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Author Topic: Do you really know how to keep score?  (Read 3452 times)

Coolerman

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Do you really know how to keep score?
« on: September 01, 2009, 03:02:20 AM »
I thought  I knew how to keep score and how many frames were in bowling but,now I'm really messed up.
For all you students of math enjoy.I know I that I'm amazed.
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BowlingPins

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 T_4.

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 11^(10) approx 2.6×10^(10) (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
66^9×241=5726805883325784576 approx 5.7×10^(18)    
(1)

(Cooper and Kennedy 1990).

Define the sets
A={(x,y):0<=x+y<=9} B={(x,y,0)x,y) in A} union {(x,10-x,z):0<=x<=9,0<=z<=10} union {(x,y,z):0<=y<=9,0<=y+z<=10} union {(10,10,z):0<=z<=10}    
(2)

and the matrices
T   =   [sum_((x,y) in A)t^(x+y) 10t^(10) t^(10) 0; sum_((x,y) in A)t^(2x+y) t^(x+10) t^(20) 0; sum_((x,y) in A)t^(2x+2y) 10t^(20) 0 t^(20); sum_((x,y) in A)t^(3x+2y) t^(x+20) 0 t^(30)]   
(3)
C   =   [sum_((x,y,z) in B)t^(x+y+z); sum_((x,y,z) in B)t^(2x+2y+z); sum_((x,y,z) in B)t^(2x+2y+z); sum_((x,y,z) in B)t^(3x+2y+z)]   
(4)
R   =   [1 0 0 0],   
(5)

then a generating function for the number of games s_n corresponding to score n is given by
P(t)=sum_(i=0)^(300)s_it^i,    
(6)

where P(t) is the entry in the 1×1 matrix
RT^9C    
(7)

(Cooper and Kennedy 1990).
BowlingDistribution

The number of possible games s_n with scores n=0, 1, 2, 3, 4, 5, are: 1, 20, 210, 1540, 8855, 42504, ... (Sloane's A060853; Cooper and Kennedy 1990). As can be seen from the figure above, the distribution of number of possible games as a function of n is not precisely symmetric about its maximum. A best-fit Gaussian is given by
s_n=ae^((n-mu)^2/sigma^2),    
(8)

where a=1.71×10^(17), mu=78.5, and sigma^2=350 (dotted blue curve above).

The mean score is given by
s^_   =   (125572265)/(1574694)   
(9)
   approx    79.7439   
(10)

(Cooper and Kennedy 1990). The mode for the score n=77, namely s_(77)=172542309343732000. For n=288, 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.
s_n   n
1   0, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300
11   289, 290
12   287, 288
13   285, 286
14   283, 284
15   281, 282

SEE ALSO: Tetractys, Triangular Number

REFERENCES:

Balmoral Software. "All about Bowling Scores." http://www.balmoralsoftware.com/bowling/bowling.htm.

Cooper, C. and Kennedy, R. E. "A Generating Function for the Distribution of the Scores of All Possible Bowling Games." Math. Mag. 63, 239-243, 1990.

Cooper, C. N. and Kennedy, R. E. "A Generating Function for the Distribution of the Scores of All Possible Bowling Games." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994a.

Cooper, C. N. and Kennedy, R. E. "Is the Mean Bowling Score Awful?" In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994b.

Sloane, N. J. A. Sequence A060853 in "The On-Line Encyclopedia of Integer Sequences."



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Weisstein, Eric W. "Bowling." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Bowling.html

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BrunsMike

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Re: Do you really know how to keep score?
« Reply #16 on: September 01, 2009, 10:52:13 PM »
quote:
People keep talking about picking off the 6/9 out of a rack to enable a 292 after throwing 11 strikes in a row. I am calling BS, an outright lie.

