#### Renslay

##### Member

I recently wrote an optimal solver in Matlab for the 2x2x2, and tested it with 3 different turn metric. Two of them are familiar, the HTM (half turn metric) and the QTM (quater turn metric), but I experienced a little with an other one, which I named TTM (tri-turn metric).

In HTM, you can do 9 movements in one step (U, U', U2, R, R', R2, F, F', F2).

In QTM, you can do 6 movements in one step (U, U', R, R', F, F').

In TTM, you can do 3 movements in one step (U, R, F).

So in TTM, the graph of the cube is directed, because for a scramble U (1 move), the inverse is U U U (3 moves).

(Note: technically, it is

We know the distribution table for HTM and QTM, even the combined of the two:

http://www.jaapsch.net/puzzles/cube2.htm

(For example, there are exactly 12 states where HTM = 7 and QTM = 13)

So, as everybody knows, every cube can be solved with 11 HTM or 14 QTM. But what about TTM? Now, here is the result:

FUN FACTS:

There is no cube state that is hardest (i.e., it is the further from the solved state) in both three metrics. Moreover, there is no cube state which is hardest in both QTM and TTM metric.

However, there are a few "almost hardest case".

Number of states where HTM = max, QTM = max, TTM = max-1:

60

Such an example is R U' F' R2 U R F2 R' U' R2 U', which has the solutions:

U R2 U R F2 R' U' R2 F U R' (11f*)

U U R U U R' U F' U R' F R U U (14q*)

U U U R R F R U F U F R U U U F U R (18t*)

Number of states where HTM = max, QTM = max-1, TTM = max:

132

Such an example is the Y PLL, like F2 U' R U R F2 R U' R' U R', which has the solutions:

R U' R U R' F2 R' U' R' U F2 (11f*)

R U' R U U F U' F' R' F F R' F (13q*) - well, actually the previous HTM solution is also an optimal QTM solution

U F U F U U R R F U U F U R R R U R R (19t*)

Later maybe I do a 3D matrix filled with bubbles (different radiuses for different amounts) showing the combined distribution table similar the 2D in Jaap's page.

In HTM, you can do 9 movements in one step (U, U', U2, R, R', R2, F, F', F2).

In QTM, you can do 6 movements in one step (U, U', R, R', F, F').

In TTM, you can do 3 movements in one step (U, R, F).

So in TTM, the graph of the cube is directed, because for a scramble U (1 move), the inverse is U U U (3 moves).

(Note: technically, it is

**not**a metric, because it's lack of symmetry...)We know the distribution table for HTM and QTM, even the combined of the two:

http://www.jaapsch.net/puzzles/cube2.htm

(For example, there are exactly 12 states where HTM = 7 and QTM = 13)

So, as everybody knows, every cube can be solved with 11 HTM or 14 QTM. But what about TTM? Now, here is the result:

Code:

```
0 moves: 1
1 moves: 3
2 moves: 9
3 moves: 27
4 moves: 78
5 moves: 216
6 moves: 583
7 moves: 1546
8 moves: 4035
9 moves: 10320
10 moves: 25824
11 moves: 62832
12 moves: 146322
13 moves: 321876
14 moves: 635632
15 moves: 988788
16 moves: 958176
17 moves: 450280
18 moves: 66420
19 moves: 1192
```

There is no cube state that is hardest (i.e., it is the further from the solved state) in both three metrics. Moreover, there is no cube state which is hardest in both QTM and TTM metric.

However, there are a few "almost hardest case".

Number of states where HTM = max, QTM = max, TTM = max-1:

60

Such an example is R U' F' R2 U R F2 R' U' R2 U', which has the solutions:

U R2 U R F2 R' U' R2 F U R' (11f*)

U U R U U R' U F' U R' F R U U (14q*)

U U U R R F R U F U F R U U U F U R (18t*)

Number of states where HTM = max, QTM = max-1, TTM = max:

132

Such an example is the Y PLL, like F2 U' R U R F2 R U' R' U R', which has the solutions:

R U' R U R' F2 R' U' R' U F2 (11f*)

R U' R U U F U' F' R' F F R' F (13q*) - well, actually the previous HTM solution is also an optimal QTM solution

U F U F U U R R F U U F U R R R U R R (19t*)

Later maybe I do a 3D matrix filled with bubbles (different radiuses for different amounts) showing the combined distribution table similar the 2D in Jaap's page.

Last edited: Sep 30, 2013