6-piece burrs

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Pio2001
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6-piece burrs

Post by Pio2001 » September 19th, 2011, 2:29 am

Hello,
I've been lucky to play with a lot of different 6-piece burrs recently. I'm especially interested in the most complex ones, but without special tricks like coordinated moves, non-standard pieces, internal mobile parts, magnets etc.

Here are the ones I own, or tried. For each one, I give the number of assemblies (the ways the pieces could fit together, but that may not be actually built), the number of solutions, and the number of moves needed to disassemble it (first piece, then second piece, etc).

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Philippe Dubois
Assemblies : 2
Solutions : 1
Moves : 6.4

It was my first burr. Very straightforward because it only has 2 assemblies. Ideal for beginners in "holey 6-piece burrs".

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Peter Marineau - Burr B (or "Piston")
Assemblies : 10
Solutions : 1
Moves : 9.3

Much later, at last, I could grab another hollow design, and to my pleasure, it was more complex than the Philippe Dubois. With 10 assemblies, the construction of the puzzle starting from the disassembled pieces is a longer process. That's with it that I started to use the "Solving burrs" method.

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Bill Cutler & Brian Young - Mega Six (Ive got three of them :D )
Assemblies : 20
Solution : 1
Moves : 10

THE hardest six piece burr, according to Mrpuzzle. I don't know if this is true, but this was indeed the first one that I failed to solve ! It features a subtlety that I failed to see entierely : two of the pieces has an ambiguous orientation (two of their sides may be the "back" side). After one hour, I realized it for one of them, but this one actually had its "natural" back outside, and after another hour, I forfeited without noticing the other one.

I liked this puzzle so much that I then bought the Craftsman edition. Later, Maurice Vigouroux made the splendid white one in curly sycamore for me. Unlike Brian Young's one, it has no glued cubes. The unmillable pieces are chiseled.

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Edward Hordern - Modification of burr B
Assemblies : 6 (the one on the right side, with three colours)
Solution : 1 (the one on the right side, with three colours)
Moves : 10

When I met Maurice Vigouroux, who had been interested in 6-piece burrs too, he gave me that one. It is a bit more difficult than burr B, because a move has to be stopped before the end for another piece to move. With the original Burr B, all moves must be performed until the end.
However, it remains much easier than the Mega Six.

*** No picture ***

Bruce Love - Love's Dozen (with three colours)
Assemblies : 4
Solution : 1
Moves : 12.3

This is the highest number of moves for a 6-piece burr. The solution is not unique, but if the pieces are made from three different woods, like the "Modification of Burr B", the 12 moves solution is enforced.
I made it with LiveCubes. I was a bit disappointed to see that it was just another variation around Burr B. I still find the Mega Six more complex.

While I was visiting Maurice, I decided to have a look at his old 6-piece crosses. We gathered every one from his attic, and got a full bucket of them in the living room. And it's not just an image, we really carried them inside a bucket !

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There were about 50 of them. I didn't have the time to solve them all, so I just looked at the way they could be disassembled. I discarded the ones with only one or two moves. Then I narrowed the selection to the ones with the most interesting or elegant disassembling sequences.

Here is what Maurice made that week :

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Bill Cutler - Computer's Choice 5 holes
Assemblies : 7
Solution : 1
Moves : 9

I think that this one is quite hard. It has the same ambiguous piece as the Mega Six. The rest of the sequence is not so intricate and bizarre, but on the other hand, it features a very nice dead end in the disassembling sequence.
The Mega Six also has one, but you can't lose yourself in it, because it happens if you deliberately leave in place the piece that should be removed and go on.
In this puzzle, you can't miss the dead end. You go straight into it after a few moves. And it even features a variation ! That's actually a move that leads into two close dead ends.
You must go back and find a more subtle move in order to disassemble the puzzle.

