Mathematical Rugby

Abstract

The December 1978 issue of the Mathematical Gazette [1] contains an elegant and humorous contribution from Anthony Hughes. His Note gives a recipe for determining the optimal spot from which to make a conversion attempt in rugby. Others then elaborated on Hughes' idea; see [2] and [3] in particular. In 1996 Isaksen [4] rediscovered Hughes' results while investigating the kicking of extra points in American gridiron. There are also a number of popularisations and summaries, in which the above results are presented, and in instances rediscovered. The ones of which we are aware are listed in the references; see [5, 6, 7, 8, 9 and 10].

Optimal Ball Placement in Rugby Conversions

“…On the other hand, the solid angle model does point to an optimal value of y of around 23m when close to the sidelines, in agreement with the player practice described in [9].…”

“…At the other end, close to the sidelines, the above expression yields an optimal value of nearly 32m for x, h, and p taking the values of 32m, 3m, and 5.6m, respectively. Since the distances for professional players reported in the last section of [9] are around 25m, we see that there is a deviation in excess of 25%.…”

“…In the former case, we know that when the try took place in front of the posts, players tend to place the ball at least 10m away from the try line [9]. Our solution yields a distance of around 5m, which is clearly too short.…”

“…To this end, assume that the try took place at a point with x-coordinate x and that we want to optimize the angle subtended by two points, one at each post and at height h, with coordinates (−p/2, 0, h) and (p/2, 0, h). In this way, we include not only Hughes's model, but also that of de Villiers [1]; to include also the modification of the de Villiers model suggested by Polster and Ross [9], it is sufficient to consider two different z-coordinates. Denoting this angle by α, its cosine is then given by…”

“…Thus, close to the sideline and for an initial speed of 20ms −1 , for instance, the maximum distance along the conversion line that one could hope to go would be just under ten meters, which is much smaller than the optimum predicted by the solid angle model. According to [9], the range of realistic initial speeds varies from 25ms −1 to 30ms −1 , while the elevation angle is close to 7π/36, and players do not tend to change either of these quantities with the location of the kick. The study carried out in [3] with elite British players indicates averages for the speed and elevation angle of 26.44ms −1 and π/6, respectively.…”