This activity has a logic and reasoning focus.
find strategies for investigating games
write a solution for a game
This 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.
Anna has a new toy. It consists of four red marbles next to four blue marbles in a U-shaped track. Anna is allowed to pick up any two consecutive marbles and place them, in the same order, at the left or at the right end of the track. Then gravity brings them all together again.
Can Anna put the marbles in order red, blue, red blue, etc., using the move described above? If so, what is the smallest number of moves that it requires?
Suppose the balls in the track alternate in colour, starting with red on the left. Can Anna find a sequence of moves that will put them back into their original positions, i.e., four red marbles followed by four blue marbles? If so, what is the smallest number of moves that she can make to do this?
- State the problem. The first part of this problem is reasonably straightforward so it may be done quickly by some of the class. You might like to solve Anna’s first problem as a discussion by the whole class. On the other hand, if you have a bright group of students you could just give it to them and let them go straight into their groups to solve it.
- If the students are not already in their groups, let them go into them now.
- 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).
By taking the two adjacent red and blue marbles and moving them to the right it only takes two moves to have the marbles alternating in colour. As it is not possible to do it in one move then this is the smallest number of moves.
We now give one possible algorithm for making the change. There are others. Starting from the right, always take the left most mixed pair of marbles in which the left most in the pair is the same colour as the right most of the eight marbles. After three moves, then move the paired blues to the right. Specifically: the capital letters indicate the marbles to be moved - always to the right hand side.
rBRbrbrb-->RBrbrbbr-->rBRbbrrb --> rBBrrbbr-->rrrBBrbb-->rrrrbbbb
Now, this straightens things out in 6 moves.