math challenge

Weeks of January 5 - 18, 1998

Wonky Ball, anyone?

A group of friends are playing Wonky Ball and have lined themselves up as shown in the diagram. In this unusual game, players stand in a circle and toss special balls to each other. A ball can only be thrown to the next person, or to the next person over, but not in a straight line across the circle. And, a ball can only be thrown clockwise.

Now here's the catch...the special balls change colour depending on who touches them. The friends discover that:

Billy turns the balls blue.
Yolanda turns the balls yellow.
Gary turns the balls green.
Randy turns the balls red.
Sally, Theresa, Ursula, and Valerie have no effect on the color of a ball.
And if more than 4 players touch the same ball consecutively, it explodes. Ouch!

The object of Wonky Ball is to take four white balls and turn two to red and another two to yellow. BUT the balls which ultimately turn red must never be green.

If Billy starts with four white balls, what is the minimum number of throws the friends will need to turn the balls the desired colors, and how do they do it?

See the solutions