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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?

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.]


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

Approximating Pi

Age 14 to 18 Challenge Level:

Why do this problem? 
This is a classic, the historical reference to Archimedes is educational, and the problem  should be part the education of every student of mathematics. To do this problem requires only very simple geometry and it introduces the idea of approximation by finding an upper and lower bounds, and refining the approximation by taking a series of values where, in this case, we use smaller and smaller edges, or more and more sides for the polygons. In addition this problem is a valuable pre-calculus experience as it uses the idea of a limiting process involving smaller and smaller ' bits'.
Possible Approach
First ask everyone to work out the perimeters of the two squares in the diagram. Then have a class discussion about what this tells us about how large the length of the circumference of a circle can be and how small. Discuss the history of this method with reference to Archimedes and introduce the idea that it is a method for finding the value of $\ pi$. Pose the problem: "How would you find the value of $\pi$ if it was not already known?"

Introduce the idea of an upper bound and a lower bound for pi and raise the question about how we might improve these bounds to get closer to the value of pi. Then ask the class to repeat the exercise using circumscribed and inscribed hexagons.and compare results.

Suggest your students researchArchimedes method for finding $\pi$ and other methods of approximating $\pi$ on the internet for themselves. Discuss the difficulties of calculation, in particular finding square roots, without modern calculating aids and refer to the problem Archimedes and Numerical Roots.

Key Questions
Can you find the perimeter of the square (or other regular polygon) circumscribing the circle?
Can you find the perimeter of the square  (or other regular polygon) inscribed inside the circle?
What can you say about the lengths of the perimeters of these two polygons and the length of the circumference of the circle?
Knowing the circumference is $2\pi r$ how does this help you find a lower and an upper bound for $\pi$.
Possible extension  
See Archimedes and Numerical Roots.  
Possible support  
Read  Eureka.