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 Post subject: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sat Jan 21, 2012 3:39 pm 
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The 9 colour Ball Sudoku Cube puzzle was already mentioned by Juanan in the marketplace section.
Since it can be considered to be a new puzzle (or at least a puzzle with new features), I am presenting some more background information about the puzzle here.

The Ball Sudoku Cube puzzle consists of 26 balls in 9 colours.

Attachment:
9 colour Ball Sudoku Cube.JPG
9 colour Ball Sudoku Cube.JPG [ 742.69 KiB | Viewed 2795 times ]


Given a situation of having maximum 3 balls of the same colour, there is a unique arrangement regarding the distribution of the colours over the centers, edges and corners of a 3x3x3 cube.
This unique arrangement is a follows:
- Colour number 1 has 2 corner balls
- Colour numbers 2-7 have 1 corner ball, 1 edge ball and 1 center ball
- Colour numbers 8 and 9 have 3 edge balls.

The reason for this unique arrangement can be explained as follows:
- There are 26 balls. Since each colour appears at maximum 3 times, there is 1 colour with 2 balls [colour number 1] and 8 colours with 3 balls
- Colour number 1 has to appear at 2 corners in diametrical positions since otherwise the 2 balls cannot cover all 6 planes
- The remaining corners have to be coloured differently [colours 2-7] since otherwise it would not be possible to position a 3rd ball of this colour
- Remaining then are 12 edge positions and 6 center positions for which we have 6 sets of 2 balls [colours 2-7] and 2 sets of 3 balls [colours 8-9]
- The colours 8 and 9 have to be at the edges since otherwise these 3 balls cannot cover 6 planes
- Colours 2-7 each occupy a corner (each covering 3 faces). So, per colour 3 faces still need to be covered which can only be realised by combinations of a center + an edge.
So, with maximum 3 balls per colour there is only one unique arrangement possible of distributing the colours over the centers, edges and corners.

Randomly trying to solve a 9 colour Ball Sudoku puzzle is probably difficult, but it becomes easier when considering the distribution of the colours over the corner, edge end center positions.
Basically, every ‘middle’ layer consists of 4 fixed center balls + the following 4 balls/colours:
- one of each of the 2 balls of the colours 8 and 9 (the colours that appear as sets of 3 edge balls)
- the colour that also appears in the center of the bottom layer
- the colour that also appears in the center of the top layer.

Assuming that the bottom layer is solved, a logical sequence of solving the edge balls in the middle layer is:
- positioning the edge ball having the same colour as the center ball of the top layer [only 1 position possible]
- positioning each of the 2 edge balls out of the 2 sets of 3 edge balls [colours 8 and 9, each time only 1 or 2 positions possible]
- positioning the edge ball having the same colour as the center ball of the bottom layer [remaining position].

This can all be achieved easily by using known sequences of e.g. the layer by layer method of solving a regular Rubik’s Cube.
The same goes for correctly positioning the remaining balls in the top layer.

There are some additional restrictions, meaning that the procedure above will not always lead to a solution.
On the other hand, there are several solutions possible.
One of the solutions, shown schematically below, is nicely symmetrical.
The colours 1-7 are listed, the colours 8 and 9 easily fit in the remaining open spaces.

Attachment:
9 colour Ball Sudoku Cube puzzle with symmetrical solution.jpg
9 colour Ball Sudoku Cube puzzle with symmetrical solution.jpg [ 27.72 KiB | Viewed 2795 times ]


The symmetry becomes visible when holding the two corner balls of the same colour [colour number 1] as north pole and south pole.
Between both poles 5 rings/levels can be observed:
Level 1: 3 edge balls
Level 2: 6 balls: 3 corner balls and 3 center balls
Level 3: 6 edge balls
Level 4: 6 balls: 3 corner balls and 3 center balls
Level 5: 3 edge balls

In this symmetrical solution level 1 and level 4 contain the same balls and the same goes for levels 2 and 5.
The balls in level 2 and 4 form a kind of circle around the poles.
Level 3 contains the 2 sets of 3 edge balls, forming a kind of equator with 2 alternating colours.

The puzzle is available for pre-order via Meffert’s and the Jade Club.
The final colours might deviate a bit from the initial sample shown.

Geert


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 Post subject: Re: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sat Jan 21, 2012 5:28 pm 
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This is starting to look a lot like Qubami - in both design/shape (rounded balls vs cubes) and 3D sudoku concept...

Anyway, here's a 3-colour sudoku version (solution), before anybody else posts it:

Attachment:
3-colour ball sudoku cube.png
3-colour ball sudoku cube.png [ 3.23 KiB | Viewed 2725 times ]

Attachment:
3-colour ball sudoku cube 2.png
3-colour ball sudoku cube 2.png [ 46.89 KiB | Viewed 2711 times ]

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 Post subject: Re: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sun Jan 22, 2012 2:32 am 
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Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Very nice Geert.
I remember three months ago you told me: "I have invented only one puzzle so far." I guess that has changed.


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 Post subject: Re: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sun Jan 22, 2012 2:49 am 
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Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Now that I see the logic behind this puzzle, I may consider picking one up at a later date. I'm always looking for new, fun, and exiting thing to do with an old cube. The solid colored balls actually look better than the multicolor versions.

