I think NxNxN cube solvers would certainly require a thread of their own

Applying brute force is probably limited to depth 12, depending on applied heuristic, size of basic moves set, cube size and computer capability.

Bi-directional search together with proper heuristic, may be a way of extending this limit.

By limiting twists to half-turns only, it might be possible to find a subset of short sub-optimal solutions.

I wrote a small javascript program just to get an idea of how many states would be needed at depth 6.

Using brute force, 8,067,360 distinct cube states were found at depth 6 for a 5x cube and from a set of 15 basic moves:
{NR2, NU2, NF2, NL2, ND2, NB2, VR2, VU2, VF2, VL2, VD2, VB2, WR2, WU2, WF2}

Notice that in bi-dir search, hash tables should be used to lower computing time. Symmetry + inversion would also lower memory requirements.

A first solver wouldn't need to give an exact solution but at least the shortest algorithm closest to the goal position. Setup + symmetry + inversion could then be used to try to find the exact solution.

Present day workstations pack a lot of power (8-CPU core) and RAM and I think a very basic 5x bi-dir (half-turns) solver could be done if good programming tricks are applied.

These are just a few thoughts and more work remains to be done.

EDIT:
I'm probably too serious here, and I would agree that it's definitely more fun to make our brain work instead of relying too much on computers

Using multi CPU and GPU for cube solving is hardly feasible for 5x5x5, except for trivial short sequences.

Back to patterns. I had a go at making grids for 7x7x7. What i have so far is this:

1. TD S2R- S3U S2F MU- S2F- MU S2F T4B2 N3D2 MF2 N3U2 TB2 N3F2 N3L2 N3B2 N3F2 N3R2 TU- TD- N3L2 N3R2 TD

2. D- WF2 WU2 R2 WF2 WU2 L2 D

This combines to:

7X: 6 Tartans (30 btm).

Hope to finish the 7X 6 Grids shortly (Possibly by some other approach ...)

Per

@ Per
Your solver is awesome! More powerful than any super computer!
I also like your Tartans.

Applied on a 7x7x7, the 'Outlined Heart' would look like this:

7x: 4 Outlined Hearts
MF2 MR2 WR2 (D WF2 D2 WF2 D) WR V3U' WR V3U2 WR' V3U' WR V3U2
NF2 NR2 N3D2 NR2 NF2 N3D2
M3R' ND M3R ND2 M3R' ND M3R MF2 MR2 (31 btm)

Per's Tartan algorithm being already darn short, some interesting variations can be obtained by replacing corner permutation with edge/center permutation to add/delete some crossings:

7x: Variation 1 (32 btm)

7x: Variation 2 (32 btm)

Shorter versions of the two following should certainly exist:
7x: Variation 3 (36 btm)

7x: 4 Small Flowers (14 btm)

A very slight improvement (by insertion):
7x: 4 Outlined Hearts (30 m)

I made a vast improvement to the variation 3 7X 6 Tartan pattern:

7X: 6 Tartans Variation 3 (22 btm).

1. T3D- MF2 MU2 T3R2 MF2 (#) MU2 T3L2 T3D
2. TD- M3F2 M3U2 TR2 M3F2 M3U2 TL2 TD

Insert MF M3U2 MF- SU MF M3U2 MF- SU- as indicated into first part ...

Quite easy construction

Per

A very slight improvement is obtained by using (MR2 M2F2 N3R2 MF2 M2R2 N3F2) directly:
7x: Variation 3 (21 btm)

By not including the 2 U/D 'flowers' in your algorithm, 8 corner loops are obtained
7x: 8 Corner Loops (15 btm)

Continuing further with 'flowers':
7x: 2 Small Flowers (6 btm)

7x: 6 Small Flowers (18 btm)

Hi

Yet again improved:

7X: 6 Tartans Variation 3 (19 btm).

Beat that!

