by Richard Taylor : 2019-02-14

Fewest Moves Challenge (FMC for short) is an official WCA event where the objective is to solve a 3x3x3 cube in the most efficient way. Speed is not an issue, other than an over all time limit of one hour.

The first rule of FMC is "There is no best method for FMC". You have to try a lot of different techniques and use your experience and intuition to pursue the directions that look most promising for the scramble at hand.

Having said that, there are numerous techniques that you can assemble into a toolkit for FMC attempts. One such technique is Block Building.

What is a block? It is just a collection of adjacent pieces that are positioned and oriented correctly relative to each other.

This picture shows the smallest block, a single corner and edge piece joined together. As the whole cube is a 3x3x3 then this block is 2x1x1.

There is another way to get a 2x1x1 block and that is to join a centre piece with an edge. If you can join both these types of block together then you get a 2x2x1 block.

Notice that the four pieces in this block are only solved correctly on the white face. To extend to a 2x2x2 block we need another edge and two centres.

How many moves should it take to get this far? I have developed some software tools on github that can help. Lets take a standard WCA scramble and test it.

```
# SCRAMBLE=\
"L F2 B2 U' B2 L D L2 F' L2 F2 U2 D2 R2 D2 L2 F' U2 F2 R' B"
# bin/block222 "${SCRAMBLE}"
INFO: Looking for a solution with 1 turns.
INFO: Looking for a solution with 2 turns.
INFO: Looking for a solution with 3 turns.
INFO: Looking for a solution with 4 turns.
INFO: Looking for a solution with 5 turns.
INFO: Solving time: 0.962193 seconds.
Solutions:
B' R' U B L'
U' R' D2 R2 B2
U L' B' R' B
R2 B2 U2 F D
```

For this scramble the shortest "algorithm" to get a 2x2x2 block is 5 moves; and there are 4 different ways to do it. Which of those is the "best" depends on what you want to do next. Also, you may be able to make additional blocks at the same time as a 2x2x2 with one or two extra moves.

But this tool allows you to see a variety of efficient ways to make a block. Studying these can help your understanding of how blocks come together.

How typical is a 5-move solution?

```
# bin/distribution222
Trials: 100000
Frequency distribution:
0 : 4 : 0.004 %
1 : 26 : 0.026 %
2 : 292 : 0.292 %
3 : 2576 : 2.576 %
4 : 19652 : 19.652 %
5 : 59961 : 59.961 %
6 : 17474 : 17.474 %
7 : 15 : 0.015 %
```

This random sample of 100,000 cube states suggests about 60% of scrambles can be solved in 5 moves. The worst case looks like 7 moves, but that is pretty rare, and about as likely as 0 moves.

Given a 2x2x2 block there are 3 dimensions in which it can be extended into a 3x2x2 block. Lets take the previous scramble and one of the 2x2x2 solutions to see what we get when we try and extend to 3x2x2.

```
# SCRAMBLE=\
"L F2 B2 U' B2 L D L2 F' L2 F2 U2 D2 R2 D2 L2 F' U2 F2 R' B"
# bin/block322 "${SCRAMBLE} B' R' U B L'"
INFO: Looking for a solution with 1 turns.
INFO: Looking for a solution with 2 turns.
INFO: Looking for a solution with 3 turns.
INFO: Looking for a solution with 4 turns.
INFO: Looking for a solution with 5 turns.
INFO: Solving time: 0.976281 seconds.
Solutions:
D R F2 R2 D
```

Again the shortest solution is 5 moves. But this time there is only one way.

How likely is a 5-move extension? Note that we are looking for solutions that extend an existing 2x2x2 block, not ones that create a 3x2x2 directly from a random position. Going straight to a 3x2x2 is probably beyond most people (and is certainly beyond me).

```
# bin/distribution322
Trials: 100000
Frequency distribution:
0 : 37 : 0.037 %
1 : 164 : 0.164 %
2 : 557 : 0.557 %
3 : 3754 : 3.754 %
4 : 14360 : 14.360 %
5 : 41094 : 41.094 %
6 : 36823 : 36.823 %
7 : 3210 : 3.210 %
8 : 1 : 0.001 %
```

The most likely shortest solution is 5 moves. But only by a small margin over 6 moves. And 2x2x2 blocks that need at least 7 moves to extend into a 3x2x2 make up about 3% of all the possibilities.

The next biggest cuboid block after 3x2x2 is 3x3x2 which is a whole two layers solved (often called F2L, for First 2 Layers). That is a big jump. And probably not all that useful as it doesn't give much room for manoeuvre when solving the remaining pieces.

Instead it is more common for an FMC solve to next add a 2x2x1 block to the 3x2x2 to make F2L-1, which is First 2 Layers minus 1 edge-corner pair.

There are 4 different 2x2x1 blocks that can be added to a 3x2x2 to make it into F2L-1. Lets take our previous 3x2x2 solution and see what we get when we try and extend to F2L-1.

```
# bin/f2L-1 "${SCRAMBLE} B' R' U B L' D R F2 R2 D"
INFO: Looking for a solution with 1 turns.
INFO: Looking for a solution with 2 turns.
INFO: Looking for a solution with 3 turns.
INFO: Looking for a solution with 4 turns.
INFO: Looking for a solution with 5 turns.
INFO: Looking for a solution with 6 turns.
INFO: Looking for a solution with 7 turns.
INFO: Solving time: 25.716 seconds.
Solutions:
F2 R F' R2 D R' D'
F R' F2 R2 U' R U
U D' F2 U' F' D R'
```

How typical is a 7 move extension from 3x2x2 to F2L-1?

```
# bin/distributionf2l-1
Trials: 100000
Frequency distribution:
0 : 120 : 0.120 %
1 : 390 : 0.390 %
2 : 810 : 0.810 %
3 : 3398 : 3.398 %
4 : 11025 : 11.025 %
5 : 29693 : 29.693 %
6 : 38893 : 38.893 %
7 : 15309 : 15.309 %
8 : 362 : 0.362 %
```

It looks like 7 is worse than average; and almost the worst case. But remember that there were 4 different ways to get our first 2x2x2 block in 5 moves... what if we try one of the others?

```
# bin/block322 "${SCRAMBLE} U L' B' R' B"
INFO: Looking for a solution with 1 turns.
INFO: Looking for a solution with 2 turns.
INFO: Looking for a solution with 3 turns.
INFO: Looking for a solution with 4 turns.
INFO: Solving time: 0.010598 seconds.
Solutions:
L' F2 L D2
# bin/f2L-1 "${SCRAMBLE} U L' B' R' B L' F2 L D2"
INFO: Looking for a solution with 1 turns.
INFO: Looking for a solution with 2 turns.
INFO: Looking for a solution with 3 turns.
INFO: Looking for a solution with 4 turns.
INFO: Looking for a solution with 5 turns.
INFO: Solving time: 0.139242 seconds.
Solutions:
D R D' R F'
U F U' R F'
```

So we can actually get to F2L-1 in 5+4+5 = 14 moves instead. Even better, the first 5 move solution for F2L-1 gives an additional 2x2x1 block on the "top" face as a bonus!

Block Building is a very flexible strategy for FMC. It gives you a few achievable way-points to aim for without restricting how to get to them or what to do next.

The cost of that flexibility is that it can be hard to know whether your progress so far can be extended further. Sometimes what seems to be a great 3x2x2 block just can't be made into a good F2L-1 in a small number of moves.