From:

~~~ ~~~ Subject:

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 24 Adjacents -- 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 ~ is another adjacent .....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.