Per

Why not?
Challenging but quite interesting

Your algorithm is based on an optimal 8-move corner permutation on a 3x cube, scaled to a 7x cube, applied 2 times plus a 'flower' pattern:

Part 1:
T3D' (MF2 MU2 T3R2 M2R2 N3F2 MR2 M2F2 N3R2 MU2 T3L2) T3D
Part 2:
TD' (M3F2 M3U2 TR2 M3F2 M3U2 TL2) TD

Part 1 can be further improved to give 1 move less:
T3D' (M2R2 N3F2 MR2 M2F2 N3R2 MF2) (MF2 N3R2 TR2 MU2 MF2 T3L2 MU2) T3D
T3D' (M2R2 N3F2 MR2 M2F2 TR2 MU2 MF2 T3L2 MU2) T3D

7x: 6 Tartans (Variation 3) (18 btm)

I like posting improvements

7X: 6 Tartans (Variaton 2) (29 btm).

I bet better could still be found, but not "trivially"

Per

New simple contribution:

5X: 8 Corner Loops (Order 3) (30 btm).

Actually this is nothing more than a trivial 8-mover applied 4 times. Again it's only my first naive attempt ...

==EDIT==

A more thoughtful approach:

5X: 8 Corner Loops (Order 3) (24 btm).

Per

A very slight improvement, gained by insertion - I agree that searching for a shorter alg would be more involved

7x: 6 Tartans (Variation 2) (28 btm)

Feeling "creative today

7X: 6 Small "Flowers" (10 btm).

Only 4 additional turns to extend from 2 to 6 faces

Per

Prepending the previous pattern with MD2 MR2 MF2 gives:

12 Extended Eedges (10 btm).

Per

A slight improvement again you could easily have found by yourself:
5x: 8 Corner Loops (Order 3) (23 btm)

Based on your algs of rings and extended edges, here are connected rings:
7x: 6 Connected Rings (14 btm)

More to come later on...

One easy Tartan pattern variation was missing:

7X: 6 Tartans (Variation 4) (22 btm).

Per

Today's last one...

Centers:
N3F' MR N3F MR'
Rectangles:
WR WF' M3R' VF NR' NL V3B'
Insertion:
WR WF' M3R' MR N3F MR' NF NR' NL V3B'

7x: 6 Handbags (18 btm)

A Twisted Master Pentultimate using a block-cycling technique I've been working on (with the help of Daniel Kwan):



The pattern is produce with 3 separate block-cycling routines:
[D&2, G, D'&2, F', D&2, G', D'&2, F, B'&2]x4, [I, K'2, I', B, I, K2, I', B]x3, [L, B'2, L', K', L, B2, L', K']x3 (84 moves).

I finally completed this new pattern:

7X: 6 Grids (51 btm).

Quite hard work to finish off the last centers after 3 repeated "corner" sequences.
Yet, i'm sure still better could be found by a different approach

By solving this pattern instead of trying to set it up i improved it drastically:

7X: 6 Grids (29 btm).

Pretty optimal now i would guess ...

Per

This is already short indeed

Re-using the last part of your algorithm (NR2 NF2 MF2 NB2)2:
Pattern 1
7x: Pattern 1 (14 btm)

Two simple new patterns (yet unoptimized):
Pattern 2
Edges
SU SF2 N3R2 SF2 N3R2 SU' N3R2 SF N3L2 SU2 N3L2 SU2 SF' N3L2
Centers
N3L2 N3R2 N3U2 N3F2 N3R2 N3L2 N3B2 N3U2 N3F2 N3B2
7x: Pattern 2 (22 btm)

Pattern 3
Pons
NR2 MR2 NL2 NF2 MF2 NB2 NU2 MD2 ND2
Lines
N3D2 N3F2 N3L2 N3R2 N3B2 N3U2
7x: Pattern 3 (13 btm)

First attempt at a new pattern:

7X: 6 Grids (Order3) (56 btm).

Of course this could be easily improved:

7X: 6 Grids (Order 3) (37 btm).

Per

@ bmenrigh:
Hey, that's an awesome pattern!
It's also great that you posted the algorithm as well.


