# 3x3x3x3¶

The 3x3x3x3 (normally referred to as the 3^{4}) is a 4-dimensional twisty puzzle in the shape of a hypercube that has 2 cuts along each of the 4 axes. It is a direct higher dimensional analogy of the 3x3x3 Rubik’s Cube.

The 3^{4} is recommended as one’s first introduction to solving higher dimensional puzzles.

## Pieces¶

The 3^{4} has 81 hypercubies, of which 72 are movable. It has 8 1c, 24 2c, 32 3c, and 16 4c pieces. The 3c pieces can be rotated in 6 different ways, and the 4c pieces in 12 different ways.

## Turning¶

Each turn of the 3^{4} is a rotation of one of its cubic cells (which can be oriented in any of 24 orientations of a cube). A normal move disturbs 8 4c, 12 3c, and 4 2c pieces.

## History¶

The 3^{4} has always been the main focus of the community. The first solve was done by Don Hatch in ~1988 using MC4D. After that, the Hall of Fame gradually began to increase in size. It eventually closed on December 7th, 2022 due to it reaching 500 solvers.

## Permutations¶

The 16 4cs of the 3x3x3x3 can be placed in any even permutation. Each 4c can be placed in \(\frac{4!}{2}\) orientations, except the last, which can be placed in only \(4\) orientations, due to the existence of monoflip. The 32 3cs can be in any permutation, and can have \(3!\) orientations, except the last, which can only have \(3\). The 24 2cs can be in any permutation, and can have \(2!\) orientations, except the last, which is determined by the others. However, the permutation parity of the 2cs and 3cs are linked. Thus, the number of permutations of the puzzle is \(\left[\frac{16!}{2} \cdot \left(\frac{4!}{2}\right)^{15} \cdot 4\right] \cdot \left[32! \cdot 3!^{31} \cdot 3\right] \cdot \left[\frac{24!}{2} \cdot 2!^{23}\right] \approx 1.76 \cdot 10^{120}.\)

## Speedsolving¶

(See the leaderboards for the current records)

There were a few hypercube speedsolving competitions during the early days of hypercubing, with the controversy at the time being if they were to allow macros or not. One of the first competitions happened in 2010, with the fastest time being 15:57 by Mateusz Burnicki (using prepared macros)
Some contestants recieved a t-shirt with an image of the 3^{4} in MC4D on it.

In mid 2017, The speedsolving scene was single handedly revived by Tetrian22, lowering his best from 37:07 to 10:11 in just under 10 months.

Starting in November 2022, a 3^{4} speedsolving frenzy began due to the popularity of the recently created Hyperspeedcube program, and its keyboard controls. The record bounced between Hactar and Grant as it smashed through all the barriers, finally lowering to sub-2 minutes in May 2023.