Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Tower of Hanoi

### Why do this problem?

### Possible approach

*These printable resources may be useful: Tower of Hanoi, *

Tower of Hanoi Building Block Cards.

*An alternative version of this problem, with videos showing the most efficient way to move the discs, can be found here.*

Start by explaining how the Tower of Hanoi game works, making clear the rules that only one disc can be moved at a time, and that a disc can never be placed on top of a smaller disc.

### Key questions

###

### Possible extension

### Possible support

*You may wish to print these handouts for teachers based on the NRICH Roadshow version of this task, with the problem on one side and the notes on the other. *
## You may also like

### Adding All Nine

### Summing Consecutive Numbers

Or search by topic

Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

The Tower of Hanoi is a well-known mathematical problem which yields some very interesting number patterns. This version of the problem involves a significant 'final challenge' which can either be tackled on its own or after working on a set of related 'building blocks' designed to lead students to helpful insights.

Initially working on the building blocks gives students the opportunity to then work on harder mathematical challenges than they might otherwise attempt.

The problem is structured in a way that makes it ideal for students to work on in small groups.

Tower of Hanoi Building Block Cards.

Start by explaining how the Tower of Hanoi game works, making clear the rules that only one disc can be moved at a time, and that a disc can never be placed on top of a smaller disc.

Hand out a set of building block cards to groups of three or four students. (The final challenge will need to be removed to be handed out later.) Within groups, there are several ways of structuring the task, depending on how experienced the students are at working together.

Each student, or pair of students, could be given their own building block to work on. After they have had an opportunity to make progress on their question, encourage them to share their findings with each other and work together on each other's tasks.

Alternatively, the whole group could work together on all the building blocks, ensuring that the group doesn't move on until everyone understands. The complete problem is available on a worksheet.

When everyone in the group is satisfied that they have explored in detail the challenges in the building blocks, hand out the final challenge.

The teacher's role is to challenge groups to explain and justify their mathematical thinking, so that all members of the group are in a position to contribute to the solution of the challenge.

It is important to set aside some time at the end for students to share and compare their findings and explanations, whether through discussion or by providing a written record of what they did.

What important mathematical insights does my building block give me?

How can these insights help the group tackle the final challenge?

Of course, students could be offered the Final Challenge without seeing any of the building blocks.

Encourage groups not to move on until everyone in the group understands. The building blocks could be distributed within groups in a way that plays to the strengths of particular students.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?