@ glazik and perfredlund:
These are further optimized algs:

7x: 6 Connected Rings
N3F2 N3R2 N3L2 N3B2 N3U2 N3F2 N3B2 N3R2 N3L2 M2U2 MR2 MF2 (12 btm)

7x: 6 Grids (Order 3)
SF (D NL2 MR2 NR2 U' D' NL2 MR2 NR2 U SR U' NL2 MR2 NR2 U D NL2 MR2 NR2 D' SR') SU SR SF
N3B' N3F N3U' N3D N3F' N3B N3U N3D' (34 btm)


... and of course, this Grid version is impossible:
7x: 6 Grids (Order 2)
NF2 NR2 MF2 NU MD' ND' MR2 NU MD' ND' NL2 NB2 (u,d) (12 btm)

Merry Christmas and the Best Wishes for a Happy New Year to everyone!

4 Christmas Trees
T3F2 MR2 T3F2 T3R2 MF2 SF2 L2 NU L2 SF2 R2 ND NF2 N3U2 VD2 NF2 NR2 NU2 MD2 N3R2 N3F2 V5U2 N3F2 (23 btm)

Happy 25th of December to all!

Fröhliches Weihnachten

I also like this grid variation:

6 Dotted Grids (Order 2) (12 btm).

Per

New contribution:

7X: 2 Chessboard Faces (18 btm).

Per

@ Per:
You certainly forget to write MR2 T3L2 as T4L2. Your algorithm actually only requires 17 moves:

7x: 2 Chessboards
N3F2 F2 NL2 MR2 NR2 N3F2 F2 T4L2 NR2 NB2 MF2 NF2 L2 N3L2 NF2 MF2 NB2 (17 btm)


Approaches to get additional chessboards:

7x: 4 Chessboards
NB2 MF2 NF2 TU2 MD2 ND2 NB2 MF2 NF2 NL2 MR2 NR2 U' D' NB2 MF2 NF2 SD (18 btm)

7x: 6 Chessboards
T3F T3B2 T3R' T3D2 T3B T3R S3D' S3R T3D' T3F' T3R2 T3D
N3F2 N3B' TD' TR2 TF TD S2R' S2D TR' TB' TD2 TR
NF' NB2 R' D2 B R SD' SR D' F' R2 D F2 B'
VF' VB NR NL' N3F N3B' WR' M3R2 MR' NF NB' N3R' N3L (51 btm)

The first algorithm can be further improved by insertion

Insert NL2 MR2 NR2 N3B2 B2 NR2 MR2 NL2 B2 N3B2 at *:
N3L2 L2 * NB2 MF2 NF2 L2 N3L2 NF2 MF2 NB2
7x: 2 Chessboards (14 btm)

@ glazik:
Impressiv! The algorithm is probably optimal now.


I found a new Corner Loops variation:

The algorithm is composed of 2 parts:
MR WD2 MR MF MD WF2 MD WR2 MF (edges)
NF2 NR2 NL2 NB2 NU2 NF2 NB2 NR2 NL2 NU2 (centers)

For the final algorithm, part 2 was insertet like this:
MR WD2 MR MF MD WF2 (NF2 NR2 NL2 NB2 NU2 NF2 NB2 NR2 NL2 NU2) MD WR2 MF

5x: 8 Corner Loops
MR WD2 MR MF MD VB2 NR2 NL2 NB2 NU2 NF2 NB2 NR2 NL2 NU2 MD WR2 MF (18 btm)


or the same on a 7x7x7:

7x: 8 Corner Loops
MR M3D2 MR MF MD M2B2 N3R2 N3L2 N3B2 N3U2 N3F2 N3B2 N3R2 N3L2 N3U2 MD M3R2 MF (18 btm)

Further improvement:

7X: 2 Chessboard Faces (10 btm).

This is surely optimal, although i cannot prove it

Per

One last progression tonight:

5X: 8 Corner Loops (16 btm).

Too lazy to write out the 7X version

Per

Here I go again:

4 Chessboards (15 btm).

I started with : NU2 MU2 ND2 T3F2 NR2 MR2 NL2 (1) T3F2 T3R2 (2) NF2 MF2 NB2 T3R2

then inserted

(1) NL2 MF2 NL2 MF2

and

(2) MR2 NF2 MR2 NF2

cancelling 6 turns (13+8-6=15)

Per

The '7x: 6 Chessboards' pattern has been improved
I guess shorter solutions could still be found.

