## Sunday, March 22, 2015

### 3x5x7

The 3x5x7 is a beautiful and complex cuboid. At the time of this post, it is not yet mass-produced. However, I have no doubt that it soon will be. This puzzle is one that has been described as an ultimate shapeshifter: it shapeshifts on every axis. Something like a 3x3x4 does not shapeshift at all. A 3x3x5 shapeshifts on one axis. But the 3x5x7 shapeshifts on all axes. This puzzle provides hours of fun in different scrambles and solves. The solving technique is essentially the same no matter what scramble is produced. I consider it to be one of the harder puzzles I've conquered.

The Basic Plot
2. Reduce and Solve Edges
3. Reduce and Solve Corners
4. Solve Centers

First, identify the top and bottom (white and yellow) centers, and therefore the 4 middle layer centers (red,orange, blue and green on this puzzle). Place the middle layer edges in position and oriented correctly. These are the single large pieces, such as blue-red. They will have other pieces attached at this stage.

Next, flip any edges around the 3x3x3 white/yellow centers that need flipping. What we want is the central 3x3 square to be unbandaged. Therefore, if there are bandaged pieces there, we need to flip them. To do this, hold the edges requiring flipping at UF and UR, and perform

FR'F'R   U'RUR'.

Next, flatten the outer corners. To do this, use the corner piece series, or the corner piece series for cuboids, or their mirrors.

U R U' L' U R' U' L  or  U' L' U R  U' L U R'  (CPS)

(U R2 U' L2) x 2  or  (U' L2 U R2)    (CPS for cuboids)

Doing this will return the puzzle to a much more recognisable cuboid shape, although there is still plenty to do.

Next, solve the blue and green middle layer pieces. These are the small pieces between the blue/green centers and the outer middle layer edges (placed first thing). Do this by turning the position of the piece up to the top layer, then rotating the top face 180° and returning the piece in its position to the middle layer. Sometimes, the desired pieces need to be moved out of the red/orange middle layer positions (in the same way). The most difficult part of this step is making sure all the protruding parts of the puzzle are pointing in the same direction, to enable this step to be performed.

Note that we don't require the blue pieces to be on the blue side, only that blue/green pieces are on the blue/green side, and not on the red/orange side.

Finally, solve the red and orange middle layer pieces. These are the small pieces between the red/orange centers and the outer middle layer edges (placed first thing). Do this by turning the position of the piece up to the top layer, then rotating the top face 180° and returning the piece in its position to the middle layer. Almost certainly, you will need to flatten the protruding parts as you go.

Do this using the CPS for cuboids (U R2 U' L2) x 2.

Once this is done, the only thing possibly left to do will be flattening protruding parts. At that point, the puzzle is returned to cuboid shape.

2. Reduce and Solve Edges

The first and very simple part of solving the edges is to correctly place all of the middle layer. This involves simple turns and takes no more than 30 seconds.

Next, reduce the edges. There are two parts to this. The first part involves attaching the white/yellow edges to their white/yellow inner parts. To do this, use either the edge piece series (EPS) or the CPS for cuboids. The puzzle will shapeshift slightly while doing this but will return once all these edges are attached.

The second part involves grouping the edges into triplets. This is done just the same as for a 5x5x5 cube. For example, all three of the yellow/blue edges must be joined into an edge triplet. Similarly, all three of the white/red edges must be joined together. Hopefully, you'll end up with three edges remaining which can be cycled home.

However, you may find you have two edges remaining, with 4 individual edges involved in a 2+2 swap.  To deal with this, turn a 5x7 slice two turns. Then re-solve the blue/green middle layer edges. However, when doing this, make sure you keep the white/yellow edges attached to their inner parts. In other words, don't just turn the outer 3x5 layer. Completing this will either reduce all edges, or else return you to a state where they may be 3-cycled home.

If you find you have two individual edges remaining to swap, a different approach is required. To deal with this situation, turn a 5x7 slice one turn, so the puzzle becomes shapeshifted. Re-solve the blue/green middle layer pieces first. Then flatten protruding parts. Finally, re-reduce the edges until all are grouped and ready to place.

Placing the reduced edges is done just the same as for a 3x3x4 cuboid. We need to keep the middle layer solved as we do it. This should take no more than 2 minutes.

3. Reduce and Solve Corners

At this point, all edges will be placed, but the puzzle may be shapeshifted, due to not all corners being placed.

Begin by solving the inner corners. These are the red/orange pieces between the outer corners and reduced edge triplets. Hold the puzzle with the 5x7 face as Up, and perform the CPS for cuboids  (U R2 U' L2) x 2.

If you find you have 2 inner corners left to swap, you will need a different approach. To deal with this, turn a 3x5 slice one turn, so the puzzle becomes shapeshifted. Re-solve the red/orange middle layer pieces first. Then flatten protruding parts. Finally, re-solve the inner corners until all are placed correctly.

Now it's time to solve the outer corners. To do this, we'll use an algorithm which is a combination of the CPS and its mirror.

To cycle the UBL->DFL->DBR corners, use
(U R2 U' L2) x 2   F2 R2 F2    U      (U' L2 U R2) x 2     U'     F2 R2 F2

To cycle the UBR->DFR->DBL corners, use
(U' L2 U R2) x 2   F2 L2 F2    U'     (U R2 U' L2)  x 2     U     F2 L2 F2

You may find you have four corners remaining, all of which are in the same orbit. If this is the case, two more algoirhtms will finish the job.

If the four corners are placed such that 2 are in one orbit and the other 2 are in the other orbit, you will need to turn a 5x7 slice 2 turns. This is the same fix as earlier, except this time, when re-solving the blue/green middle layer edges, we deliberately want to separate the outer 3x5 face from its inner parts. From here, re-reduce edges and re-solve inner corners. Finally, re-solve outer corners and the job will be done.

3. Solve Centers

The centers are simple to solve. We'll use the same algorithm we've been using all the way through. This time, though, we'll carry it out using slice turns.

(U r2 U' l2)  x 2  will cycle both center-edges and center-corners, depending on which slices are turned.

And that's it. Your 3x5x7 is now solved. I trust this site has been helpful. If you have any questions or want some clarifications, please use the comments to do so.

#### 1 comment:

1. Hello! First of all I'd like to say your approach to the puzzles is the most interesting I've found (I just started cubing 9 months ago) - it's brilliant!
Anyway I just got myself a 2x4x6 and learned how to solve it using your youtube video. BUT the second time I got round to solving it I got stuck with just two outer corners needing swapping.
You don't seem to include a fix for that situation so I'm wondering if I'm misunderstanding something or doing something wrong...
Cheers. Will