Free Throw Percentage Puzzle
June 10, 2006 am30 10:56 am
A basketball player has .750 free throw percentage. (that's what they call it, even though we have a 3 decimal digit number – oh well.)
And for those of you who don't know, basketball players take one free throw at a time, and either make them, or not. So someone could go 1 for 1, then 1 for 2 (miss), 1 for 3 (another miss), 2 for 4 (make), 3 for 5 (make), 4 for 6 (make), 5 for 7 (make), etc.
So here's the puzzle: A basketball player has a .750 free throw shooting percentage. Some time later he has raised his percentage to .900 (from 3/4 to 9/10 for those of you who prefer fractions). Did he have to be at .800 (4/5) somewhere in between?
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JD2718 
This is a beautiful question, modified somewhat from its original form on the Putnam exam.
It was a Putnam? I caught it as a hand-me-down. (thus the alterations). Do you know the name of the author?
It was 2004 A1. The original statement was,
“Basketball star Shanille O’Keal’s team statistician keeps track of the number, S(N), of successful free throws she has made in her first N attempts of the season. Early in the season, S(N) was less than 80% of N, but by the end of the season, S(N) was more than 80% of N. Was there necessarily a moment in between when S(N) was exactly 80% of N?”
I don’t know who the 3 authors of the 2004 contest were, although I feel like that information should be available somewhere. None of the obvious websites seem to have the information.