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Derwin Brown - Puzzle #23
Assemblies : 9
Solution : 1
Moves : 6

I liked the way this puzzle opens. It separates into two halves after the 6th move.
To be honest, The Mega Six, the Computer's Choice 5 holes, the love's 1 and Abad's level 6.7 also do.
This one also have an ambiguous piece, but it is maybe not very difficult to spot.
This is the puzzle that I chose for the article about Solving Burrs, because following the method, we find the right assembly quite fast.
Overall, I prefer the Computer's Choice 5 holes and the Mega Six.

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Bruce Love - Love's 1, with three colours
Assemblies : 4
Solution : 2
Moves : 6, or 6.2

I selected this one because the disassembling sequence looked nice. It must be manufactured with three colours for the solutions to have at least 6 moves.
It also has an ambiguous piece, but unfortunately, it must be placed in its "natural" orientation.
I still prefer Mega Six and Computer's Choice.

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Bill Cutler - L46AA with dowels
Assemblies : 1
Solution : 1
Moves : 6 with 8 units pieces or 10 with 10 units pieces

This one is very interesting. Its disassembling sequence features a lot of dead ends. It is not very long, not very hard, but awfully confusing ! Today, I have disassembled it about 30 times, and I'm still unable to find the right moves without making a mistake first.
The bare design features many uninteresting assemblies. So many that coloured pieces are not enough to enforce the level 6 one. Maurice found a smart way to solve this problem. Each piece is marked with a dowel on its side, and the dowels must be gathered in two opposed groups.
We later identified this burr as being a small variation of Bill Cutler's L46AA notchable... The only difference is that Maurice can mill the pieces, so this one has one more voxel and is not notchable. The extra voxel does not change anything.

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Bill and Jerry McFarland draw a black line around the puzzle in order to enforce the solution.
When we made it with Maurice, we had to use coloured pieces in addition to the dowels. I later realized, completely by chance, that choosing another location for the two groups of dowels allowed to use only one colour.

I learned online, in Ishino's website, I think, that if the pieces are 10 voxels long, the solution requires 10 moves instead of 6. I then built this bigger model with livecubes.

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The missing voxels at the end of the pieces were a study about beveling the end of the pieces, like in "Diagonal Burrs". It would have looked like this :

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I also found that to get 10 moves, only one pair of pieces have to be 10 voxels, and another can be 6 voxels short :

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But I don't like the final shape.

I think that this burr is brillant. It may become my prefered one, above the Mega Six, once I get one in wood.

Lately, I was intrigued by another design listed in Ishino's website :

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Rafael Juan Guarinos Abad - Abad's level 6.7
Assemblies : 25
Solution : 1
Moves : 6.7

With 25 assemblies, and one ambiguous piece (which unfortunately goes in the "right" position), it compares to the Mega Six for the difficulty of the assembling challenge.
There is no problem with the ambiguous piece, but on the other hand, only one of the extra assemblies of the Mega Six allows some moves that leads the puzzlist to think that it may be the right solution, while in most of Abad's 6.7 assemblies, some moves can be performed, and each time we find a new assembly, we must take a moment to ponder if it is a solution or not.

For me it looks as difficult as the Mega Six.

It seems very hard to manufacture, because it has 8 internal corners (un-notchable, un-millable), including two opposite ones in a single voxel, the worse configuration for chiseling the pieces.

One interesting feature is that the total number of moves is 16. This is huge. More than the 14 moves of the Mega Six (which is nothing else than a variation of the Computer's Choice level 10, highest level with a unique solution).
Love's Dozen has 18 moves. Burr B has 15. The other high level burrs that I know have 14 moves or less.

I can't be sure, but Abad's level 6.7 may well be the 6-piece burr with the highest total number of moves and a unique solution (the Love's Dozen needs coloured pieces to have a unique solution).

This feature has never been completely analyzed because Bill Cutler's gigantic analysis of all 6-pieces designs only took into account the number of moves for the first piece. In that analysis, Abad's level 6.7 is just a random level 6 puzzle among millions of others.
It stands out because the second piece needs 7 more moves to be taken out.

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Paradox
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Re: 6-piece burrs

Post by Paradox » September 21st, 2011, 9:40 am

Very interesting! I've always thought that it's amazing how puzzles that look so similar can have completely different solutions. You should get yourself a set of boxed burr pieces!

