### All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

### Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

### The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

# Disappearing Square

##### Stage: 3 Challenge Level:

Several teachers didn't like this trick but is it really a trick? When the rotation takes place the grid lines up along the slopping sides but does it exactly fit? When you count the squares the second time what are you counting? The counting may not be accurate. Are the 'half' squares all exactly half?

Many of you were puzzled by this. What is really going on here? We can't 'make' an extra square of area just by chopping a shape up and putting the pieces together differently so what is happening? William from Tattingstone describes it like this:

I think he is trying to tell us that the rectangle C doesn't quite fit in there. The line on the sloping side of the triangle isn't really a straight line is it? Measuring is never absolutely accurate is it?