Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
Can you explain why a sequence of operations always gives you perfect squares?
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?
Conjectures are important, and should be encouraged, but along
with a challenge to really explain why any claim might be true
We hope the problem will give students a genuine pleasure in
discerning real structure, and lead their interest on into Number
The articles on the NRICH site (see link from problem page) are
an excellent follow-on.