Conjecturing and generalising

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Which armies can be arranged in hollow square fighting formations?

Two ladders are propped up against facing walls. At what height do the ladders cross?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

What is the largest number which, when divided into these five numbers in turn, leaves the same remainder each time?

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?