Edges
WF [U' NR2 MR2 NL2 U D NL2 MR2 NR2 D', SR'] WF'
Centers
WF NU' MU' ND N3B N3F' N3U2 WU N3D2 NF' MF' NB N3U N3D'

7x: 6 Chessboards (Order 3) (36 btm)

Two additional (unoptimized) patterns:

7x: 6 Distorded Checkerboards (Order 3) (36 btm)

7x: 6 Checkered Diamonds (Order 3) (28 btm)

Minor tweak:

Insert

(1) NR- MR- NL N3U N3D- WR N3R2 N3L2 NU MU ND- N3R N3L- WD

into

SF (D NL2 MR2 NR2 U- D- NL2 MR2 NR2 U SR U- NL2 MR2 NR2 U D NL2 MR2 NR2 (1) D- SR-) SF-

7X: 6 Chessboard Faces (Order 3) (35 btm).

Was hoping for one more cancelled turn ... Oh well, maybe Glazik will find out how

Per

It seems that 35 btm would be minimal for edges + centers...

This is a hashstag-like pattern (needs to be optimized):
7x: 6 Hashtags (Order 3) (12 btm)

It IS optimal and i guess you know it Nice pattern

Per

Some of these are improvements and some are new:

7x: 8 Small Crosses new:
NR2 NL2 NB2 T3F2 T3R2 NB2 NF2 NL2 R2 N3R2 N3F2 F2 (12 btm)

7x: 16 Small Crosses (Order 2) new:
NR2 NL2 N3D2 D2 NR2 NL2 NF2 NB2 D2 N3D2 NF2 NB2
MR2 ND2 MR2 MF2 NU2 MF2 (18 btm)

7x: 16 Small Crosses (Order 4) improvement:
U2 N3U2 NL2 NR2 U' N3U' NL2 NR2 T3U' T3D' MR2 NU MR2 NU ND MR2 ND' NL2 MR2 NR2 D N3D NR2 NL2 (24 btm)

7x: 24 Small Crosses (Order 3) improvement:
U2 N3U2 NL2 NR2 U' N3U' NL2 NR2 T3U' T3D' MR2 NU MR2 NU ND MR2 ND' NL2 MR2 NR2 D N3D NR2 NL2
F2 N3F2 NU2 ND2 F' N3F' NU2 ND2 T3F' T3B' MD2 NF MD2 NB NF MD2 NB' NU2 MD2 ND2 B N3B NU2 ND2 (48 btm)

7x: 24 Small Crosses (Order 6) new:
NB2 NF2 L2 N3L2 NB2 NF2 NU2 ND2 L2 N3L2 NU2 ND2
MF2 NL2 MF2 MU2 NR2 MU2
NL2 NR2 D' N3D' NL2 MR2 NR2 ND MR2 NU' ND' MR2 NU' MR2 T3U T3D NL2 NR2 U N3U NL2 NR2 U2 N3U2 (42 btm)

Two algorithms have been improved and a third added.
As usual, those should still be improvable.

1.
Insert (NF2 NB2 NL2 NR2 N3U2 U2 NL2 NR2 NB2 NF2 U2 N3U2) at *:
MR2 ND2 MR2 MF2 * NU2 MF2
7x: 16 Small Crosses (Order 2) (16 btm)

2.
Centers:
[N3F N3B', NU' ND]
[NF NB', N3U' N3D]
Insertion:
WF MF' ND NU' N3F' N3B NU N3U' ND' N3D NB NF' N3D' N3U
Edges:
WF [U' NR2 NL2 U D NL2 NR2 D', SR'] WF'
7x: 24 Small Crosses (Order 3) (32 btm)

3.
Centers:
WF MF' ND NU' N3F' N3B NU N3U' ND' N3D NB NF' N3D' N3U
Edges:
WF [U' MR2 U D MR2 D', SR'] WF'
7x: 6 Diamonds (Order 3) (28 btm)

My latest contibutions for 2012...

