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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.
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?
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.
Choose any three by three square of dates on a calendar page...
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?