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## 'Solving the Net' printed from http://nrich.maths.org/

This problem follows on from the

net
puzzle problem
Full Screen Version
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A Rubik's cube can be rotated
about three perpendicular axes,vertical, horizontal and facing you
as you look at the cube. In these planes any one of three layers
may be rotated in either the clockwise or anticlockwise directions.
On the net of the cube these directions are labelled with 2 letters
for Horizontal, Vertical, Facing, Left Right, Up Down, Clockwise,
Anticlockwise and a number 1, 2, 3 to show which layer to
permute.
Although randomly applying
operations quickly permutes the net into a seeming tangle, certain
combinations of operations permute a small subset of the squares on
the net.
This problem is an investigation
of these sequences of operations
Edge switcher 1 is the
sequence PP W GGG BB W GGG YY W GGG RR W GGG. It permutes four
middle edge squares only and leaves the
whole of the white and green faces unchanged

In solving the net we will need to be able to repeat a similar
permutation whilst leaving other faces unchanged. Consider this
configuration:

In solving the net we will need to be able to repeat a similar
permutation whilst leaving other faces unchanged.Consider this
configuration:

How is it similar to the previous net configuration?

How is it different?

Can you find the sequence which creates this configuration from an
un-mixed net?

How many different configurations of this type are there?

Edge switcher 2 is the
sequence P WWW P W PP R P RRR PPP GGG R G

The effect on the unpermuted net is as follows:

This permutes 2 middle edge squares and 4 corner squares.

Can you generate a similar sequence which leaves the yellow and
green faces unchanged from an unpermuted net? How many different
permutations of this type are there?

Tip: You may wish to click on the
corners to black them out: this will help you to visualise the
effect on the edges .

Once you have a feel for the way the edges permute under these
operations the final challenge is to try to solve the net into this
configuration, in which all of the edges are in the correct places
(we are not interested in the location of the corners for this
challenge, so they have been blacked out in this picture)

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