# Generalising - May 2004, Stage 4&5

## Problems

### Pair Products

##### Stage: 4 Challenge Level:

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

### Three Frogs

##### Stage: 4 Challenge Level:

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

### Pencil Turning

##### Stage: 4 Short Challenge Level:

Weekly Problem 37 - 2011
Rotating a pencil twice about two different points gives surprising results...

### Perfectly Square

##### Stage: 4 Challenge Level:

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

### Squirty

##### Stage: 4 Challenge Level:

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.

### Generally Geometric

##### Stage: 5 Challenge Level:

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

### Golden Fibs

##### Stage: 5 Challenge Level:

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

### Water Pistols

##### Stage: 5 Challenge Level:

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?