Where are the traffic jams when you need them?
“Here we go again” thought a morose Justin as he stepped out of his mother’s car onto the curb. Winter break was over, his trip back to New York a mere memory and his dread of stepping back into the Willowside social scene was almost too much to bear. He was still in Mr. Bott’s horrible math class (“We think it is best for you right now,” the school counselor had explained distractedly) and he hadn’t quite figured out how to fit in with the flannel shirt and baseball cap crowd that dominated this school. Talking about this to his parents was a lost cause: his father was rarely home anymore and his mother was either working or on the phone as she plopped down the next microwaved dinner for him in the evenings.
He looked around at the cliques of kids milling around the entrance of the school, excitedly jabbering about their winter breaks, their new jackets or other nonsensical mush. Most of them probably just stayed around the area during the break; hanging out at home, playing video games or calling each other on the phone. Few, if any, would have left the area, much less cross the country, as he had. Once again he felt out of place.
Justin saw Tyrone, Juli and Wendy over by Mrs. E’s classroom. He walked over towards them just as the bell rang. They were all in Mrs. E’s English class together. They waited for him to catch up, quickly said hi and then tucked in the door. Mrs. E, in her customary way, was not there yet. She always came in late, with stacks of papers under her arms and a huge, airy smile on her face. She was, in many ways, the opposite to Mr. Bott’s militancy. But that did not necessarily make her class any easier on Justin. He never knew what to expect from such a loosely organized teacher as Mrs. E. But even if she did confuse him, she never offended, which was a major relief.
“How was New York?” asked Juli as they sat down.
“Great, snowy, but great...” Just then Mrs. E breezed into the room, laid down her stacks of papers, smiled broadly at the class, sighed, and said “Isn’t it great to be back, you guys.” General, good nature groaning filled the room.
“I’m glad you’re back, Justin,” Juli whispered secretly. Justin smiled shyly.
“Welcome back to a bright new year,” continued Mrs. E. “I am so excited. Every year is a gift, you know. Remember that. Now, we have a new student, in case you hadn’t notice. Her name is Cassidy and she comes to us from New York City, can you believe that?”
Justin’s felt himself jerk to attention. Did she just say New York City? He looked over to see that, indeed, a new girl was sitting on the other side of the classroom. She had long, shiny black hair and a fancy pair of lavender tinged glasses. She looked calm, poised and above all, sophisticated.
“I think it is so wonderful to get new students this time of year. It is a gift, you know. Remember that” Everything was a gift to Mrs. E. But this time, Justin agreed with her.
“Let’s do an activity to help us reconnect and bond with our peers, ok? After all, we are social animals. We need our peers and they need us. Cooperation is a gift. Remember that. Let’s work on that. I have this game called Traffic Jam. It’s played first in groups of 4, then groups of 6, then 8 and so on. There are five stepping stone in a row. On the two left-hand stones, facing the center, stand two people. Justin and Juli, why don’t you be those two people?”
Justin knew Mrs. E would pick him. She always did for some reason. He and Juli walked up to the front of the class.
“On the two right-handed stones stand two other people. Let’s have Doni and Cassidy stand there, ok?”
Suddenly Justin was much more interested in the outcome of this game.
“The center stone is not occupied. The point of this game is that everyone on the left side must somehow move to the right side, and vice versa, with the center stone left unoccupied in the end. Here are some basic rules: after each move, each person must be standing on a stone. If you start on the left, you can only move towards the right. You you’re on the right, you can only move towards the left. You can leap frog a person if the stone on the other side is empty. You cannot jump over more than one person at a time. And finally, only one person may move at a time.”
“The winning team will show us how to change positions in the least number of moves. Then we’ll try it with 6 people and 7 stones (always the center stone unoccupied). This will be so much fun! Let’s start!”
Justin wasn’t sure how much fun the game was going to be, but he was sure interested in knowing more about Cassidy.
Your task: what are the minimum moves for 4, 6, 8 and 10 people games? Is there a pattern that helps?