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

Three frogs started jumping randomly over any adjacent frog. Is it possible for them to finish up in the same order they started?

Weekly Problem 37 - 2011

Rotating a pencil twice about two different points gives surprising results...

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

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

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?

David has given a neat solution and also written a program to calculate big powers in modulus arithmetic.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Can you beat Piggy in this simple dice game? Can you figure out Piggy's strategy, and is there a better one?