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

This problem follows on from the net puzzle problem

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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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