From Per's algorithms scaled up to a 7x7x7 cube:
7x: 8 Corner Loops (Order 2) (16 btm)
7x: 6 Small Rings (f,b)(r,l)(u,d) (Order 2) (10 btm)

Three new/improved patterns:
1. 14 Rings
14 = 8 + 6 is the maximal number of 'isolated' ring-like patterns on a 7x cube.
8 Corner Loops:
NB2 NF2 ND2 NU2 WR2 M3U M3R2 M3U' WU WF2 WU' V4R2 NB2 M3R2 V4F2 NR2
6 Small Rings:
M2R2 N3D2 MF MR2 MF' M2U2 N3R2 M3F M3U2 M3F'
7x: 14 Rings (Order 2) (25 btm)

2. 6 Small Flowers + 6 Grids
6 Small Flowers:
M2F2 M2R2 MU2 MF2 N3R2 M2F2 M2U2 MF2 MR2 N3U2
6 Grids:
NU2 NF2 NR2 NL2 NB2 ND2
7x: 6 Small Flowers + 6 Grids (15 btm)

3. 24 Crosses
6 Grids:
NU2 NF2 NR2 NL2 NB2 ND2
6 flowers:
ND2 NB2 MF2 NR2 MU2 MF2 NR2 MR2 NB2 MF2 ND2 MF2 MR2 MU2
7x: 24 Crosses (Order 2) (16 btm)

Happy new Year 2013 to all!

Nice contributions and improvements Glazik.

Today i have a new pattern to contribute also:

6 Hashtags (Order 2) (16 btm).

Per

Hi Per,

Nice pattern too!

The following is a simple variation of the hashtag pattern, with some centers removed.
Patterns on faces L,D,B are obtained from those of faces F,R,U by point reflection.
Centers:
VF N3D NF' N3B' ND N3B VD' NU' N3F' NB' N3U' NB VU
Small Rings:
M3R M3F' M3R' N3F N3B' MR MF MR'
Insert centers at *
M3R M3F' M3R' N3F N3B' MR MF * MR'
7x: 6 Four-Blade Rings (Order 3) (20 btm)

Hi Glazik !

I am surprised you did not find this easy breakdown:

7X: 6 Four-Blade Rings (Order 3) (16 btm).

Per

Oops...I completely missed this shorter algorithm!

Yet another simple pattern:
NR2 NU2 NF2 (MR2 MF2 MU2) NF2 NU2 NR2
M3R2 M3U2 M3F2
7x: 24 Ys (Order 2) (11 btm)

Hmm. I can see 24, 4 on each of the 6 faces of the cube

Per

This algorithm can be further improved:

7x: 6 Four-Blade Rings (Order 3)
V4U V3L N3D NL' MR V3U' N3L'
V4D' V3R' N3U- NR MR V3D N3R (14 btm)

I guess optimizations of the alogrithms below should still be possible:

7x: 2 X
M3F2 VR2 MF2 M2R2 M2F2 MR2 M2B2 NL2 NF2 NR2 NL2 F2 SR2 F2 NB2 R2 SF2 R2 (18 btm)

7x: 6 X (Order 2)
M3L NU2 ND2 M3L
M3U NF2 NB2 M3U
M3F NR2 NL2 M3F
MF2 M2R2 M2D2 N3F2 MR2 MD2 M2F2 M2D2 N3R2 MD2 (22 btm)

A very light improvement (by insertion):
M3L NU2 ND2 M3L
M3U NF2 NB2 M3U
M3F NR2 M2D2 M2F2 MR2 MU2 N3F2 M2D2 M2L2 MU2 MF2 VL2 M3F
7x: 6 Xs (Order 2) (21 btm)

Same process applied to the following:
M3L NU2 ND2 M3L
M3U NF2 NB2 M3U
M3F NL2 NU2 NF2 MU2 MF2 MR2 NF2 NU2 NR2 M3F
7x: 6 Uneven Checkerboards (Order 2) (19 btm)

Shorter versions should certainly exist

Easy new contribution:

7X: 6 Wide Yahtzee 5's (order 2) (10 btm).

Per

Ane one more:

7X: 6 Prison Windows (Order 2) (12 btm).

Per

And finally:

7X: Sparse Grids (Order 2) (14 btm).

Unfortunately it cannot be "scaled down" to 5X.

Per