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'Classic Cube' printed from https://nrich.maths.org/
These solutions came from Fred from Albion Heights Junior Middle
School and Low Zhen Lin from Garden International School. Both
contributed to this solution. From Low Zhen Lin's diagram you will
see that he had the clever idea of cutting one of the faces into
four pieces.
To maximize the size of the net, we transform it so that as many of
the vertices of the net as possible touch the edge of the paper.
This is to maximise the length of the sides and therefore, the
volume.
For the traditional net, the most efficient way is to have the
lines parallel to the diagonals of the square, as shown in the
diagram on the left.
To find the length $E$ of an edge of the cube divide the paper into
5 by 5 and use Pythagoras' theorem. So $E^2 = 20^2 +20^2$ and
$E=\sqrt{800}$ or approximately 28.3, and the volume is 22627. Six
of the eight non-retroflex vertices touch the edge of the square,
and the net fills 48 per cent of the square.
However, this is not the most efficient net. The traditional net
has a 3 $\times$ 4 bounding rectangle - the smallest rectangle that
it will fit in. The most efficient net will have a bounding
rectangle that is a square. That net is shown in the diagram on the
right which fills 75 per cent of the square.
The edge of this cube is $\sqrt{1250}$, or approximately 35.4, and
the volume is 44194.