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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?

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Rotating Triangle

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

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Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Threesomes

Stage: 3 Challenge Level: Challenge Level:1

Solutions received gave part of an insight into the problem but ommitted to consider triangles on the dotty grid that do not have a base of whole unit length. It is possible to draw triangles whose bases and heights are neither horizontal or vertical. These triangles have bases that are not a whole number or units. A complete solution needs to consider these.
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

It is possible to demonstrate that it is also possible to obtain triangles with multiples of half a sq unit but there is still work to do.