THERE are 35 players standing in a circle. As the games wizard walks round the circle she kills every second player until only player survives. The players are numbered one through 35. Which player lives?
The guesses came fast. The first one. The last one. The one next to the last one. Then some students started to draw circles and cross off players.
Answers varied. Musbah smiled and said, “number 7, I did it in my head.” Musbah never fails to delight. But what if the number of players changes I challenged him.
Emma, Brittany and Julia worked together drawing circles and carefully working, talking and comparing results. They confirmed Musbah’s answer.
Jason, at the back of the room, asked do the dead ones get dragged away?
Emma was quick to add, “it must be a prime number.” Then Jeremiah tried a circle that gave the answer nine. Haley added, “you have to be an odd number to survive.”
Heaven, Jessica and Celeste raced as they took up the suggestion to try some smaller numbers in the circle. They quickly listed the survivor for a page full of circles, busily crossing off imaginary players. “There’s no pattern,” Heaven declared, giving me an accusatory look.
“You’ve destroyed your evidence,” I commented. “What if you write out the numbers as you cross them off,” I suggested.
Taylor was sufficiently intrigued to not need to be reminded to put away her lines for the school play… each of her players was reprsented by an open circle. As each player was killed off by the game wizard she filled in the circle. While others students were crossing off dots, Taylor’s method was simple and clear and a guard against confusion.
“Taylor, show your method at the board.” She responded, “I’ll do 35.” I resisted suggesting she did a small circle, beginning to think the pattern might reveal itself easier with larger numbers of players.
Emma and friends urged her on and soon Taylor was also listing the numbers as she filled in circles. In a stroke of pure genius Julia suggested Taylor colored the circles a different color on the second revolution by the wizard. Soon the numbers were color coded too.
By the sixth revolution we’d exhausted the supply of colors.
On the second revolution “it’s every fourth player… the next every eighth… then every 16,” Brittany pointed out.
Hadassah and Amanda, on the far side of the room, pointing to their list with numbers outlined in boxes said there was a “doubling” as the wizard circled.
As the 50 minutes ended, Brittany asked, “you do know the answer, right?” No, I confessed.
Not everyone in class was as enthused by the problem. Everyone attempted to answer the straight question, but a significant number then tried to take the question further. A pleasingly large minority were still working on the problem as the lesson ended.
Significantly the students who persevered were mostly girls, and not girls who always stay so focused in class.
❏ The use of an open-ended question for a whole lesson was inspired by The Elephant in the Classroom: Helping Children Learn and Love Maths by Jo Boaler, to be published next March, and Boaler’s US classic What’s Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject
Prof Boaler, who has done research on effective math teaching involving thousands of students in schools in both the USA and UK, argues for open-ended projects, mixed-ability group work and students talking math.
❏ The problem has a name: the Josephus problem.
The UK’s Royal Institution has used it as the basis for one of its master-class series. I got the idea to use the problem from the November 2009 issue of Mathematics Teaching, the journal of the UK Association of Teachers of Mathematics (MT216).
The “best” solution is to use binary numbers.