In the first part of this analysis, we’ve come up with the move counts for solving the first two layers in one step after solving EOLine. In this post we’re going to split that up in to two steps by solving two 3x2x1 blocks, one after the other.
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| Solving two 3x2x1 blocks | ||||
Again, for each step we will use a computer program to generate all possible cases for that step and see how many moves are required to optimally solve them.
Solving first 3x2x1 block
In this step, we’re trying to solve a 3x2x1 on one side of the cube next the Line we’ve already built, completing a full 3x2x2 block.
One could always solve the first 3x2x1 block on a fixed side. For instance, you could decide to always build the left 3x2x1 first so that the remainder of the cube can be solved using mostly, or even only, R and U moves.
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| Solving the left 3x2x1 block | ||
In this scenario, there are 2 corners that can be placed in 8 positions and 3 edges that can be placed in 10 positions, leading to a total of 32 x (8 x 7) x (10 x 9 x 8) = 362,880 cases.
| Moves* | # cases | Distribution | Cummulative |
| 0 | 1 | <0.01% | <0.01% |
| 1 | 3 | <0.01% | <0.01% |
| 2 | 9 | <0.01% | <0.01% |
| 3 | 48 | 0.01% | 0.02% |
| 4 | 182 | 0.05% | 0.07% |
| 5 | 769 | 0.21% | 0.28% |
| 6 | 2,938 | 0.81% | 1.09% |
| 7 | 10,783 | 2.97% | 4.06% |
| 8 | 35,330 | 9.74% | 13.80% |
| 9 | 89,070 | 24.55% | 38.34% |
| 10 | 136,195 | 37.53% | 75.87% |
| 11 | 77,681 | 21.41% | 97.28% |
| 12 | 9,769 | 2.69% | 99.97% |
| 13 | 102 | 0.03% | 100.00% |
| Weighted average: 9.69 moves* | |||
| *Quarter turn metric using only R, U and L faces | |||
Instead of blindly solving the first 3x2x1 block on the same side, one could solve the block on whichever side that has the shortest solution. This will give you more options and a lower move count, but is also harder to work out because you need to consider more pieces.
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| Solving the left or right 3x2x1 block | ||||
In this scenario all 4 corner pieces and 6 edge pieces of the first two layers are involved, which means there are 20,575,296,000 cases in total.
| Moves* | # cases | Distribution | Cummulative |
| 0 | 113,399 | <0.01% | <0.01% |
| 1 | 340,185 | <0.01% | <0.01% |
| 2 | 1,020,515 | <0.01% | 0.01% |
| 3 | 5,442,002 | 0.03% | 0.03% |
| 4 | 20,624,126 | 0.10% | 0.13% |
| 5 | 86,992,281 | 0.42% | 0.56% |
| 6 | 330,431,260 | 1.61% | 2.16% |
| 7 | 1,188,117,385 | 5.77% | 7.94% |
| 8 | 3,629,618,668 | 17.64% | 25.58% |
| 9 | 7,393,787,191 | 35.94% | 61.51% |
| 10 | 6,613,172,985 | 32.14% | 93.65% |
| 11 | 1,281,665,000 | 6.23% | 99.88% |
| 12 | 23,963,478 | 0.12% | >99.99% |
| 13 | 7,525 | <0.01% | 100.00% |
| Weighted average: 9.09 moves* | |||
| *Quarter turn metric using only R, U and L faces | |||
As expected this gives us a slightly lower move count.
Solving second 3x2x1 block
After solving the first 3x2x1 block we can complete the first two layers by solving another 3x2x1 block on the opposite side. This can be done by using just 2 faces.
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| Solving the second 3x2x1 block | ||
There are 2 corner pieces involved, which can be placed in 6 positions, and 3 edges pieces, which can be placed in 7 positions. Hence, the total number of cases for this step is 32 x (6 x 5) x (7 x 6 x 5) = 56,700.
| Moves* | # cases | Distribution | Cummulative |
| 0 | 1 | <0.01% | <0.01% |
| 1 | 3 | 0.01% | 0.01% |
| 2 | 9 | 0.02% | 0.02% |
| 3 | 27 | 0.05% | 0.07% |
| 4 | 73 | 0.13% | 0.20% |
| 5 | 215 | 0.38% | 0.58% |
| 6 | 614 | 1.08% | 1.66% |
| 7 | 1,730 | 3.05% | 4.71% |
| 8 | 4,374 | 7.71% | 12.43% |
| 9 | 9,620 | 16.97% | 29.39% |
| 10 | 17,082 | 30.13% | 59.52% |
| 11 | 17,005 | 29.99% | 89.51% |
| 12 | 5,593 | 9.86% | 99.38% |
| 13 | 344 | 0.61% | 99.98% |
| 14 | 10 | 0.02% | 100.00% |
| Weighted average: 10.03 moves* | |||
| *Quarter turn metric using only R and U faces | |||
The move count is slightly higher compared to solving the first 3x2x1 block, which can be explained throught the fact that we’re more limited in our movements with the 3x2x2 block being already there.
Thoughts
Let’s add up the results we’ve collected so far:
- Solving two 3x2x1 blocks in a fixed order averages 19.72 moves (= 9.69 + 10.03 moves)
- Solving two 3x2x1 blocks in any order averages 19.12 moves (= 9.09 + 10.03 moves)
- Solving the first two layers in one step averages 15.64 moves
Finding and solving 5 pieces at a time is certainly a lot more manageable than solving 10 pieces at a time and it only comes at the slight cost of ∼3.5-4 added moves.
Other blockbuilding oriented methods, like Petrus and Roux, also have similar steps involving the same amount of pieces although they don’t promote trying to work out an optimal solution for them. It is generally advised to build blocks in steps of 2 to 3 pieces at a time.
The next step
The move counts we have established so far are useful indicators of the potential of this method. However they’re still rather optimistic since we’re not following any real practical approach for building the blocks.
In the following posts we’re going to split up the building of the two 3x2x1 blocks even further. This is when we will start getting a good view of the typical move counts you can expect from a human solver who has mastered ZZF2L to a high degree.