Bowling

Bowling, known as "ten pins" throughout most of the world, is a game played by rolling a heavy ball down a long narrow track and attempting to knock down ten pins arranged in the form of a triangle with its vertex oriented towards the bowler. The arrangement of the 10 bowling pins is that of a tetractys and also represents the triangular number .

Up to two balls (or "bowls") are allowed per "frame," and a game consists of ten frames (with a special rule being used for the number of balls awarded in the last frame). If all pins are knocked down on the first ball, the result is called a "strike," no second ball is awarded for that frame (except in the case of a strike being obtained in the tenth and last frame, in which case two extra balls are awarded), and the number of points tallied is 10 plus the number of pins knocked down on the next two balls. If some or none of the pins are knocked down on the first bowl, then a second ball is awarded. If all the remaining pins are knocked down on the second ball, the result is called a "spare," and the number of points tallied is 10 plus the number of pins knocked down on the bowl of the next ball. If all the pins are not knocked down after bowling two balls in a frame, then the score for that frame is tallied as the total number of pins knocked down.

In the tenth frame, a spare earns one extra ball. Pins are reset after a spare or strike, including a strike on an extra ball.

The maximum number of points possible, corresponding to 12 strikes, is 300.

A game is distinguished here by the number of pins knocked down on each ball, without distinguishing which physical pins fell. The total number of possible bowling games is quite large; there are eleven possibilities for the first ball thrown in the first frame (gutter, 1, 2, ..., 9, strike), and the same possibilities occur for each of the other nine frames. So without considering the second ball in each frame, at a minimum, there are (Balmoral Software). In fact, the true number of games is much larger due to the effect of the second ball in each frame. The total number of possible games is

(Cooper and Kennedy 1990).

For the first nine frames there are 55 open outcomes, ten spare outcomes, and one strike outcome. The tenth frame has 55 open outcomes, 110 spare outcomes, 65 strike outcomes whose next ball is not a strike, and 11 outcomes beginning with two strikes, giving 241 outcomes in all.

Define the sets

where all coordinates are nonnegative integers. The set contains the open-frame outcomes, and contains all tenth-frame outcomes, with a zero third coordinate used for an open tenth frame. The four states have no outstanding bonus, one bonus ball, two bonus balls, or bonuses from two consecutive strikes. Their transition and terminal matrices are

A generating function for the number of games corresponding to score , for , is given by

where is the entry in the matrix

(Cooper and Kennedy 1990).

The number of possible games with scores , 1, 2, 3, 4, and 5 is 1, 20, 210, 1540, 8855, 42504, ... (Cooper and Kennedy 1990; OEIS A060853). As can be seen from the figure above, the distribution of number of possible games as a function of is not precisely symmetric about its maximum. An approximate least-squares fit of a Gaussian function to these game counts is

where , , and (dotted blue curve above). Here is a width parameter equal to twice the variance of the corresponding normalized Gaussian function.

Giving every pin-count game equal weight, the mean score is given by

(Cooper and Kennedy 1990). The mode is the score , with . For , 289, ..., 300, the totals are 12, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1.

Scores that have the same number of possible ways to be bowled are summarized in the following table.