### 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?

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

# Screwed-up

##### Stage: 3 Challenge Level:

We received two different explanations to this problem.

Danny Ng, 16, from Milliken Mills High School in Canada sent us this solution:

"Turning a left-handed helix over does not make it become a right handed helix.

When you observe a bolt at eye level, you see the parallel threads.
There are two kinds of threads:
1) go "down-right", from upper-left down to lower-right
2) go "down-left", from upper-right down to lower-left.
If you draw a line that is "down-right" on a piece of paper(the line goes from upper-left down to the lower-right) and turn the paper upside down (just like turning a left-handed helix over) you will notice the line is still "down-right" and it never goes "down-left"."

Harry (Xiaotian) Li (Age: 15) from Riccarton High School, Christchurch, New Zealand, sent us this solution:

"Consider, for example, the spiralling direction of a right-handed one:
To screw down (from top) turn clockwise.
To screw up (from top) turn anti-clockwise.
After that we turn it over (top to bottom).
To screw it down, still turn clockwise.
To screw it up, still turn anti-clockwise.

And also, two kind of helix look different (opposite each other). When you turn just one of them over they are still opposite, which confirms that it has not changed.

Therefore, when we turn a left-handed helix over (top to bottom). It does not become a right-handed helix."

Paul Marcinkowski, aged 14, from the Harwich School in GB convinced himself by making a helix, and contributed:

"No because I have made a left handed helix and whatever way you turn it up it is still a left handed helix."