Hypercuboids¶

Introduction¶

A hypercuboid is the multi-dimensional version of a cuboid.
In a general context, we define a hypercuboid as an \(n\)-dimensional puzzle denoted by \(a_1 \times a_2 \times \dots \times a_n\).
For the sake of clarity and consistency, we will use \(a_1,a_2, \dots a_n\) as non-decreasing values.

Structure¶

A hypercuboid, as defined, is composed of \(2n\) cells, each of which is \((n-1)\)-dimensional.

Given \(k \geq 0\) and \(n \geq 1\), the elementary symmetric polynomial \(e_k(x_1, x_2, \dots, x_n) = \sum_{Y} \prod_{y\in Y} y\), where \(Y\) ranges over subsets of \(\{x_1, \dots, x_k\}\) where \(|Y| = k\). In other words, it is the sum of all terms, each of which are product of distinct \(x_i\) taken \(k\) at a time.

For example: \(e_1(x_1,x_2,x_3,x_4,x_5) = x_1 + x_2 + x_3 + x_4 +x_5\), i.e. the sum of terms of 1 element, chosen in \(x_1, x_2, x_3, x_4, x_5\).
Another example: \(e_2(x_1,x_2,x_3,x_4) = x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4\), i.e., the sum of the products of all possible unordered pairs made with \(x_1, x_2, x_3, x_4\).

Note that \(e_k(x_1, \dots, x_n)\) has \(\binom{n}{k}\) terms. From this, we can also see that \(e_0(x_1, \dots, x_n)=1\).

Using the above notation, the \(a_1 \times a_2 \times \dots \times a_n\) hypercuboid, where \(a_i > 1\), has \(2^k\cdot e_{n-k}(a_1-2, \dots, a_n-2)\) pieces which are \(k\)-colored.

For example, consider the \(2 \times 3 \times 5 \times 7\) hypercuboid.

For 1-colored pieces we have:
\(2^1 \cdot e_3(0,1,3,5)=2^1\cdot (0\cdot1\cdot3 + 0\cdot 3 \cdot 5 + 1 \cdot 3 \cdot 5 + 0\cdot 1 \cdot5 )=\)
\(=2 \cdot (0+0+15+0)=30\) pieces.
For 2-colored pieces we have:
\(2^2 \cdot e_2(0,1,3,5)=2^2\cdot ( 0\cdot 1+ 0\cdot 3 +0 \cdot 5 + 1\cdot3 +1\cdot 5 + 3\cdot 5 )=\)
\(=4 \cdot (0+0+0+3+5+15)=92\) pieces.
For 3-colored pieces we have:
\(2^3 \cdot e_1(0,1,3,5)=2^3\cdot (0+1+3+5 )=\)
\(=8 \cdot 9=72\) pieces.
For 4-colored pieces we have:
\(2^4 \cdot e_0(0,1,3,5)=2^4\cdot 1= 16\) pieces.

If \(a_1, \dots, a_m\) are all equal to \(1\) and \(a_{m+1} > 1\), the cuboid is a floppy cuboid. In this case, the number of \(k\)-colored pieces is equal to the number of \((k-2m)\)-colored pieces on the \(a_{m+1} \times \dots \times a_n\) hypercuboid, when \(k \geq 2m\). This includes the case where \(k = 2m\), in which case the floppy cuboid has pieces corresponding to the 0-colored pieces of the lower dimensional hypercuboid.

4D Hypercuboids¶

In 4 dimensions, a hypercuboid is denoted as \(a \times b \times c \times d\).
\(a \times b \times c \times d\) is composed of 8 cells: 2 \((a \times b \times c)\)-cells, 2 \((a \times b \times d)\)-cells, 2 \((b \times c \times d)\)-cells and 2 \((a \times c \times d)\)-cells.
In the following sections, we will denote some of these cells using the classic 3-dimensional puzzle names, in particular:

“tower cell” will indicate a \(2 \times 2 \times 3\)-cell;
“domino cell” will indicate a \(2 \times 3 \times 3\)-cell;
” \(n\) -cubic cell” will indicate a \(n \times n \times n\)-cell.

General solving strategies¶

Hypercuboids in the form \(1 \times a \times b \times c\) can be solved by first orienting the \(a \times b \times c\)-cells, then solving the puzzle like a 3-dimensional \(a \times b \times c\).
Hypercuboids in the form \(2 \times a \times b \times c\) can be solved first by solving the \(a \times b \times c\)-cells and then solving the opposite, eventually adapting the solution for new possible cases.
If 2 dimensions have the same values, the puzzle can be seen as a duoprism.
If 3 dimensions have the same values, i.e. there is a couple of \(n\)-cubic cells, RKT can be used on these cells.
If 4 dimensions have the same values, we have a hypercube.

Some notable 4D hypercuboids¶

In some cases an idea of a possible solution method provided by Ema will be present but not spoiled.

1x3x3x3¶

Puzzle	4c pieces	3c pieces	2c pieces	1c pieces
1x1x3x3	16	24	12	2

Solve idea (click to reveal)

Orient both cubic cells.
Solve 3^3 cube, paying attention to corner orientation.

