-----Original Message----- From: math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On Behalf Of Michael Reid Sent: Tuesday, October 25, 2005 1:00 PM To: math-fun@mailman.xmission.com Subject: RE: [math-fun] basketball shooting problem

the solutions in the monthly appear to strive for conciseness, which sometimes yield less insight.

I agree. Here is my solution, which I found simple enough to solve it mentally. Instead of considering the ratio (Hits/total throws), consider the ratio (Hits/misses). Since the latter is a strictly monotonic function of the latter, you can pass from below the target to above the target of one of these ratios exactly if you can do so with the other one. If the target (hits/misses) ratio is K, then the goal is to find integers M and H such that H is less than MK, but H+1 is greater than MK. It is clear that this is possible only if MK is not an interger, which in turn is only possible for some choice of M if K is not an integer. So it is impossible to cross the target without hitting it if the target for hits/misses is an integer K, or equivalently, if (hits/(hits + misses)) is of the form K/K+1 for some integer K. Moral: choosing the right parametrization makes things easier. Andy Latto andy.latto@pobox.com