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Pio2001
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Re: 6-piece burrs

Post by Pio2001 » September 21st, 2011, 11:58 am

I have got one. But burr sets only allow to make burrs with no internal holes. I find them less interesting, because except the "Impossible Second Move" of Bill Cutler, they can be disassembled in one move...

There are still other burrs that look interesting and that I have not tried, like Abad's level 4.9, Notchable 10 moves 11 holes (Ishino) etc.

It is possible also that burrs with more assemblies are the hardest. L5 Notchable most assemblies (Cutler), Notchable unique most assembles (Ishino), and Abad's Level 5 Most Assemblies have respectively 480, 688 and 896 assemblies for one solution.

They may be not so difficult, if we can find a clue thinking at the way they should be disassembled, but they may as well be insanely difficult, if we have to explore manually hundreds of possible assemblies until we find the right one.

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Pio2001
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Re: 6-piece burrs

Post by Pio2001 » September 24th, 2011, 10:46 pm

Writing my method about Solving Burrs, I was guessing that the sorting order of the puzzle pieces was important, and that using the most difficult pieces first allowed to reduce the solving time.

I didn't discuss it because I was not certain.

I have drawn diagrams showing the list of configurations through which the puzzles goes while the solving method is used. I took the Philippe Dubois for this purpose.

I can now confirm my guess. The exploration tree has 8 configurations and one dead end before getting to the solution with a given order for the pieces, and 15 configurations and 7 dead ends if the pieces are sorted differently.

Therefore using this kind of method, it is more effective to try first the pieces that seem to fit nowhere.

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Re: 6-piece burrs

Post by jack » September 29th, 2011, 11:21 pm

Pio2001 wrote: ... There are still other burrs that look interesting and that I have not tried, like Abad's level 4.9, ...
I was intrigued also and made this one. By 'spending' an extra move to free the first piece, the second piece can be removed by two moves (level 5.2).
A bit disapppointing, but still a nice design.

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Re: 6-piece burrs

Post by Pio2001 » September 30th, 2011, 12:14 pm

I see. I guess that the 4 official moves only serve the purpose of freeing the first piece, and then you have to return to the initial configuration, and perform 5 others until you reach the configuration of the 5.2 sequence, right ?

That's an intersting observation about the number of moves of a given puzzle. The shortest way to free a piece is not necessarily the most effective.

I see that you created a new 18-piece design (Century). How many assemblies does it have, taking the colours and edges into account ? If it has more than one, do you think that some clues would allow a puzzlist who is trying to assemble the pieces to eliminate the wrong ones ?

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Re: 6-piece burrs

Post by jack » September 30th, 2011, 7:15 pm

Hi Guillaume,
you're right about Abad's 4.9 puzzle. That's just what is going on.

Century has 8 resp. 12 assemblies in the two orientations of the colors. I haven't looked for clues for detecting unsolvable positions. You can take a look yourself shortly; I sent the design to Ishino to be published.

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Pio2001
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Re: 6-piece burrs

Post by Pio2001 » October 21st, 2011, 12:04 am

Hello,
I've been trying some other designs using LiveCubes.

First, Bill Cutler advised me to test Bill Baffling Burr, Love's Dozen, and Programmer's Nightmare.

Bill's Baffling Burr (Bill Cutler) :
Length : 6
Assemblies : 24
Solutions : 1
Moves : 5

This one is simpler than the ones that I tried above. I didn't keep it in my shortlist.

Love's Dozen (Bruce Love) :
Length : 6
Solutions : 1 with three colours
Moves : 12

I had already tried that one some time ago, and didn't keep it either, because it looked too close to the Modification of burr B and to the Peter Marineau (burr B). But this time, must admit that it is more confusing and interesting than the two previous ones.
The only drawback is that it is easier to assemble than the Peter Marineau thanks to the three colours necessary to enforce the unique solution.

Programmer's Nightmare (Bill Cutler) :
Length : 8
Assemblies : 102
Solutions : 1
Moves : 5 (rotation required)

What's special about this one is that it cannot be solved using only translations, a rotation is required. Besides this, it is rather easy to disassemble, compared to the Mega Six or the Love's Dozen, for example.

