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 is also 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.

Ten frames are bowled, unless the last frame contains a strike or spare, in which case an additional bowl is awarded.

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

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).

Define the sets

and the matrices

then a generating function for the number of games corresponding to score 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, 5, are: 1, 20, 210, 1540, 8855, 42504, ... (OEIS A060853; Cooper and Kennedy 1990). 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. A best-fit Gaussian is given by

where , , and (dotted blue curve above).

The mean score is given by

(Cooper and Kennedy 1990). The mode for the score , namely . For , 289, ..., 300, the totals are 12, 11, 11, 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.