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calendar   Friday - November 07, 2008

Oooh, you’re a cheater!

This is what a score sheet looks like for league bowling:











bowler Game 1 Game 2 Game 3 Total
Bob 125 147 159 431
Carol 155 163 114 432
Ted 215 246 211 672
Alice 113 151 132 394
Total 608 707 611 1929
Handicap 15 15 15 45
Grand Total 623 722 631 1974



A series in bowling for the bowler is 3 games. Since the score can be from 0 to 300, each game has 301 possible scores. Since each score is unique to it’s placement - Game 1, Game 2, Game 3 - all scores are unique to the series, even if your game scores are 174, 174, 174. The series totals are the ones added up horizontally

Thus there are 301 x 301 x 301 = 27,270,901 possible game score combinations in a bowler’s series.

On a bowling league with teams of 4, the team’s game total is the total of each bowler’s score for that game. Plus any handicap,which is always a static amount, so we can ignore it. These scores are added vertically.

Thus a 4 person team’s total is the sum of one of the 301 x 301 x 301 x 301 = 8,208,541,201 score combinations. Do these numbers matter at all?

The scores are written in pencil. Then pencils do not have erasers. Let’s consider that someone tried to cheat by adding 2 points to one of the bowler’s scores after the score sheets were signed off and handed in. In the above example it was Alice in Game 1. She actually rolled a 111, but didn’t want to catch any grief for rolling yet another “shithouse” score this season. (It’s a bowling thing.) So she changed the trailing 1 to a 3. Then she changed her team’s total for that game from 606 to 608. Then she changed to total with handicap from 621 to 623 [all in green] Can she get away with it?

You can fudge a 1 to look like a 3. You can fudge a 6 to look like an 8. You can fudge a 7 to look like a 9. Trying to fudge any of the other digits by sneaking in an extra 2 points gives you something weird looking. That won’t work.

To be effective, a good “+2” cheat must change the score for that bowler’s game, that bowler’s series total, the team’s game total, that games Grand Total, and the overall Grand Total for the night. In this example, her cheat is the number in green. All the red numbers are the ones that also have to be changed to cover her tracks. Bowling score sheets are in a grid for exactly this reason. If the scores add up going across, but not down, or down but not across, it’s a math error. This happens all the time. Especially after several beers.

To narrow things down even further, a “+2” cheat would not only have to change these 6 numbers, those numbers would have to all end in either 1, 6, or 7 for the fudging not to stand out.
Maybe she doesn’t have to change all 6 numbers, but she will have to change her score for that game, her series total, and one of the team totals. The rest could be written off to poor math. Could they? Or would her changes stick out like a sore thumb?

In this example she only changed 3 of the numbers. The errors stand out. In most “cheating” circumstances only one of the numbers is changed. What are the odds that Alice can come across a perfect scoring opportunity that would let her skillfully changed all the right numbers and get away with it? Pretty darn slim. And she’d have to have that numeric event happen at the same time she had access to the score sheet with nobody looking. Even less likely.

Well, of those 301 possible scores per game, you can meet the “ends in 1, 6, or 7” criteria with any of these scores: {x1, x6, x7}, where x goes from 0 to 290 in increments of 10. Which means 29 ranges. With 29 ranges you thus have 87 matches out of 301 choices. That’s 28.97%. If I get to multiply that out 3 times, once per game, I should find the percent of series totals that match the criteria:  0.2897^3 = 0.024147 or 2.41% rounded off for now. About 1 chance in 40.

If I get to multiply it across, I can also multiply it down, so a “cheatable” total for a 4 person team can happen in 0.2897^4 = 0.6995% of all possible totals. Around 1 chance in 150.

For the Overall Grand Total, find the odds by adding up the exponents and raising the number to that power; 0.2897^7 = 0.000171253504046166, then taking the inverse, 1/0.000171253504046166, and rounding off, which gives the odds: less than 1 in 5840. Pretty long odds. And that’s with perfect pencil skills.

So it is possible to do a perfect “+2” cheat when nobody is looking, but the perfect opportunity will never actually arise; you will either leave obvious math errors or some strange looking squiggles that are obviously not numbers. Cool. Good to know. Always good to have this kind of ammunition in the back of my mind. Yes, I’m still pissed over those accusations. Can you tell?

See what happens when my wife goes on a business trip? My mind just wanders, and tries to eff with the ineffable.


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Posted by Drew458   United States  on 11/07/2008 at 12:15 AM   
Filed Under: • Bowling Blogging •  
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