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Why do this problem?

This short problem provides an interesting insight into the history of mathematics and into the nature of hypotheses: a hypothesis cannot be proved correct, merely reinforced or rejected on the acquisition of new information.

Possible approach

Trying to spot number patterns is an easy activity, in this problem we would like students to make clear hypotheses which are consistent with the data. In some sense, any hypothesis which the numbers do not contradict is equally valid, although some might seem more reasonable than others.

Students could try to consider possible patterns in pairs and then to share their ideas. As a class, can the group decide on the best hypotheses? Encourage students to write down their hypotheses clearly.

Key questions

What patterns do these numbers exhibit?
Can you write down a clear hypothesis stating the rules underlying the structure of the numbers?
Discuss the question 'How would we be able to work out whose hypothesis is correct?' (we can't!)

Possible extension

Interested students might like to follow the historical links. Alternatively, they might like to consider the creation of extreme, outlandish explanations for the numbers.

Possible support

Provide calculators to allow students to test number rules.