A computer simulated 3x3x3x3 in a scrambled state.

The 4D Rubik's Cube is, as the name says, a virtual "Rubik's Tesseract"; a 3x3x3x3 octachoron with 27 3D-cube stickers per face. It is of course presented as a projection onto 2-space of a projection onto 3-space of a 4-cube; in this case, using a "looking into the cube through one face" perspective, in which the tesseract appears as a cube within another cube, with the other 6 faces distorted into truncated square pyramids. This is the most sensible perspective to use, as 7 of the 8 faces can be seen at once.

It used to be available to play online, and although this has ceased because most browsers no longer support Java, it is still available for download as a program in the .jar format which can be run on any system which supports a Java virtual machine. This program can simulate puzzles based on several different polychora, from the "Pentaminx" (a pentachoron, the five faces of which are each a Pyraminx) to the dodecaplex or dodecacontachoron, the 120(!) faces of which is each a Megaminx. The linked page has further details, and other interesting stuff including a video by Mathologer on how to solve a 3x3x3x3 (the techniques you already know can be adapted).

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