Message #241

From: Don Hatch <>
Subject: 2^d and 3^d solve function
Date: Mon, 15 May 2006 04:21:10 -0700

Hi folks,

Sorry I’ve been out of the loop, I have not tackled my spam problem yet
and have been simply avoiding my mailbox :-(
But Melinda showed me Roice and Charlie’s 5d program the other day
when I was visiting… awesome, you guys :-)

Attached is a general function to solve the n^d puzzle
(length n = 2 or 3, any number of dimensions d >= 3).
It should be easy to plug this into
Melinda’s and/or Roice&Charlie’s program, if you are so inclined.
(I wrote it in java this time to make it easy on you :-) )

A couple of surprising facts I discovered while doing this–
not sure whether they are already well-known?
- The notion of parity (i.e. whether the solution will have
an even or odd number of twists)
does not exist for 2^d puzzles with d >= 4;
that is, for such puzzles, there is a sequence of an odd number
of 90 degree twists that will bring the solved puzzle
back to solved state (unlike the 2^3, and n^d with n odd,
all of which have a parity restriction).
Not sure whether this is true for n^d for other even n
(I didn’t look at trying to solve anything with n>3).
- There is no "twirl modulus" restriction on corner cubies
for n^d puzzles with d >= 5:
that is, you can cycle 3 stickers
on a single corner cubie (without having to anti-cycle
3 stickers on some other corner cubie like you have to do
in 3 or 4 dimensions).
This is fairly easy to see if you think about it–
it’s exactly the same reasoning that lets you
twirl a single 3-sticker cubie on the 3^4 puzzle.
The crux of the matter is that
in >=5 dimensions you can’t tell the difference
between a cycle and an anti-cycle (of 3 stickers on a corner cubie);
that is, one can be rotated to the other.
So, to cycle 3 stickers a,b,c on a single corner cubie:
1. cycle a,b,c and anti-cycle 3 stickers on some other cubie
2. twist a face containing a,b,c and not the other cubie so that:
a goes to a
b goes to c
c goes to b
d goes to e
e goes to d
(where d,e are two more stickers on the same corner cubie
as a,b,c – this is where d>=5 is needed)
(you can’t just swap b and c without swapping some other d and e,
since that would turn the cubie inside out)
3. undo 1.
4. undo 2.
The result is that a,b,c got forward-cycled twice and nothing else
on the puzzle was touched– i.e. they got backward-cycled once.
So do all of the above in reverse to cycle a,b,c forward, as desired :-)