# Message #146

From: guy_padfield <guy@guypadfield.com>

Subject: Cubes, beer and Beethoven

Date: Thu, 05 May 2005 19:05:23 -0000

My solutions of the 3^4 and 4^4 are probably the least elegant and

most long-winded of anyone’s. Instead of wracking my brains for the

shortest way to bring 3 distant pieces into alignment and

orientation and memorising cunning preliminary moves, I sit back at

the computer with a beer, chewing on a pipe and listening to

Beethoven, gently reversing entropy with minimum energy expenditure.

In terms of time, this is actually quite a quick method (and very

enjoyable), but the solution files are horrific.

When I embarked upon the 5^4 in the same spirit I met a new problem -

I can’t see all the interior cubies without twisting the cube back

and forth and find myself peering at the computer screen like an old

lady looking for her glasses. The perspective distortion also means

I often mistake their positions. The notes say you can change the

size of the cubies with a command line but being unfamiliar with

Linux I don’t know how to do this (I am using Knoppix). Does making

them smaller increase the transparency of the faces and make this

easier? If so, how do I do this (answers phrased for a graduate of

philosophy/ancient languages rather than maths/IT please)?

I have much enjoyed the mathematical discussions on this site

recently, particularly concerning the question of whether the

available rotations would be physically possible in a ‘real’ 4D cube

(my conclusion, like yours I think, is that they would). Could the

mathematicians answer another question, relating to the 4^4? In the

3D version there are two independent parity problems if you follow

the ‘ultimate solution’: the last pair of edges is inverted 50% of

the time, resolvable by reconstructing the faces and the last two

corners are switched 50% of the time, resolvable by reconstructing

the edges. I guessed there might be three parity problems in the

4^4: that the faces would only be realisable if the centres were

correct, the edges only if the faces were and the corners only if

the edges were. When I solved it I met just one of these problems: I

had three intractably inverted edge cubies and found they dropped

into place only after I had re-aligned the faces (I switched the

position of a pair of adjacent face pieces on two separate sides).

The corners were right. Was I lucky? I don’t propose to test this

empirically, for obvious reasons.

Would a layer by layer solution, as Marc Guegueniat has found,

remove the parity issues?

Guy Padfield