Date: Wed, 31 Dec 80 12:10:00 -0500 (EST)
From: David C. Plummer <DCP@MIT-MC >
~~~ ~~~ Subject: the 5x5x5 [133 lines]

OK, folks, I'm considering going further than 4x4x4 and entering
the realm of the 5x5x5.

```Cubies:
C := Corner
X := aXis (center)
E := Edge (outside center)
L := Left (external edge)
R := Right (external edge)
D := Diagonal (internal [to the face] corner)
A := Adjacent (to the center, thanks to WER)
(internal [to the face] edge)
```

A 3-D view would look like this

```                z=-5   +---+---+---+---+---+
/   /   /   /   /   /|
/ C / L / E / R / C / |
+---+---+---+---+---+  |
/   /   /   /   /   /|C +
/ R / D / A / D / L / | /|
+---+---+---+---+---+  |/ |
z=0     /   /   /   /   /   /|R +  |
/ E / A / X / A / E / | /|L +
+---+---+---+---+---+  |/ | /|
/   /   /   /   /   /|E +  |/ |
/ L / D / A / D / R / | /|D +  |
+---+---+---+---+---+  |/ | /|E +
/   /   /   /   /   /|L +  |/ | /|
/ C / R / E / D / C / | /|A +  |/ |
y,z=5   +---+---+---+---+---+  |/ | /|A +  |
|   |   |   |   |   |C +  |/ | /|R +
| C | L | E | R | C | /|D +  |/ | /|
|   |   |   |   |   |/ | /|X +  |/ |
y=3     +---+---+---+---+---+  |/ | /|D +  |
|   |   |   |   |   |R +  |/ | /|C +
| R | D | A | D | L | /|A +  |/ | /
|   |   |   |   |   |/ | /|A +  |/
y=1     +---+---+---+---+---+  |/ | /|L +
|   |   |   |   |   |E +  |/ | /
y=0     | E | A | X | A | E | /|D +  |/
|   |   |   |   |   |/ | /|E +
y=-1    +---+---+---+---+---+  |/ | /
|   |   |   |   |   |L +  |/
| L | D | A | D | R | /|R +
|   |   |   |   |   |/ | /
y=-3    +---+---+---+---+---+  |/
|   |   |   |   |   |C +
| C | R | E | L | C | /
|   |   |   |   |   |/
y=-5    +---+---+---+---+---+
x=-5  -3  -1   1   3   5
```

LOVE THAT ASPECT RATIO !!!!

```All in all there are
6 aXis faces
8 Corners
12 Edges
24 Left/Right type edges
24 Diagonals
--
98 = 5^3 - 3^3 = 125-27 visible cubies
```

Computation (inaccurate, but within a couple orders of magnitude)
of the number of reachable positions:

```Axes:	lets not hack the extended problem yet	->	1
Corners:8 of them anywhere			->	8!
each can take 3 orientations		->	3^8
parity of the corner			->	1/3
Edges:	12 of them anywhere			->	12!
each can take 2 orientations		->	2^12
position/orientation restriction	->	1/4
L/R:	24 of them anywhere			->	24!
orientation defined (they cannot flip)	->	1
parity (cannot swap only two)		->	1/2 (I think)
Adjac:	24 of them anywhere:			->	24!
one edge always touches a face center	->	1
parity					->	1/2 (at least)
Diags:	24 of them anywhere			->	24!
one corner always touches a face center	->	1
parity					->	1/2 (at least)

It may not be accurate, but this computation gives
1.291318 * 10^90
```

A slice through the center (z=0) probably looks something like

```y=5\
/    ..XXXXXXXXXX++++++++++EEEEEEEEEE
..XXXXXXXXXX++++++++++EEEEEEEEEE
.....XXXX++++++++EEEEEEEEEEEEEEE
.....XXXX++++++++EEEEEEEEEEEEEEE        X is an axis cubie
y=4\    .....XXXX++++++++EEEEEEEEEEEEEEE        E is an edge cubie
/    .....XXXX++++++++EEEEEEEEEEEEEEE        + is one adjacent cubie
.....XXXX++++++++EEEEEEEEEEEEEEE
.....XXXX++++++++EEEEEEEEEEEEEEE
y=3\    .....XXXX++++++++EEEEEEEEEEEEEEE
/    .....XXXX++++++++EEEEEEEEEEEEE~~
.....XXXX++++++++EEEEEEEEEEEEE~~
.....XXXX++++++++EEEEEEEEEEEEE~~
.....XXXX++++++++EEEEEEEEEEEEE~~
y=2\    .....XXXX++++++++EEEEEEEEEEEEE~~
/    .....XXXX+++++++/~~~~~~~~~~~~~~~
.....XXXX++++++/~~~~~~~~~~~~~~~~
.....XXXX+++++/~~~~~~~~~~~~~~~~~
\....XXXX++++/~~~~~~~~~~~~~~~~~~
y=1\    .\...XXXX+++/~~~~~~~~~~~~~~~~~~~
/    ..\..XXXX++/~~~~~~~~~~~~~~~~~~XX
...\.XXXX+/~~~~~~~~~~~~~~~~~~~XX
....\XXXX/~~~~~~~~~~~~~~~~~~~~XX
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
y=0\    XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
/    XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
..../XXXX\....................XX
.../.XXXX.\...................XX
y=-1\   ../..XXXX..\..................XX
/
/\   /\   /\        /\        /\
x=-1    0    1         3         5
```

This time the central axis is rigid in the sense that it does
form a cross, but each of the six spokes can rotate as in the
3x3x3 cube. The curvature and tolerances of some of the pieces
gets a little hairy, but I'm working with graph paper and looking
at the other slices through the cube. Wish me luck -- I have
thoughts of construction.