Automata Antics: Ball Throwing

I recently played this game with my students, once again found in Solve This by J.S. Tanton. They had a good time exploring a problem to which the answer was unknown and taking a break from the everyday math to play a game. Play went as follows:

A number of students stand in a circle, each with a card showing either “left” or “right”. John begins a ball game by tossing a ball across the circle. If Beatrice catches the ball she throws it back across the circle one place to the left (in her perspective) of John if she is holding left, or one place to the right if holding right. Then Beatrice flips her card over (“right” becomes “left” and vice versa) and waits for another turn. Each person receiving the ball operates in this way.

There is one complication: If Maxine is holding “right” and receives the ball from Christian directly to her right, the rules do not make sense. We make the convention that in this predicament Maxine holds the ball, turns in place to the right more than 180⁰, and throws the ball to the first person she sees (and still changes the word she holds). This is of course, equivalent to Maxine passing the ball to the person directly on her left.

Similarly, if Keone is holding “left” and receives the ball from Shannon directly to his left, he turns left and throws the ball to the person on his right. This is a little confusing at first, but it doesn’t take long to get the hang of it.

Here is the question: In this game is everyone guaranteed a turn? Will the ball eventually reach everyone, no matter the choice of word initially on everyone’s card?

(A lightweight ball is recommended.)

Note: The book with which this game was found often presents similar problems in the same section. There were two related problems presented before this one which involved a grid-like playing board and proposed the situation of an ant walking along that grid alternating vertical and horizontal steps or a grid with random “L” and “R” which dictate which direction the ant is to turn 90⁰. In both the these the question is similar: will the ant walk through every box in the grid, and will the ant ever return to a box from the vertical direction if it originally left that box from the horizontal? Having done these activities first, we might have had an idea of what would happen in the circle, however I chose the circle because I wanted a whole class activity. Have fun exploring any or all of these and report your findings!

My class has yet to discover a "solution," and I'm not sure what we did find out, but many predictions were made, variables changed, and a modeling tool like Excel has yet to be utilized to try multiple cases quickly and efficiently.