Visualising and representing

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find there are 2 and only 2 different standard dice. Can you prove this ?

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it is facing the other way round.

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

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400 metres from B. How long is the lake?