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.
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
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!)
Interested students might like to follow the historical links.
Alternatively, they might like to consider the creation of extreme,
outlandish explanations for the numbers.
Provide calculators to allow students to test number rules.