This activity has a logic and reasoning focus.
find strategies for investigating games
write a solution for a game
This game is very much like Rubic’s cube. It is another game that requires logic to solve it. Hence in this way it is linked to the Lake Crossing I, Lake Crossing II and Space Crossing problems. However, it is not directly related to any of the non-Mathematical Processes strands.
This particular problem can be attacked by a logical approach. Clearly equipment is a useful strategy to use but another one that is valuable is ‘use a smaller case’. So if your students have trouble getting started they might try three marbles of each colour first. This might give them a clue as to how to approach the eight ball situation.
In recent years games have been given a lot of attention by mathematicians. In fact there have been a wide variety of games analysed and solved. For instance, it is well known that the first player in noughts and crosses cannot lose if she plays optimally. On the other hand, the second player, playing optimally can always at least force a draw. Other games are more difficult. Although computers can play a very good game of chess, it is still not known what is the best strategy for either the first or second players.
But analysing games goes back to at least the 17th Century. The famous French mathematician Blaise Pascal (of Pascal’s Triangle fame) spent a great deal of time studying card games. As a result he laid the foundation for probability and statistics.
A great deal of theoretical work has now been established. So much so that there is now a branch of mathematics called Games Theory. Generally speaking, very little of this has so far found its way into the school curriculum, though.
Richard finds 9 dice arranged in a 3 by 3 square. He’d like to rearrange the dice so that only 6's are showing. But there are rules as to how he can move the dice. He is only allowed to pick up a row of three dice or a column of three dice and rotate them altogether in the same way. Is it possible for Fred to accomplish this task? Can you tell him how?
- State the problem. Get the class to think about it. Ask
How would you get a 6 to the top?
Can you get a whole row of 6s?
- After some discussion, let the class go into their groups.
- Help groups that need it. For the quicker groups, encourage them to try the Extension.
- Let a few groups report back to the whole class.
- Leave time for students to write up their solutions (or let them do this as homework).
The algorithm has two steps and is based on the fact that when a cube is rotated a face that is perpendicular to the axis of rotation does not change location. We use ‘left’ to mean the left of the dice; ‘away’ to mean the top in the diagram; and the columns go down the page, while the rows go across.
Step 1: Work along a row. For each die pick up the column that it is in and rotate until either the 6 appears in which case leave it showing, or after three rotations of 90 degrees it has not appeared. In this case pick up the row and rotate the row until the 6 appears on top. In both cases the 6 is now on top, pick up the row and rotate it so that the 6 is facing left. (This last move ensures that we can move the 6s in a row to the top by a rotation of the columns.)
Step 2: After each dice in the row has the 6 facing left, pick up each column in turn and rotate so the 6's are facing up. Pick up the row and rotate so that the 6's are facing away.
Algorithm: Start with the top row using Step 1. Once a die has its 6 facing left, that column is not used again for the other dice in the row.
Do Step 2. Because the 6's are facing away, they are not moved when the column is rotated. It is not necessary to use the first row again. Repeat the algorithm for row 2. Repeat Step One for row 3 but the 6's need only be rotated to the top in Step 2. Rotate Rows 1 and 2 to bring all the 6's to the top.