Half of a spring
(a) Suppose you cut a mass less ideal spring in half. If the full spring had a force constant k, what is the force constant of each half, in terms of k? (Hint: Think of the original spring as two equal halves, each producing the same force as the entire spring. Do you see why the forces must be equal?)
(b) If you cut the spring into three equal segments instead, what is the force constant of each one, in terms of k?