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Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

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Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

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There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Data Chunks

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Systematic working and recording of results help a lot here.

Conjectures are important, and should be encouraged, but along with a challenge to really explain why any claim might be true generally.

We hope the problem will give students a genuine pleasure in discerning real structure, and lead their interest on into Number Theory.

The articles on the NRICH site (see link from problem page) are an excellent follow-on.