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 Post subject: Pentultimate vs. Icosamate?Posted: Sun Aug 19, 2012 2:44 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Without spoiling anything, which is more difficult?

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 Post subject: Re: Pentultimate vs. Icosamate?Posted: Sun Aug 19, 2012 3:07 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Jared wrote:
Without spoiling anything, which is more difficult?

Taking everything into account, I think they're about equal. It depends on what aspect of the solve you find the hardest.

The centers of the Pentultimate are the corners on the Icosamate.
The corners of the Pentultimate are the centers on the Icosamate.

The Pentultimate corners on the Icosamate are much easier because they lack orientation but this is somewhat balanced out by the centers on the Pentultimate being harder because they now have orientation.

Worse, overall twist is not maintained for Pentultimate centers / Icosamate corners so this is possible:
Attachment:

icosamate_corner_twist.png [ 17.22 KiB | Viewed 984 times ]

Unless you have a pure sequence for fixing this issue, you need to check for and fix it early in the solve.

Looking at the "difficulty" purely from a number of distinct positions:

Pentultimate positions:
? (20! / 2 * 3^20 / 3 * 12! / 2) / 60
% = 5643573414231758192239391539200000

Icosamate positions:
? (20! / 2 * 12! / 2 * 5^12) / 60
% = 1185469517577945600000000000000000

So the Pentultimate : Icosamate ratio is 4.76 : 1

_________________
Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Pentultimate vs. Icosamate?Posted: Mon Aug 20, 2012 9:37 am

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Thank you! Of course the best shape would be something like a Rhombultimate so that both types of pieces have orientation. I hope I can get all three someday...

I was aware of the single twist problem (all face-turning dodecahedral puzzles with orientable centers have that ability). It sounds a bit like the difference between solving a 4x4 and a 5x5 cube with reduction - one has more positions, but the other has parity problems.

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