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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

' Tis Whole

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Remove three consecutive numbers from the set of natural numbers from $1$ to $n$. The mean of the remaining numbers is $7.5$ . Find $n$ and the numbers that were removed.

What if any three numbers are removed, not necessarily consecutive numbers?

Notes for students

A true problem solver is like a detective. When there is an obvious or routine method of solution anyone can follow the usual steps and get a solution. Such mathematical 'problems' don't even need an intelligent human being, they can simply be fed to a computer to do the work automatically.

When solving real problems you have to search like a detective for all the information you can gather and then try to make sense of what you have found out.

Some solutions can be found to this problem by trial and improvement. That is just the start of the investigation! What other methods can be used? Which is the best method? What can we discover about the mathematical ideas involved and how can we use these ideas to find out more? When we have some solutions can we prove that there are no remaining undiscovered solutions? Can we pose and solve similar problems? What generalisations can we make?

It appears at first sight that there is too little information given about this problem so we have to find out all we can. The problem involves triangle numbers and the arithmetic mean and we know it is about whole numbers. We can construct and simplify some algebraic expressions.

Can we prove that $n$ cannot be very small or very large so we only have to test a limited number of cases?

Can we use a spreadsheet or other computer program to help search for solutions?

When we find some solutions can we be sure we have found them all?