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.

Threesomes

Here is a synopsis of the solutions offered for the cases considered so far (i.e. it does not consider triangles that have non-horizontal bases):

The smallest triangle it is possibkle to draw has a base of 1 unit and a height of 1 unit. So the smallest area is $\frac{1}{2}$ sq. unit.

There are an infinite number of triangles that can be drawn with these diagonals (see the problem "Shear Magic" )

There are two ways of creating a triangle of area 1 sq and with a horizontal base:

Base 1 unit; height 2 units
or
Base 2 units and height 1 unit, again

For an area of 2 sq units there are three families of triangles with a hoirizontal base::

Base 1 unit and height 4 units
or
Base 2 units and height 2 units
or
Base 4 units and height 1 unit

For each family there are an infinite number of triangles