Reasoning, convincing and proving

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

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

Can you find a rule which relates triangular numbers to square numbers?

The squares of this grid contain one of the letters P, Q, R and S. Can you complete this square so that touching squares do not contain the same letter? How many possibilities are there?

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it works?