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Problem of the Week
Problem B and Solution
Sorting Marbles

Problem

Dr. Cy Linder has tubes containing marbles of different colours which he would like to sort. He pours marbles from one tube to another until all the marbles of the same colour are in their own tube. For example, Dr. Linder starts with blue and red marbles in the three tubes shown.

Tube 1 contains 3 stacked marbles; the top and middle marbles are blue and the bottom marble is red. Tube 2 contains 2 stacked marbles; the top marble is red and the bottom marble is blue. Tube 3 is empty.

He can sort the marbles by first pouring both blue marbles from Tube 1 to Tube 3. Next he pours the red marble from Tube 2 to Tube 1. Finally he pours the blue marble from Tube 2 to Tube 3. The marbles are now sorted, and it took 3 pours in total. Note that this is not the only way to sort these marbles.

A description of the tubes follows.

  1. Write down steps to sort the marbles in the given tubes. How many pours did you need in total?

    Three tubes. Tube 1 contains 4 marbles: from top to bottom, the colours are red, blue, red, blue. Tube 2 contains 4 marbles: from top to bottom the colours are blue, red, blue, red. Tube 3 is empty.

  2. Write down steps to sort the marbles in the given tubes. How many pours did you need in total?

    Five tubes. Tube 1 contains 4 marbles: from top to bottom they are blue, red, grey, blue. Tube 2 contains 4 marbles: from top to bottom they are blue, red, grey, grey. Tube 3 contains 4 marbles: from top to bottom they are grey, red, blue, red. Tubes 4 and 5 are empty.

Extension: Create your own marble sorting problem that requires 5 pours to be sorted.

Solution

Students may find it helpful to to use coloured strips of paper or coloured cubes to model the marbles in the tubes. Note that other solutions are possible.

  1. Using the following 7 steps, we can sort the marbles.
    1. Pour the red marble from Tube 1 to Tube 3.
    2. Pour the blue marble from Tube 2 to Tube 1.
    3. Pour the red marble from Tube 2 to Tube 3.
    4. Pour two blue marbles from Tube 1 to Tube 2.
    5. Pour the red marble from Tube 1 to Tube 3.
    6. Pour three blue marbles from Tube 2 to Tube 1. All the blue marbles are now in Tube 1.
    7. Pour the red marble from Tube 2 to Tube 3. All the red marbles are now in Tube 3.
  2. Using the following 10 steps, we can sort the marbles.

    1. Pour the blue marble from Tube 1 to Tube 4.
    2. Pour the blue marble from Tube 2 to Tube 4.
    3. Pour the red marble from Tube 1 to Tube 5.
    4. Pour the red marble from Tube 2 to Tube 5.
    5. Pour the grey marble from Tube 1 to Tube 2.
    6. Pour the grey marble from Tube 3 to Tube 2. All the grey marbles are now in Tube 2.
    7. Pour the blue marble from Tube 1 to Tube 4.
    8. Pour the red marble from Tube 3 to Tube 5.
    9. Pour the blue marble from Tube 3 to Tube 4. All the blue marbles are now in Tube 4.
    10. Pour the red marble from Tube 3 to Tube 5. All the red marbles are now in Tube 5.

Solution to Extension:

Answers will vary. Here is one marble sorting problem that requires 5 pours to be solved. We leave it up to the reader to verify this.

Tube 1 contains 4 marbles: from top to bottom they are red, blue, red, blue. Tube 2 contains 4 marbles: from top to bottom they are red, red, blue, blue. Tube 3 is empty.