I have been bowling for 50 years now. I have yet to see someone pick out the 6/9 ever. While it is more possible with today's balls, the way they make drastic and dramatic turns, I have yet to see one or even hear of one, except here on br.com

You throw a ball at the 6/10 and you ALWAYS pick off the 9 pin, in addition. You throw at the 4/7 and you ALWAYS pick off the 8 pin, in addition.

When I see it on YouTube.com or another video or in person, I will take it back. Until then, I say BS!
--------------------
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While picking out the 6-9 only in a full rack isn't a normal thing to do, it can be done! I've done it 3 time's in the last 20 year's of bowling. All 3 time's were not on purpose or even an attempt. I got the ball too far outside and the ball reacted too late to the backend and tried to recover, when it tries to recover the angle created is enough to cleanly pick out just the 6 and 9 pin. It's the same concept behind leaving the 1-2-4-6-10 because both use the same entry angle to achive that kind of pin leave.
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Mike Zadler

MI 2 AZ

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Re: Do you really know how to keep score?
« Reply #17 on: September 02, 2009, 01:04:19 AM »
Jeff,

I saw a bowler on the next pair pick out the 6-9 on a full rack.  Of course, it was the Chameleon pattern, but still it happened.  


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Adrenaline

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Re: Do you really know how to keep score?
« Reply #18 on: September 02, 2009, 02:16:13 AM »
I left a 1,3,6, and when I shot at it, managed to pick off just the 3 and went right between the 1,6 without even wiggling them.

As for picking up the 5,6 without going in between, I just managed to do it today by nicking the left side of the 5 pin, and slid it straight over into the 6.

But shots right handed with my VG.
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srlunatic

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Re: Do you really know how to keep score?
« Reply #19 on: September 02, 2009, 06:33:39 AM »
Charlest,

Have you ever seen someone go between the baby split 3-10?? Then you have seen exactly the same shot as what is necessary for the 6-9 only to be taken out.  Happens infrequently but have seen it quite a few times especially on longer patterns and the PBA patterns.
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Nbgiant25

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Re: Do you really know how to keep score?
« Reply #20 on: September 02, 2009, 07:29:49 AM »
quote:
People keep talking about picking off the 6/9 out of a rack to enable a 292 after throwing 11 strikes in a row. I am calling BS, an outright lie.

I have been bowling for 50 years now. I have yet to see someone pick out the 6/9 ever. While it is more possible with today's balls, the way they make drastic and dramatic turns, I have yet to see one or even hear of one, except here on br.com

You throw a ball at the 6/10 and you ALWAYS pick off the 9 pin, in addition. You throw at the 4/7 and you ALWAYS pick off the 8 pin, in addition.

When I see it on YouTube.com or another video or in person, I will take it back. Until then, I say BS!
--------------------
"None are so blind as those who will not see."


During a tournament last year in College Station, a member of our team picked the 6-9 off the full rack TWICE.  Bad spacing of the pins is what I'm chalking it up to...
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trash heap

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Re: Do you really know how to keep score?
« Reply #21 on: September 02, 2009, 10:40:06 AM »
Some things just don't need that kind of detail and bowling is definitely one of them. All I state is WOW!!

Talkin' Trash!

backupball

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Re: Do you really know how to keep score?
« Reply #22 on: September 02, 2009, 10:54:11 AM »
quote:
I think it's a 282 thats not possible.


As Dan mentioned, 282 is possible by spare, 10 strikes, 2. It can also be done by:

10 strikes, 2-spare
10 strikes, 4-4
10 strikes, 6-0

SleepOnIce

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Re: Do you really know how to keep score?
« Reply #23 on: September 02, 2009, 02:40:07 PM »
I have a 283, front 10-bucket-took the 4 pin off of the bucket, around 1 board from getting a 282. Every score can be bowled, whether or not it has/will happen is a different story.

I've also seen (and have myself) taken the 6-9 off of the rack on the older version of the Shark. King Prebowl also has a sanctioned 292, if that means anything.
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Edited on 9/2/2009 2:41 PM