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Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

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Two Cubes

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

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Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Unit Interval

Stage: 4 and 5 Challenge Level: Challenge Level:1

Why do this problem?
The problem gives practice in solving linear and quadratic inequalities.

Possible approach
Use this as a lesson starter. If learners do not know how to start let them use the Hint.

Then discuss the problem as a class but try to elicit ideas from the learners themselves.

Key questions
How do we make use of the information given?

If we are not making progress, then have we used all the information given?