Bowling scores

Some years back Anna got a new family tradition started: bowling on Thanksgiving morning. We all gather at Bountiful Bowl just prior to their 10 AM so we're among the first in the door when they open.

A perfect (or near perfect) game would only have 12 balls thrown (one each in the first nine frames and three in the tenth). It would be possible to bowl an 11-ball game by scoring nine consecutive strikes and then missing a strike or a spare in the tenth frame. A strike-free game would generally have 20 balls thrown though a strike or spare in the final frame would boost the number up to 21.

This past Thanksgiving I bowled a (for me) respectable 136. I didn't get a strike in any of the first nine frames and then got a turkey (three consecutive strikes) in the tenth frame. A 21-ball game! Go me!

I recall discussing (in passing) whether there was any bowling score that was "impossible" to bowl. Clearly there is only one way to score 0 (all gutters) and 300 (all strikes). There would be one way to score 291 thru 299 (11 strikes and 1-9 pins on the final ball).

What about 289? Without scratch paper & a pencil we didn't bother to figure it out. I assumed there would probably be ways of scoring that high. (And it turns out there are eleven different ways, but more on that in a moment.) If there was some score that was impossible to get it seems like it would make a good plot device in a detective story—some intrigue at the local bowling alley (league rivalry turns violent?) and the villan's alibi falls apart when our able gumshoe realizes the mathematical impossibility...

Yeah, well... guess it'd be a real turkey of a plot device after all. (There is probably a reason I've never written more than five pages during NaNoWriMo :-)

While catching up on Project Euler (I've slipped to 72% with the rate they've been adding new problems lately) I looked a sequence my programs was generating (1, 20, 210, 1540, 8855, ...) up in the On-Line Encyclopedia of Integer Sequences and found out that the sequence I was calculated (at least in the first five terms) matched the "number of possible games of 10-pin bowling with a score of n."

The OEIS entry had links to Balmoral Software's All About Bowling Scores page, which is full of great mathematical insight into the game of bowling, as well as easy to understand charts and graphs. (Definitely worth checking out.)

For instance, did you know that the number of possible different games is roughly six quintillion? Or that an "average" 280-point game has more strikes than a 281-point game? Other fun facts:

• The most common score is (i.e., the mode) 77
• The mean score rounds up to 80
• Bowling a 98 (or higher) puts you in the 90th percentile (or greater)
• There are 17,590,903,116 different ways Becca could have played her 18-point game

Now if I could just swap my bowling & golf scores I'd be good at both!

—Michael A. Cleverly
Monday, January 07, 2008 at 23:39

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