# Message #1761

From: schuma <mananself@gmail.com>

Subject: Notes on 3^4 alt, 2x2x3x3 and 2x3x4x5

Date: Wed, 08 Jun 2011 02:31:00 -0000

Hi everyone,

In the last three days I solved 3^4 alt, 2x2x3x3 and 2x3x4x5 on MPUlt. Actually before that I tried the 120-cell on MPUlt. After spending 3 hours on it, I decided I was not patient enough to do it. So I switched to small puzzles. The common property that three puzzles share is that they are all "subgroups" of tesseract puzzles. However, I have quite different experience solving them.

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- 3^4 alt (turning around 2C by 180 deg and 4C by 120 deg)

When I first thought of this puzzle, I was imagining the 180-deg-only Rubik’s Cube. Please try to scramble and solve a Rubik’s Cube using only 180 deg turns. It’s pretty simple and very different from the common Rubik’s Cube. 3^4 alt, however, is not that different from 3^4.

2C pieces behave in the same way as in 3^4, except odd permutations are forbidden. Flipping two 2C’s is possible.

Two 3C pieces can be rotated simultaneously (the permutation of stickers of each piece is a 3-cycle), but not flipped ((a,b,c)->(b,a,c) is impossible) in place. Note that in 3^4 the latter thing is possible.

4C vertices belong to two orbits. Before solving it I only prepared the algorithms for rotating two 4C pieces in the same orbit (the permutation of the stickers on each 4C is a 3-cycle). But I met the situation that I needed to rotate two 4C in different orbits. It was a surprise and I had to re-solve a large fraction of the puzzle.

Overall, 3^4 alt is quite similar to 3^4.

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- 2x2x3x3 (turning around square by 90 deg and rectangles by 180 deg)

This puzzle is similar to the duoprisms in MC4D. The cells are of two kinds: 2x3x3 cells and 2x2x3 cells. Stickers in one type of cells never visit the other type of cells. The above properties are shared with all the duoprisms.

Compared with the duoprims in MC4D, 2x2x3x3 is a really small puzzle. Once the algorithms are ready, it doesn’t take long at all to solve it.

———-

- 2x3x4x5 (180-deg only)

This puzzle is truly similar to the 180-deg only 3D puzzles: each sticker can only stay in the original cell or the antipodal cell.

The direct analog in 3D should be 2x3x4 and 3x4x5. I have solved 2x3x4 and 3x4x5 on a simulator. So I pulled out my notes for them. What I found was pretty vague, like, "use algorithms like [R,u]x3 cleverly". I don’t have a systematic analysis for this kind of puzzle. Usually I just try my best to solve one or two pieces, and sometimes magically the others fall into the correct slots. It seems like many pieces are strongly correlated. But I don’t understand the relation between pieces. Sometimes I solve centers first and sometimes corners first.

So I had to boldly go to 2x3x4x5 and hoped magic happened again. I was not that lucky.

It took me a long long time to solve it. The method was again very heuristic. At the end I was solving 3C pieces. Once I saw a situation, I had to look for algorithms ad hoc. Application of macros is limited in this puzzle, because a macro recorded on a, say, 2x3x4 cell cannot be used on a 2x3x5 cell. So I alternated between a testing log file and a solving log file several times.

## To give you an idea how frustrated I was, let me tell you this: just now I tried to revisit 2x3x4x5. But after two minutes I was lost again. I didn’t know what to do, because I’ve never found a systematic solution.

This is my story of solving the three "subgroups" of the cube. 3^4 alt is very similar to 3^4; 2x2x3x3 is basically a duoprism; 2x3x4x5 is solved heuristically, with pain.

Nan