2x2x2x3¶

Puzzle	4c pieces	3c pieces	2c pieces	1c pieces
2x2x2x3	16	8	0	0

Solve idea (click to reveal)

Solve the middle 3-colored pieces of a tower cell (similar to solving a \(1 \times 2 \times 2 \times 2\) ).
Orient both \(2\)-cubic cells at the same time, slicing the solved part for exchanging pieces,being careful to use an even number of slice moves.
Use RKT to solve the cubic cells, using the same tower cells as R.
Fix tower cell middle layer.

2x2x3x3¶

Puzzle	4c pieces	3c pieces	2c pieces	1c pieces
2x2x3x3	16	16	4	0

Solve idea (click to reveal)

Solve a domino cell.
Orient the opposite domino cell, potentially re-solving the first cell.
Move pieces on the correct layers of the last cell.
Solve last domino cell using 3-dimensional cuboid algorithms an even number of times and conjugating between them.

2x3x3x3¶

Puzzle	4c pieces	3c pieces	2c pieces	1c pieces
2x3x3x3	16	24	12	2

Solve idea (click to reveal)

Orient both 3-cubic cells at the same time.
Solve first cubic cell.
Solve the second cubic cell using RKT.

2x2x2x4¶

Puzzle	4c pieces	3c pieces	2c pieces	1c pieces
2x2x2x4	16	16	0	0

2x3x4x5¶

Puzzle	4c pieces	3c pieces	2c pieces	1c pieces
2x3x4x5	16	48	44	12

The smallest 4-dimensional “brick” hypercuboid.

4D hypercuboids in MPUlt¶

Here is a way to create your own 4D hypercuboid in MPUlt.
The result would not be isometric, but still working.

Step 1: Recognize the form of your hypercuboid in one of the following

\(a \times b \times c \times d\),
\(a \times a \times b \times c\),
\(a \times a \times b \times b\),
\(a \times b \times b \times b\),
\(a \times a \times a \times a\).

Step 2: Recognize the values of the letters, then substitute the letter with the corresponding string from the following table:

Value	String
2	0.0
3	0.333 -0.333
4	0.5 0.0 -0.5
5	0.6 0.2 -0.2 -0.6
6	0.667 0.333 0.0 -0.333 -0.667
7	0.714 0.429 0.143 -0.143 -0.429 -0.714
8	0.75 0.5 0.25 0.0 -0.25 -0.5 -0.75
9	0.778 0.556 0.333 0.111 -0.111 -0.333 -0.556 -0.778

So if \(a=3\), you need to change “CUT-A” with “0.333 -0.333” in the general puzzle code, and so on.

Step 3: Insert the created code in “MPUlt_puzzles.txt” file, save and enjoy your puzzle.

Case axbxcxd¶

General code:

Puzzle NAME_AXBXCXD
Dim 4
NAxis 4
Faces 1,0,0,0 0,1,0,0 0,0,1,0 0,0,0,1
Group 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1
Axis 1,0,0,0
Twists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1
Cuts CUT-A
Axis 0,1,0,0
Twists 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1 0,0,1,0/0,0,0,1
Cuts CUT-B
Axis 0,0,1,0
Twists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0
Cuts CUT-C
Axis 0,0,0,1
Twists 1,0,0,0/0,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0
Cuts CUT-D

Case axaxbxc¶

General code:

Puzzle NAME_AXAXBXC
Dim 4
NAxis 3
Faces 1,0,0,0 0,0,1,0 0,0,0,1
Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 1,0,0,0/0,0,0,1
Axis 1,0,0,0
Twists 0,1,0,0/0,0,1,0 0,1,0,0/0,0,0,1 0,0,1,0/0,0,0,1
Cuts CUT-C
Axis 0,0,1,0
Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,0,1 0,0,0,1/0,1,0,0
Cuts CUT-B
Axis 0,0,0,1
Twists 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,1,0,0
Cuts CUT-A

Case axaxbxb¶

General code:

Puzzle NAME_AXAXBXB
Dim 4
NAxis 2
Faces 1,0,0,0 0,0,1,0
Group 1,0,0,0/1,1,0,0 1,0,0,0/0,0,1,0 0,0,1,0/0,0,1,1
Axis 1,0,0,0
Twists 0,0,1,0/0,0,1,1 0,1,0,0/0,0,1,0 0,1,0,0/0,0,1,1
Cuts CUT-A
Axis 0,0,1,0
Twists 1,0,0,0/1,1,0,0 0,0,0,1/1,0,0,0 0,0,0,1/1,1,0,0
Cuts CUT-B

Case axbxbxb¶

General code:

Puzzle NAME_AXBXBXB
Dim 4
NAxis 2
Faces 1,0,0,0 0,0,0,1
Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/0,0,0,1
Axis 1,0,0,0
Twists 0,1,0,0/0,1,1,0 0,1,0,0/0,0,0,1
Cuts CUT-B
Axis 0,0,0,1
Twists 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0
Cuts CUT-A

Case axaxaxa¶

General code:

Puzzle NAME_AXAXAXA
Dim 4
NAxis 1
Faces 1,0,0,0
Group 1,0,0,0/1,1,0,0 1,0,0,0/1,0,1,0 1,0,0,0/1,0,0,1
Axis 1,0,0,0
Twists 0,1,0,0/0,1,1,0 0,1,-1,0/0,0,0,1 0,2,-1,-1/0,1,1,-2
Cuts CUT-A

5D+ Hypercuboids¶

These hypercuboids haven’t been studied yet, except for some “simpler” versions with lots of \(1\)’s.