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'Black Box' printed from http://nrich.maths.org/

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Why do this problem?

This problem is best suited to tenacious problem solvers who will relish the intellectual challenge in making sense of the numbers. Although determining the function rules will be very challenging, on the technical side the mathematics required to verify the rules is elementary (KS3). Interestingly, the ideas raised will actually have relevance in university number theory and, as such, could provide important insights for those going to university to study mathematics.

Possible approach

This challenge will require experimentation to gain sense of the numbers involved as, after a few trials, it will likely seem that there is little obvious pattern in the numbers produced.

Part of the fun of this problem is to be faced with a straight 'black box' and being left to decide how it works, so you may wish to hold back from offering hints to students.

Once a feel for the numbers is found, students might start to make conjectures concerning the function rules which can then easily be tested: new examples will either add weight to the conjectures or cause them to be rejected. Students might need a spreadsheet or calculator to help in this regard. Alternatively, they could use the Wolfram Alpha computational knowledge engine http://www.wolframalpha.com/.

Once students have solved the problem they should write up their rules clearly to allow others to independently verify them. Of course, it is not possible to prove that the Black Box follows certain rules but, once discovered, evidence will soon build up in their favour.

Finally, we don't suggest using this task with students who are unlikely to enjoy this style of working: reserve it for very keen students who like individual challenges.

Key questions

What sorts of numbers does the problem involve?

After a few trials, are you getting a feel for the sorts of numbers produced?

Do you have any rough ideas for how the numbers are related?

Possible extension

Students could read about the ideas raised. The links will be posted following the publication month for this problem

Possible support

Support questions will be posted following the publication month for this problem.