Editor’s note: The material below is adapted from ‘Book 0 Chapter 1: Operational Systems’ of the Elements of Mathematics series. The EM series of secondary school mathematics textbooks is a 27-volume collection written and published by the IMACS Curriculum Development Group and serves as a basis for the high-level online math courses available through eIMACS, the distance-learning division of IMACS.
In our previous IMACS blog post on modular arithmetic, we introduced the Clock Game as a fun way to teach children about modular addition. Now, we’ll introduce a few variations on the Clock Game that make it more interesting. If you’re not familiar with the Clock Game, we suggest that you first read our introduction to modular addition before adding on with this post. Have fun!
The Double Game
In the double game, there are two pieces at each number at which to play instead of just one.
The game described in our introduction to modular addition can also be played as a “double game.” The only difference is that when playing double games, a player has more choices. For example, if, as a result of a particular move, a player has to place a checker at a number, one of whose circles is unoccupied and one of whose circles is occupied by the opponent, the player has the choice of either taking the opponent’s checker by replacing it with one of his or her own, or placing a checker on the unoccupied circle and leaving the opponent’s checker on the math. Double games, therefore, last longer, since all the available places must be occupied before the game is over.
Kings and Double Kings
Another variation of the games which can make them more interesting involves the use of kings and double kings. Only one place at each number is used in these games. In this variation, if the move requires it, you may place a second checker on top of one of yours already occupying a particular position, thus creating a “king checker.” If you have a king checker already occupying a certain position, then, if the move requires it, you may place a third checker on top of the king, thus making a “double king.” If a double king occupies a particular position, no further checkers may be added to the pile and so the play continues as in the simplest version of the game.
If your opponent would normally land on a place where you have a double king, then that double king cannot be removed, and the opponent cannot place a second checker. The hour hand, however, should still be moved so as to point at the position occupied by the double king even though your opponent could not place a checker there. If, however, your opponent lands on a place where you have a king, then both your checkers may be removed and replaced by one of your opponent’s checkers. The game ends when all of the positions are occupied by at least one checker upon the completion of a turn. This type of game is scored by counting 3 points for each double king, 2 points for each king, and 1 point for each single checker.
Kings, Double Kings and the Double Game
This variation utilizes the rules of the last two sections simultaneously. The first checker of each move must be placed on an unoccupied position, but there may be several choices for the second checker. For example, suppose your second checker has to be placed at “3,” your opponent has a king on one of the positions labeled “3,” and you have a checker (or a king) on the other position labeled “3.” You may either remove your opponent’s king and replace it with one of your own checkers, or you may form a king (or a double king) of your own. In such a situation, you may always choose either one of the two possibilities offered you by the two positions corresponding to the number at which your second checker is to be placed.