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Here's a simple 2x2x3 cuboid solving method I came up with while solving the QiYi one for the first time. It only uses one or two algorithms and is square-1/Ortega inspired (although I can't remember how to solve a sq1 now). The cuboid is always held vertically with left hand holding middle layer.

1. Solve middle layer using R2 and F2 moves. This should be easy and doable in at most 3 moves.

2. Solve white and yellow top and bottom faces (ignoring corner permutation) using only R2 moves. If you have ever solved 4x4 centers the idea is very similar.

3. Now comes the main algorithm: (R2 D' R2 D)2 R2. This swaps top face left corners and bottom face upper left and bottom right corners. It also swaps the middle right edges which can be flipped again with another application of the algorithm or with (R2 U2)3. If the entire middle layer seems flipped then also do that on left side: (L2 U2)3.

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Bottom

Call this algorithm A. By performing (A, some combination of U and D, A, some combination of U and D,) many different results can be achieved.
For example, A U A U' does an Aa perm on the top layer.

The method isn't very efficient but it's good enough to solve for fun. The same method can probably be used for larger 2x2xn cuboids.

Here's a simple 2x2x3 cuboid solving method I came up with while solving the QiYi one for the first time. It only uses one or two algorithms and is square-1/Ortega inspired (although I can't remember how to solve a sq1 now). The cuboid is always held vertically with left hand holding middle layer.

1. Solve middle layer using R2 and F2 moves. This should be easy and doable in at most 3 moves.

2. Solve white and yellow top and bottom faces (ignoring corner permutation) using only R2 moves. If you have ever solved 4x4 centers the idea is very similar.

3. Now comes the main algorithm: (R2 D' R2 D)2 R2. This swaps top face left corners and bottom face upper left and bottom right corners. It also swaps the middle right edges which can be flipped again with another application of the algorithm or with (R2 U2)3. If the entire middle layer seems flipped then also do that on left side: (L2 U2)3.

Call this algorithm A. By performing (A, some combination of U and D, A, some combination of U and D,) many different results can be achieved.
For example, A U A U' does an Aa perm on the top layer.

Yeah, this is pretty much what everyone uses, I Prefer to use Square-1 Cp algs since my execution is extremely fast on those to begin with. It's a very fun puzzle, probably my 3rd favorite non-wca event.

Yeah, this is pretty much what everyone uses, I Prefer to use Square-1 Cp algs since my execution is extremely fast on those to begin with. It's a very fun puzzle, probably my 3rd favorite non-wca event.

I saw some others that used a T perm or something to swap two edges, but there are some minor benefits to only using R2 moves (not as move efficient tho)
IDK any square-1 algs but my guess is that they might need extra care to keep the cubic shape.

I saw some others that used a T perm or something to swap two edges, but there are some minor benefits to only using R2 moves (not as move efficient tho)
IDK any square-1 algs but my guess is that they might need extra care to keep the cubic shape.

Not really, most square-1 algs are actually more move efficient than your average 3x3 alg, U perms on square-1 are only 8 slices compared to the regular 11 move RU U perms. Square-1 algs are just as efficient as they normally are even if they go out of cube shape. JJ is only 8 moves if you use HTM, not to mention they are extremely ergonomic because they are RUD instead of a possible RUF or RUDF alg that would be more move efficient but they would suck to execute because F2 is bad from the home grip and bad in general due to the shape changing during a double flick. Also, your "A" Algorithm is actually NJ on square-1. Actually you would only need to know 5 square-1 algorithms (J, N, JJ, NJ, NN) for a 1-alg PBL, so if you want to get fast on 2x2x3 it's worth learning 5 relatively short algorithms.

The reason why I use PBL on 2x2 instead of CP is because of the lack of a middle layer to hold on to while doing RUD, so RUF is better in that case, but on 2x2x3 RUD is definitely better than RUF or RUB in every case except for maybe N/N but that requires a rotation for it to be fast.