by Dave Youngs.
This month's Puzzle Corner is designed to be done in conjunction with the Maximizing Math activity since both deal with the same mathematical game. The Puzzle Corner activity introduces students to the rules of the game and gives them the opportunity to play it a number of times. After students are familiar with the game, move on to the Maximizing Math activity which challenges them to examine the mathematics inherent within it.
Taking Away by Ones and Twos is similar to the historic nim* games which have their origins in ancient China. There are dozens of different versions of nim, but each involves two players who take turns removing objects from several separate piles with the winner being the person who takes the last object(s). (In a few versions, the loser is the one who takes the last object.) While Taking Away by Ones and Twos is related to the nim games, it differs in one significant way the objects are all placed in only one pile. This difference makes the game easier to figure out from a mathematical standpoint and is, therefore, more appropriate for elementary students.
In Taking Away by Ones and Twos, two players start with a pile of 15 small objects, such as beans, pennies, or plastic chips. The players take turns removing either one or two objects from the pile on each move. The player to take the last one or two objects is the winner. Players alternate going first and should try to develop a winning strategy.
Once students have played the game enough to have learned some of its unwritten rules (such as, you will lose the game if there are three objects left when it's your turn), they are ready to begin the Maximizing Math activity which challenges them to look at the game from a mathematical standpoint. You will need to decide if you want to do the Puzzle Corner and the Maximizing Math activities on the same day or on separate days. My preference is to let students have several days to play the game before starting a formal exploration of its mathematics. This way, students are more likely to discover some of the underlying mathematics on their own through repeated playing of the game.
Activity Page | PDF File (165K)(This file contains the Puzzle Corner and Maximizing Math sections.)