A Stacking Problem

Task 149 ... Years 4 - 10

Summary

Six numbered cubes stand in a tower on Cell A. The cubes are in numerical order from the top down. The challenge is to shift the cubes so they stand in three mini-towers on Cells A, B & C (as shown on the card) by moving only one cube at a time and never placing a higher number on top of a lower number.

This cameo includes an Investigation Guide and a From The Classroom section with a video from a Year 8 class which challenges students to constantly review the conditions of the problem.

 

Materials

  • 6 numbered cubes

Content

  • spatial thinking/visualisation
  • logical reasoning and problem solving strategies
  • sequencing and patterning

A Stacking Problem

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

This is a tough problem for most students (and teachers). It certainly doesn't need to be solved in one sitting and, equally, we need not offer hints too soon. It relates to Tower of Hanoi, but it is not identical.

In the early stages of exploration students (in pairs or in a whole class investigation) could benefit from discussing the video provided to Cube Tube by Vis Chetty's Year 8. See From The Classroom below.

When the minimum number of moves is eventually counted (the answer being 60 moves), the challenge and the patterns in the problem may seem surprising for a problem so easy to begin.

Action Moves Total So Far
Blocks 1 & 2 to Position C. 3  
Blocks 3 to Position B. 1  
Blocks 1 & 2 back onto Block 3. 3 7 moves
Block 4 to Position C. 1  
Blocks 1 - 3 onto Block 4. 7 15 moves
Block 5 to Position B. 1  
Blocks 1 - 4 onto Block 5. 15 31 moves
Block 6 to Position C. 1  
Blocks 1 - 4 to Position A. 15 47 moves
Block 5 onto Block 6. 1  
Blocks 1- 3 to Position C. 7 55 moves
Block 4 to Position B. 1  
Blocks 1 & 2 to Position A. 3  
Blocks 3 onto Block 4. 1 60 moves

The turning point of the problem is when Block 6 is the only one left on Square A, and Blocks 1 through 5 are in order from 1 down to 5 on Square B. Even when students have reached this stage and continued to the solution, it may be necessary to repeat the solution several times to realise the pattern in the movements.

Once a problem is understood, a mathematician has the responsibility of explaining the solution to others. Encouraging this could lead to listing as above, diagrams, or perhaps use of algebraic notation.

Extensions

The solution above encourages the mathematician's strategy of breaking a problem into smaller parts. Further investigation develops by asking about whether similar problems (3, 9, 12, 15... blocks) on three squares, can be solved, and, if so, what is the minimum number of moves. For example, begin with blocks 1, 2, & 3 stacked on A and finish with 1 on A, 2 on B and 3 on C.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

If you have cubes which don't click together (separating and rejoining cubes becomes a bit annoying when using linking cubes) the problem is easy to turn into a whole class investigation. If you don't have cubes an equivalent can be produced by each pair tearing a piece of scrap paper into 9 pieces. Three are marked A, B, C. The others are marked from 1 to 6. The numbered pieces are then lined up above A, rather than stacked on top of it. The problem becomes two dimensional but retains the same characteristics.

The problem is used to highlight mathematical reasoning and strategies and encourage communication of mathematics. It is easy to state and easy to start, but soon proves not so easy to conquer. Allow time for the students to try the main problem, but as frustration grows, lead a discussion to try to summarise what students have noticed so far.

From here the lesson is perhaps best tackled through this Investigation Guide with appropriate sharing times to extend the list of what the students notice.

The Investigation Guide begins with an equivalent problem which comes from asking What happens if we change...?. In this variation the movement rules and number of bases remain the same, the equal distribution of blocks between the bases is still the aim and the left to right sequencing of the blocks is also expected. What has been changed is the number of blocks.

  • Students might also ask, What happens if we change the number of bases to 6 and keep the other conditions unchanged?.

To conclude the lesson it will be important to review it from the point of view of working like a mathematician. A particularly important part of that work in this case is to be able to explain to someone else how to do it. Students could use, words, diagrams, slides or video.

At this stage, A Stacking Problem does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

The A Stacking Problem task is an integral part of:

  • MWA Pattern & Algebra Years 5 & 6
  • MWA Space & Logic Years 9 & 10
This task is also included in the Secondary Library Kit. Solutions for tasks in the latter kit can be found here.

From The Classroom

Living Waters Lutheran College

Vis Chetty
Year 8
Hi Doug,
Thank you for this really interesting problem. I gave this to my Y8 students as a fun problem solving activity and I had one student complete the puzzle in 16 moves and another in 18. I will try to send you a video clip of the solution.
Kind regards,
Vis Chetty

Vis was able to send a video. You can find it in Cube Tube and it asks the question: Does this solve A Stacking Problem?. Thanks Vis, we love the way this snippet challenges students to check the conditions of a problem and wonder about what mathematicians do when they make mistakes.

Green Line
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