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 Post subject: Some new puzzle ideas
PostPosted: Wed Nov 12, 2003 4:13 pm 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
I've come up with some neat twisty puzzle concepts in the last few days. Here are some of the better ones, starting with the most obscure symmetry groups and working backwards.

Take an icosidodecahedron. Place a very shallow pyramid on each pentagonal face. If you make it the exact right height, it will be possible to rotate a set of six triangles an exact half rotation. Two types of pieces, valences 1 and 3, numbering 60 and 20. 30 slices. A bit big, but it's interesting to have anything resembling a workable puzzle based on this symmetry group.

Start with a dodecahedron. On each face put a shallow pyramid. A piece comprises the two triangles which meet at one edge of the dodecahedron. If you make the pyramids the exact right height, it will be possible to rotate any three pieces which meet at a point. One type of piece, valence two, numbering 30. 20 slices. This would probably be a nice puzzle, kind of like a very large pyraminx.

Take a small stellated dodecahedron. Actually, that's pretty much it. To see where a slice goes, look at the top of one of the pyramids head-on. Pick two non-adjacent corners of the base of the pyramid. Now make a slice going through the top vertex and the other two corners you just picked. One type of piece, valence 1, count 60. 30 slices. I suspect others have thought of it before. This looks like an interesting puzzle, probably very difficult to design a mechanism for it though.

This next one is I think my most innovative idea. Please bear with me, it's a bit weird :-). Start with a dodecahedron. Cut out an antiprism below each face. A piece consists of two triangles which meet at the same edge of the dodecahedron. The exact depth of the antiprisms is such that if you extend the line from a corner of the dodecahedron to the nadir of the antiprism, it hits another vertex of the dodecahedron. Specifically, if point X is on face A and edge Q is the edge of A opposite X, then if face B also has edge Q then the antiprism line will extend through the corner of B which is opposite Q. The motion of this one is a little hard to describe, best you just build a model of it yourself and look at it. One type of piece, valence 2, count 30. 12 slices.

Both of the last two puzzles are probably easier than they appear at first blush, because it's possible to rotate two opposite slices together, only moving the middle rim.

Take a cuboctahedron. Make slices which connect opposite corners of one square face, go down edges of two neigboring triangles, and cut across two opposite corners of a neigboring square face. Two kinds of pieces, valence 1 and three, count 24 and 8. 12 slices. I'm sure someone has thought of this before (I first thought of it years ago) but I haven't seen it mentioned before and think it would make a nice puzzle. The first puzzle I mentioned is really a larger (and probably inferior) version of this one.

On a sphere, cut thin ribbons along great arcs which would be halfway between the faces of an octahedron. This puzzle is sort of like equator in that only the ribbons are allowed to move, the bulk of the puzzle is a solid object. One kind of piece (not counting the ones which can always have their positions fixed trivially, albeit with moving the others) valence 2, count 12. Four slices (well, ribbons anyway). I think this is a very nice puzzle, with some interesting and unique properties. It's also the only one of the puzzles I've mentioned with an obvious mechanism.

There are dodecahedral and icosahedral versions of that last puzzle, but they're large and I think don't add much.

Well, that's all for today. I could go on, but I think that's the pick of the litter.

Thanks to Jaap for helping me figure these out.


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 Post subject: Re: Some new puzzle ideas
PostPosted: Wed Nov 12, 2003 4:56 pm 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Here are some pages I should have linked to with neat applets to show the archimedean and kepler-poinsot solids -

http://mathworld.wolfram.com/ArchimedeanSolid.html

http://mathworld.wolfram.com/Kepler-PoinsotSolid.html

And of course right after posting I figure out a much simpler way of visualizing one of the puzzles.

Start with a great stellated dodecahedron. For each pair of adjacent prisms, add a moveable piece which has four vertices - the two apexes of the prisms and the two pase vertices they both share. These pieces will completely cover up the original great stellated dodecahedron. A slices moves all the pieces which touch all the prisms which meet at one point.

This is a surprisingly easy puzzle. There are pairs of slices which only share a single piece between them. Flipping the pieces is much more involved than with the pyraminx though.


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 Post subject: Re: Some new puzzle ideas
PostPosted: Thu Nov 13, 2003 3:09 am 
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Joined: Mon Sep 09, 2002 2:19 pm
Location: Yaroslavl, Russia and Maryland, USA
This looks like an "edges only" megaminx. Am I right? As for the other puzzle ideas, I am having hard time figuring them out. I would be best if you could draw the outlines of the puzzles. On the other hand, it's 1am here, and I might not be thinking right...


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 Post subject: Re: Some new puzzle ideas
PostPosted: Thu Nov 13, 2003 10:08 am 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
No, the cuts are deeper than they would be on an edges only megaminx. You move ten pieces at once, instead of just five.

Yeah, it would be a lot easier to see what I'm talking about with some nice animations.


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 Post subject: Re: Some new puzzle ideas
PostPosted: Thu Nov 13, 2003 5:44 pm 
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Joined: Mon Sep 09, 2002 2:19 pm
Location: Yaroslavl, Russia and Maryland, USA
Bram, after visiting the mathworld site, I can visualize the puzzle now. Yes, 10 pieces would move at once. Looks nice, and probably is not a complicated puzzle. I will try the same approach to try to visualize the other ideas of yours. :-) Thanks a lot for sharing!


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 Post subject: Re: Some new puzzle ideas
PostPosted: Fri Nov 14, 2003 10:06 am 
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Joined: Fri Dec 08, 2000 2:32 am
Location: San Francisco, CA
For this puzzle:


Take a cuboctahedron. Make slices which connect opposite corners of one square face, go down edges of two neigboring triangles, and cut across two opposite corners of a neigboring square face.


Is this the same as this idea for a puzzle on Jaap's site, but with truncated corners?



Puzzles with 12 axes of rotation, around the edges of the cube or octahedron.



1. Cube



In this shallow cut puzzle a move involves giving the triangular prism along any edge a half turn. I suspect it is quite a tough puzzle. Its mechanism could be based on a 12-armed spider.


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 Post subject: Re: Some new puzzle ideas
PostPosted: Fri Nov 14, 2003 10:40 am 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
I said 'prism' a bunch of times when I meant 'pyramid'. Please substitute all such occurences as you read them :-)


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 Post subject: Re: Some new puzzle ideas
PostPosted: Fri Nov 14, 2003 5:44 pm 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Now that you mention it, it probably isn't too hard to build. The interior great stellated dodecahedron could be made as a build-up of a megaminx, and the actual pieces could have grooves which fit onto hooks from the interior. It would need fairly good tolerances to keep the pieces from falling off though.


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 Post subject: Re: Some new puzzle ideas
PostPosted: Sat Nov 15, 2003 10:06 am 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Yep, same puzzle. I realized the similarity after posting.

I suspect that it wouldn't be all that bad to solve, since such a small proportion of all the pieces in the whole puzzle move with each turn, and once the corners are solved there are plenty of subgroups which can change the corners's position but not their orientation.


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