three-point games”. Prior to the “three-point games”, a perfectly average team that had played N games would have N points. Simple. Cuz you used to get two points for a win, one point for a tie, and no points for a loss. Overtime was not played in regular season games. We used to say a team was “over 500” if they had more points than the number of games they’d played.

We still do, actually. But if you look at the standings, you see there are far more teams “over 500” than there are “500” or “under 500”. As of Nov 12/2010, 19 teams out of 30 are “over 500”. It’s no coincidence that there are so many teams “over 500”. Cuz the “three-point game” introduces ‘inflation’ into the standings.

Now there are no ties in the NHL. Ties are decided in overtime with five minutes of extra play and, if no one scores in the extra five minutes, a shoot-out occurs. In any case, the team that wins overtime gets two points. The team that loses overtime gets one point. That’s why overtime games are called “three-point games”; cuz three points are awarded in games that go to overtime.

So it’s no longer the case, having played N games, that a perfectly average team should have N points. A perfectly average team, having played N games, will have more than N points. But how many more? Well, if the team loses y games in overtime, then they should have N + y points to be perfectly average.

That’s it. That’s yer brain trinket. If you look at the NHL standings, as they’re published in newspapers or on the web, you see that they tell you the number of games, the wins, the losses, the overtime losses, and the points. So all you do is add the number of games to the number of overtime losses. Then you compare that with the number of points the team has. They should have more than N + y points to be better than perfectly average.

Now I will prove the brain trinket is true. Yes it is a true mind trinket. It comes with a mathematical guarantee. You don’t get that for cheap usually. But, you know, it’s fire-sale here all year round.

Here’s what a perfectly average team is. It wins as many games in regulation time as it loses in regulation time, and wins as many games in overtime as it loses in overtime. Right? Right.

Suppose a team is perfectly average according to our definition and that it wins x games in regulation time and wins y games in overtime. Then, since it’s perfectly average, it loses x games in regulation time and loses y games in overtime.

How many points did it earn? From its x wins in regulation time, it earns 2x points. From its y wins in overtime, it earns 2y points. From its y losses in overtime, it earns y points. So, overall, it earns 2x + 2y + y = 2x + 3y points.

How many points does an average team earn in N games? Suppose it wins x games in regulation-time and wins y games in overtime. Then, since it’s average, it loses x games in regulation-time and loses y games in overtime. Notice that any game we look at is either won in regulation-time, lost in regulation-time, won in overtime or lost in overtime. So the below equation just says that if we add up the number of games won in regulation-time, lost in regulation-time, won in overtime, and lost in overtime, that’d be the total number of games. Right? O ya.

x + y + x + y = N

Now we do some algebra on the equation. First we gather similar terms

2x + 2y = N

Then we divide each side of the equation by 2.

x + y = N/2
x = N/2 – y

We have seen that a perfectly average team earns 2x + 3y points.

Points = 2x + 3y

Now we substitute:

Points = 2(N/2 – y) + 3y
Points = N – 2y + 3y
Points = N + y

So there you are. Your mathematically certified NHL brain trinket. Get one for every hockey-loving, statistics-consuming person you know for Christmas. They’ll think just a little bit more clearly.

One Response to “NHL Brain Trinket”

  • Here is a different derivation of the same result.

    As before, we’re after the number of points a perfectly average team has after N games.

    Let rt be the number of games decided in regular time.

    Let ot be the number of games decided in overtime.

    N = rt + ot
    ot = N – rt

    Let P be the total points awarded in N games. Not just to the perfectly average team but to their opponents.

    2 points are awarded in a regular time game. 3 points are awarded in an overtime game. So

    P = 2*rt + 3*ot
    P = 2*rt + 3*(N – rt)
    P = 2*rt + 3*N – 3*rt
    P = 3*N – rt
    P = 2*N + (N – rt)
    P = 2*N + ot

    Since our team is perfectly average, they get half of these points. If PA stands for the number of points the perfectly average team gets, then

    PA = N + ot/2

    ot/2 is the number of overtime losses, since our team is perfectly average.