# Basketball: Mathematicians Prove Umps Not Fair

And how to succeed at free throws - It's not occult magic, it's straight science: Lots of interesting things have arrived and one of these is the basketball season. Two engineers have figured out the best way to shoot a free throw, a frequently underappreciated skill that gets more important as the game clock winds down.

Chau Tran and Larry Silverberg say, "To get a swish rather than a brick, you need the best possible conditions for releasing the basketball from your hand." They used hundreds of thousands of three-dimensional computer simulations of basketball free-throw trajectories to arrive at their conclusions. After running the simulations, Tran and Silverberg arrived at a number of major recommendations to improve free-throw shooting.

The engineers assumed that the basketball player doing the shooting was 6 feet 6 inches tall, and that he released the ball 6 inches above his head, so the "release height" was set to 7 feet. First, they say that shooters should launch the shot with about three hertz of back spin. That translates to the ball making three complete backspinning revolutions before reaching the hoop. Back spin deadens the ball when it bounces off the rim or backboard, giving the ball a better chance of settling through the net.

Where to aim? Tran and Silverberg say you should aim for the back of the rim, leaving about 2 inches between the ball and the back of the rim. According to the simulations, aiming for the center of the basket decreases the probabilities of a successful shot by almost 3%.

The ball should be launched at 52 degrees to the horizontal, meaning that the shot should, at the highest point in its arc to the basket, be less than 2 inches below the top of the backboard. Free-throw shooters should also release the ball as high above the ground as possible.

Silverberg says, "Our recommendations might make even the worst free-throw shooters break 60% from the free-throw line. A little bit of physics and a lot of practice can make everyone a better shooter from the free-throw line."

A new study suggests that officials often are NOT objective in their efforts to be fair to both teams. An examination of 365 major conference games played during the 2004-05 college men's basketball season found a clear pattern of an increased probability of a foul on the team with fewer fouls, the visiting team and the team that was leading.

Kyle J. Anderson says, "Whether consciously or subconsciously, officials seem to show a pattern where they try to make the number of fouls called on each team come out approximately even. That is seen as being objective or fair."