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Rolling marbles

Student Activity: 

Anna has a new toy. It consists of four red marbles next to four blue marbles in a U-shaped track.
Anna can 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?

Achievement Objectives:

Purpose: 

This activity has a logic and reasoning focus.

Specific Learning Outcomes: 
find strategies for investigating games
write a solution for a game
Description of mathematics: 

This is a logic problem, and logic is an important part of mathematics and of the curriculum.

If students choose to use equipment, a careful record of each step should be made. Using a simpler problem, for example with 3 marbles of each colour, is another approach students may choose. As in any problem solving situation, a written record should be encouraged in order for students to be able to justify their solution, or to identify an error made in the process.

Other logic problems include: Lake Crossing I , and Space Crossing.

Note: In recent years mathematicians have analysed a wide variety of games. 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 and for a branch of mathematics called Games Theory. 

Required Resource Materials: 
Two colours of marbles, counters or blocks.
Activity: 

The Problem

Anna has a new toy. It consists of four red marbles next to four blue marbles in a U-shaped track. Anna can 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?

Teaching sequence

  1. Pose the problem and check that it is understood.
  2. Have students work on this in groups, taking care to keep a record of their solution as they do so.
  3. Support as required and make the Extension available when appropriate.
  4. As students share their solutions ask: 
    Is this the only solution? 
    Is this the smallest number of moves?

Solution

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

Anna can accomplish this in 6 moves.

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RollingMarbles.pdf144.49 KB