After that, I tried some other high level designs.

Abad's Level 5.7 Improved Burr (Rafael Guarinos Abad)
Length : 6
Assemblies : 11
Solutions : 1
Moves : 5.7

The disassembling sequence of this one seems based on the same principle as Abad's 4.9, tested by Jack. After the 5 first moves are performed, we must go back to the initial position, and start another move to go on.
Actually, the same thing happens in Love's Dozen, if I'm not mistaken, but without any piece coming out in the middle of the sequence.
It has a unique solution, however. So, though the design is completely different, is compares with the Peter Marineau, for example.

Notchable 10 Moves 11 Holes (Keiichiro Ishino)
Length : 10
Solutions : 1 if six dowels are used, like with the L46AA.
Moves : 10

I was surprised to see that this one is actually very close to Bill Cutler's L46AA Notchable. Actually, it is the same, with two voxels missing.
I didn't check with Burrtools yet, but it looks like the same six dowels in strategic positions allow to enforce a unique level 10 position, with, consequently, a quite easy assembling challenge. Like with L46AA.
It is also possible to keep two more voxels without changing the solution. The puzzle is then no more notchable or millable, but has less holes.

The voxels removed by Ishino allows to increase one of the dead ends of the L46AA. Therefore I think that this design is slightly better than the original. I hope to have this one made in wood one day.

Last, I tried
Notchable 9 Moves 7 Holes (Keiichiro Ishino)
Length : 6
Assemblies : 26
Solutions : 16
Moves : 9.2 max

I found this one to be quite close to the Burr B (Peter Marineau), Modification of Burr B, and Love's Dozen. I found the level 9 solution assembling the pieces like in these other designs. I didn't check if three colours were enough to have one solution only. I still find Love's Dozen to be the more interesting of this small family.

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Re: 6-piece burrs

Post by Pio2001 » November 27th, 2011, 2:09 am

Some quick notes about the crosses that I tried recently.

Abad's level 9
Length : 6
Assemblies : 2
Solutions : 1
Moves : 9

It has an ambiguous piece. The right assembly is not too difficult to find. The disassembling sequence has no real dead end except a very short one at the beginning. A bit further, there is a move of two pieces at once that is not obvious. That's more interesting than the variations around the Piston (Burr B, Love's Dozen...), but I still prefer the brillant dead end of Cutler's Computer's choice 5 holes.
It seems very difficult to manufacture. It has 6 internal corners, including 2 around the same voxel.

D Super (Peter Rösler)
Length : 6
Assemblies : 38
Solutions : 1
Moves : 9

Another level 9 burr. A bit less interesting than Abad's level 9 and Computer's choice 5 holes, I'd say. It hasn't got the subtle move of the first, nor the dead end of the second. But the disassembling sequence still looks a bit more complex than the Piston and Modification of Burr B ones.
The assembling is long to find because of the high number of assemblies.

Abad's level 5 most assemblies
Length : 8
Assemblies : 896
Solutions : 1
Moves : 5

This is certainly the most difficult 6-piece cross ever !
The designers of six-piece crosses have often looked for the most possible assemblies. Cutler found 480 (L5 Notchable Most Assemblies), and Ishino 688 (notchable Unique Most Assemblies).
It means that starting from the disassembled pieces, you may try hundreds of possibilities until you find the solution.
The disassembling sequence is not difficult, but quite nice, with two groups of 3 pieces getting apart. Each group is made of completely intricated pieces, which makes it easy to build, unlike the Mega Six, where the pieces don't stand together well before being assembled.
The pieces look ordinary, which makes difficult to find a strategy to sort them and try to optimize the search for the right assembly. If I sort them in the order that should give the fastest resolution, it turns out that the second piece is actually the last one of my list.
In short, no clue seems to help the resolution, which borders the inhuman.

An interesting game would be, Knowing the solution first, to find an intuitive way to assemble the pieces instead of following Burrtools and preparing two groups of 3 pieces in bizarre positions.

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