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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.

For a general element \(a_i\), we will define \(\bar{a_i}= \max (0, a_i-2)\).

Given \(k>0\) and \(n \geq 1\), we can denote the cyclic sum of the products of elements \(\bar{a_1}, \bar{a_2}, \dots, \bar{a_n}\) taken \(n-k\) at a time by \(Cyc(n,k)\).
- For example: \(Cyc(5,4) = \bar{a_1} + \bar{a_2} + \bar{a_3} + \bar{a_4} +\bar{a_5}\), i.e. the sum of groups of 5-4=1 elements, chosen in \(\bar{a_1}, \bar{a_2}, \bar{a_3}, \bar{a_4}, \bar{a_5}\).
- Another example: \(Cyc(4,2) = \bar{a_1} \bar{a_2} + \bar{a_1} \bar{a_3} + \bar{a_1} \bar{a_4} + \bar{a_2} \bar{a_3} + \bar{a_2} \bar{a_4} + \bar{a_3} \bar{a_4}\), i.e., the sum of the products of all possible ordered pairs made with \(\bar{a_1}, \bar{a_2}, \bar{a_3}, \bar{a_4}\).

Note that \(Cyc(n,k)\) has \(C_{n,k} = \binom{n}{k}\) terms.
We can also define \(Cyc(n,n)=1\)

Using the above notation, the \(a_1 \times a_2 \times \dots \times a_n\) hypercuboid has \(2^k\cdot C(n,k)\) pieces which are \(k\)-colored.

For example, consider the \(2 \times 3 \times 5 \times 7\) hypercuboid, in this case:

  • \(a_1\) =2, so \(\bar{a_1}=0\),
  • \(a_2\) =3, so \(\bar{a_2}=1\),
  • \(a_3\) =5, so \(\bar{a_3}=3\),
  • \(a_4\) =7, so \(\bar{a_4}=5\).

So there will be:

  • For 1-colored pieces we have:
    \(2^1 \cdot Cyc(4,1)=2^1\cdot (0\cdot1\cdot3 + 0\cdot 3 \cdot 5 + 1 \cdot 3 \cdot 5 + 0\cdot 1 \cdot5 )=\)
    \(=2 \cdot (0+0+15+0)=30\) pieces.

  • For 2-colored pieces we have:
    \(2^2 \cdot Cyc(4,2)=2^2\cdot ( 0\cdot 1+ 0\cdot 3 +0 \cdot 5 + 1\cdot3 +1\cdot 5 + 3\cdot 5 )=\)
    \(=4 \cdot (0+0+0+3+5+15)=92\) pieces.

  • For 3-colored pieces we have:
    \(2^3 \cdot Cyc(4,3)=2^3\cdot (0+1+3+5 )=\)
    \(=8 \cdot 9=72\) pieces.

  • For 4-colored pieces we have:
    \(2^4 \cdot Cyc(4,4)=2^4\cdot 1= 16\) pieces.

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.