# Difference between revisions of "Physical Puzzle"

## Definition

In the context of hypercubing, a physical puzzle refers to an N+1 dimensional puzzle projected in N dimensions, such that the projection is operationally equivalent, i.e. the projection "emulates" the true puzzle. Usually, if not specified, it is used to refer to a 4D puzzle projected into 3-space.

## 2D Physical Puzzles

Drawing showing (from top to bottom, left to right)2^3, 2x2x3, 2x3x3, 3^3, 2x2 pyraminx, 3x3 pyraminx, bandaged 3x3 pyraminx, bandaged 4x4 pyraminx

When designing 3D physical puzzles, it's a good idea to try and step down the problem by first looking at what the 2D physical analog would look like. For example, the 2^3, or by it's more recognizable name, the 2x2x2, can be projected down into 2D by splitting it in 2 halves and putting them next to each other along the X or Y axis. By doing this, the Z axis will coincide with the X or the Y axis, which will make certain moves inaccessible without the use of a gyro algorithm to reorient the puzzle (the gyro algorithm does a cube rotation). We can also see that the projected cubies aren't square shaped, and that is because a square doesn't have 3 fold symmetry.

By the same logic, we can then build a 3D physical 2^4/2x2x2x2 - split the hypercube in 2 halves and put them next to each other along X, Y or Z axis, making certain moves inaccessible without the use of a gyro. We can see that the 2^4 can have the cubies be cube shaped because (by mathematical coincidence) a cube has 4 fold symmetry (a 2^4 cubie has 4 colors).

Another thing to note is that physical puzzles don't have a true mechanism/way to hold the pieces together, so magnets are used to hold the puzzle together. Because we don't have a true mechanism, we need to limit the moves we are allowed to do by the canonical moves to make sure we aren't doing illegal moves on the puzzle.

Another example is the 3x3 pyraminx. In this case we can take a top-down orthographic view of the puzzle and then project it, making sure each piece type has the same shape in the projection so they are interchangeable. Doing this, we get an interesting 2D physical puzzle that looks like the wireframe of a tetrahedron projected in 2D, but with some of the lines disconnected. This is because we can't project a tetrahedron into 2D without unevenly distorting the shape or without having breaks in the shape. This can then be scrambled and solved using its canonical moves (rotate 3C pieces with its 2Cs, by first making sure all its associated 2Cs are touching it). By the same logic one can construct a 3D physical simplex.

Image showing the correlation between the top down orthographic view of the pyraminx, the 2D physical pyraminx and the wireframe of the pyraminx

## 3D Physical Puzzles

Render showing (from left to right, top to bottom) Bandaged void simplex, simplex, 1x2x2x2, 1x3x3x3, 2^4, 2x2x2x3, 2x2x3x3, 2x3x3x3, 3^4

The first physical puzzle was the 2^4, designed by Melinda Green in 2017. After this Grant, Luna, Hactar and Markk designed and built many other 3D physical puzzles based on Melinda's 2^4 design, like the 3^4, the simplex, and hypercuboids like the 2x2x3x3. There where attempts at designing shapeshifting physical puzzles with no success.

## 3D Physical designs list

Below is a table of physical puzzles, both produced and unproduced.

Real Physical Puzzles
Puzzle Name(s) Date Finished
2x2x2x2 Melinda Green 2017
2x2x2x3 Luna & Grant December 2021
2x2x3x3 Grant & Hactar May 2022
2x3x3x3 Grant July 2022
3x3x3x3 Grant July 2022
Puzzle Name(s) Date of thought
1x1xnxn series Grant
Simplex Markk August 2022
pretty much any cuboid Grant

## Physical Shapeshifting Puzzles

Physical shapeshifting puzzles are hard to design, if not impossible, because of their shapeshifting nature. All the current 2D and 3D designs don't fully work, in some solved states appearing as they're scrambled.

2D physical 2^3 mirror before and after performing the gyro algorithm
Drawing showing all the current 2D physical mirror cube designs(NOT fully functional)
Render showing the current 3D physical 2^4 mirror design(NOT fully functional)