Bowling Scores

Bowling is played in ten frames, each of which allows one or two deliveries. If all ten pins are bowled over in the first delivery, there is no second delivery.

Each frame has a score based on the delivery in that frame, as well as the next one or two deliveries. This means that the score for a frame may not necessarily be posted at the end of the frame. It also means that the tenth frame may require a total of three deliveries to resolve the scoring.

• Rule A. The score for a frame is the total pins bowled over during that frame, if the number is less than ten (an open frame, or error or split depending some other rules beyond the scope of this problem).
• Rule B. If all ten pins are bowled over on the first delivery (a strike), the score for that frame is 10 + the next two deliveries.
• Rule C. If all ten pins are bowled over between the first two deliveries (a spare), the score for that frame is 10 + the next delivery.

A game can be as few as twelve deliveries: ten frames of strikes require two additional deliveries in the tenth frame to resolve the rule B scoring. A game can be as many as twenty-one deliveries: nine open frames of less than 10 pins bowled over during the frame, and a spare in the tenth frame requiring one extra delivery to resolve the rule C scoring.

There is a relatively straight-forward annotation for play. Each frame has two characters to describe the pins bowled during the delivery. The final frame has three characters for a total of 21 characters.

Rule A: If the frame is open, the two characters are the two deliveries; the total will be less than 10. If a delivery fails to bowl over any pins, a "-" is used instead of a number.

Rule B: If the frame is strike, the two characters are "X ". No second delivery was made.

Rule C: If the frame is a spare, the first character is the number of pins on the first delivery. The second character is a "/".

For example:

"8/9-X X 6/4/X 8-X XXX"

This can be analyzed into ten frames as follows:

Frame	First delivery	Second delivery	Scoring rule	Frame Score	Total
1	8	/, must have been 2	C- spare = 10 + next delivery	19	19
2	9	-, 0	A- open = 9	9	28
3	10	(not taken)	B- strike = 10 + next 2 deliveries	26	54
4	10	(not taken)	B- strike = 10 + next 2 deliveries	20	74
5	6	/, must have been 4	C- spare = 10 + next delivery	14	88
6	4	/, must have been 6	C- spare = 10 + next delivery	20	108
7	10	(not taken)	B- strike = 10 + next 2 deliveries	18	126
8	8	-, 0	A- open = 8	8	134
9	10	(not taken)	B- strike = 10 + next 2 deliveries	30	164
10	10	10 and 10	B- strike = 10 + next 2 deliveries, two extra deliveries are taken during this 10th frame.	30	194

Each of the first nine frames has a two-character code for each delivery. There are three forms:

• "X ".
• "n/" where n is - or 1-9.
• "nm" where n and m are - or 1-9. The two values cannot total to 10.

The tenth frame has a three-character code for each of the deliveries. There are three forms:

• "XXX".
• "n/r" where n is -, 1-9 and r is X, -, 1-9.
• "nm " where n and m are - or 1-9. The two values cannot total to 10.

Write a valid(game)() function that will validate a 21-character string as describing a legal game.

Write a scoring method, scores(game)(), that will accept the 21-character scoring string and produce a sequence of frame-by-frame totals.

Write a reporting method, scoreCard(game)(), that will use the validation and scoring functions to produce a scorecard. The scorecard shows three lines of output with 5 character positions for each frame.

• The top line has the ten frame numbers: 2 digits and 3 spaces for each frame.
• The second line has the character codes for the delivery: 2 or 3 characters and 3 or 2 spaces for each of the ten frames.
• The third line has the cumulative score for each frame: 3 digit number and 2 spaces.

The game shown above would have the following output.

 1    2    3    4    5    6    7    8    9   10
8/   9-   X    X    6/   4/   X    8-   X    XXX
 19   28   54   74   88  108  126  134  164  194