Expanding and factorising quadratics

If you plot these graphs they may look the same, but are they?

Weekly Problem 26 - 2008
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

How do scores on dice and factors of polynomials relate to each other?

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.