The 3x3x3x3 (normally referred to as the 34) 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 34 is recommended as one's first introduction to solving higher dimensional puzzles.
The 34 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.
Each turn of the 34 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.
The 34 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.
(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 34 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 34 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.