Mathematics of hockey

Summary: Playing hockey is an application of geometry, as players in constant motion determine angles of approach, plot routes through opponents, and visualize the vector of the puck.

Ice hockey is a team sport played on an ice rink by skating players using sticks to move a rubber disk called a “puck” into the opposing team’s goal. Field hockey and street hockey are usually played on foot, either on grass fields or street surfaces, using a ball. There is evidence that hockey-style games have existed for millennia, and ice hockey has long been popular in parts of the world that are cold enough for long-lasting seasonal ice. The basic rules of modern ice hockey were developed in Canada in the late 1800s, and the National Hockey League of North America (NHL) dates back to the early 1900s. The growing prevalence of indoor ice rinks has allowed hockey to expand into warmer places, like Florida and California, with mixed success. Ice hockey is highly geometric, in terms of both player action and the surface on which it is played. Mathematics and statistics are also used to model various aspects of game play and to develop improved equipment.

Geometry

A hockey rink is in some ways more geometric than other sports surfaces. Overall, the ice is essentially rectangular. North American professional rinks have corners that are rounded on a circle with a radius of 28 feet. Rinks have mirror symmetry end-to-end and side-to-side, including five circles used for face-offs. The goalie primarily occupies the space in front of the goal known as the “crease,” which is a half-circle with a six-foot radius in international play. In North American professional rinks, the crease is truncated to eight feet wide by transecting lines drawn one foot on either side of the six-foot-wide goal. Aside from the crease, goalies in some professional leagues may play the puck only in the goaltender’s trapezoid. This symmetrical region has one 18-foot base formed by the goal line and another 28-foot base determined by the boards (the wall behind the goal).

Hockey also requires an awareness of geometry for competitive play. Players are in constant motion and thus always calculating the best angle at which to approach an opponent, based on the opponent’s speed and trajectory, as well as the best route through the moving players. Turning and stopping on ice require different applications of forces than sports played on foot, with arcing turns or various radiuses being more common than point pivots and sudden reversals. Being a hockey goalie is an ongoing exercise in mathematics and physics. Geometric ideas like circumferences, radiuses, and angles are very important, as is the ability to visualize vectors. Goalies shift within the crease in response to the continuously changing locations of other players in the plane of the rink to simultaneously minimize opponents’ possible angles of attack and maximize their ability to intercept the puck. Time series analyses of several decades of data have shown that NHL games steadily average about 30 shots on goal per 60-minute game. There have been vocal critics of the artificial intelligence used for hockey goalies in some video games, with assertions that the programming fails to accurately mimic the sort of continuous precision adjustments used by real goalies. Hockey terminology has been used with some students to motivate and teach geometric concepts.

Statistics

Sports fans have become increasingly interested in studying sports statistics for prediction and deeper analyses. Operations researchers Jack Brimberg and William Hurley investigated the common belief that the first goal in the game “sets the tone” for the rest of the game. They calculated that the team that scored first was more likely to win, especially if the first goal was scored later in the game. Others have analyzed the way in which the NHL determines which teams will compete in the play-offs. There are 82 games in the regular NHL season. Points are awarded to the teams as follows: two points for winning the game, zero points for losing in a regulation 60-minute game, but one point for losing if the game went to overtime. No other league rewards a team differentially for losing in overtime. The intent is purportedly to keep tied teams playing competitively in the third period. However, data suggest that teams tend to rein in play and allow the game to go into overtime, which mathematical game theory suggests is the better move, because the reward for winning is the same, but the penalty for losing is reduced. A European system changes optimal strategy because the winner gets only two points in overtime versus three.

Other Connections to Mathematics

In climate science, Michael Mann, Raymond Bradley, and Malcolm Hughes quantitatively reconstructed temperature trends for the last 1000 years, producing a controversial graph called the “hockey stick graph,” since its changes in slope resemble the bend of a hockey stick. One theorem regarding diagonals in Pascal’s Triangle, named for Blaise Pascal, is also sometimes known as the “hockey stick theorem” for the shape it produces.

Bibliography

Brimberg, Jack and W. J. Hurley. “A Note on the Importance of the First Goal in a National Hockey League Game.” International Journal of Operational Research 6, no. 2 (2009).

Gill, Paramjit. “Late-Game Reversals in Professional Basketball, Football, and Hockey.” The American Statistician 54, no. 2 (2000).

Hache, Alain. The Physics of Hockey. Baltimore, MD: Johns Hopkins University Press, 2002.