And yay for a sudoku cube with no gosh awful numbers on it!

This puzzle kind of reminds me of a game I used to play with scrambled cubes, which was to arrange the cube so that no two same-colored squares are allowed to touch each other (cube must remain scrambled the whole time and pretty patterns are not allowed).

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 Post subject: Re: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sun Jan 22, 2012 6:03 am 
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By the way, do all solutions also have 8 different colours on each layer, as well as 9 different colours on each face? It must be the case for the six outer layers as they are formed by each face, but I was wondering about the 3 middle layers. I hope that is the case, as it would be more like an original sudoku puzzle... :D

PS. The balls look like scoops of ice cream in different flavours, or snooker balls - snookerdoku! 8-)

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 Post subject: Re: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sun Jan 22, 2012 8:21 am 
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About the question by Kelvin: “do all 3 middle layers have 8 different colours on each layer?”, the answer is yes.
Take e.g. a center in the top layer. The corresponding corner of the same colour will then have to be in the bottom layer since all other 4 corners are in the top layer.
Now only 2 faces are not covered which can only be covered with an edge of the same colour in the middle layer.
So all middle layers contain all 6 colours of the centers.
All middle layers also contain 2 of the edges out of the 2 sets of 3 edges of the same colour.
No middle layer can have 2 of such identically coloured edge parts since in such a case the third edge part of the same colour would not be able to cover the remaining top and bottom layer.
So indeed, all 3 middle layers have 8 different colours on each layer.

Geert


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 Post subject: Re: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sun Jan 22, 2012 8:43 am 
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Joined: Mon Oct 22, 2001 12:45 pm
Actually, I should not get credit for having ‘invented’ this puzzle since the concept was already known and applied to other puzzles.

The oldest example of a 9 colour 9 piece distribution over a 3x3x3 cube known to me is a puzzle by Sivy Farhi.
He used 9 sets of 3 wooden blocks glued together to form “L” shapes with the objective to create 9 colours at each of the 6 faces.
So, this is a kind of assembly puzzle (not a twisty puzzle).
Attachment:
9 colour assembly cube by Sivy Farhi.jpg
9 colour assembly cube by Sivy Farhi.jpg [ 990.39 KiB | Viewed 2493 times ]


Another example is the beautiful 3x3x3 cube made by Jean-Louis Mathieu which he presented at the French Cube Day in 2008.
Attachment:
French Open 2008 9 colour cube by JLM.jpg
French Open 2008 9 colour cube by JLM.jpg [ 520.18 KiB | Viewed 2493 times ]


There are also commercially available 3x3x3 cubes called ‘8-colour cube’ which apply 8 colours (the centers remain blank). These puzzles have corners that have 3 faces of the same colour but edges that still have 2 colours. So, that is more limited.
Attachment:
2 types of 8 colour 3x3x3 cubes.jpg
2 types of 8 colour 3x3x3 cubes.jpg [ 255.22 KiB | Viewed 2493 times ]


Two weeks ago (January 4th), I received a package containing the Welness Ball cube. I like the puzzle a lot but I considered it to be a bit of a pity that the nice spheres of the puzzle were divided into 2 or 3 parts to apply different colours in order to create the 6 colour cube pattern.
It would seem more elegant to me to keep all spheres in uniform colours. Clearly in that way a Rubik’s Cube pattern would no longer be possible but the Sudoku 9-colour pattern would be an alternative.

So, I suggested this option to Uwe mentioning at the same time the examples listed above.
Uwe decided to immediately make some samples and to show these at the Toy Fair of last week.
I was really impressed by this quick action.

Knowing that puzzle samples would be made with the 9 colours, I started thinking about the solutions and possible arrangements. I discussed this with Jacques Haubrich who is a close friend from NKC and the Cube Days and an expert in pattern puzzles and matching puzzles. He is also a former teacher and professor in mathematics and my math teacher at high school (indeed a couple of years ago).
Jacques was the one who pointed out to me that the distribution of the 9 colours over the centers, edges and corners of a 3x3x3 cube would be unique in case per colour a maximum of 3 balls would be used. Actually, I was a bit surprised by that at first and I believe that this fact adds to the uniqueness of the puzzle.

Jacques also investigated other solutions.
Including the already mentioned symmetrical solution, he came to the following 5 solutions (not counting rotations/mirroring):
Attachment:
9 colour Ball Sudoku Cube puzzle 5 solutions.jpg
9 colour Ball Sudoku Cube puzzle 5 solutions.jpg [ 156.6 KiB | Viewed 2493 times ]


Uwe mentioned to me to be looking into additional challenges for the Ball Sudoku Cube puzzle, e.g. in terms of other, easier patterns etc.
This may also include different number of colours and/or more than 3 balls per colour.
So, the proposal by Kelvin is very useful.

One pattern I proposed (using the same colour distribution of the 9 colour Ball Sudoku Cube puzzle) is the following:
It is based on the fact that there are 2 (corner) balls of 1 colour and 8 sets of 3 balls.
Of these 8 sets of 3 balls, 2 sets only appears as ‘edges’ and 6 sets appear as sets of ‘corner-edge-center’ pieces.
Grouping 3 edge pieces having the same colour can be done around one corner. There are 2 sets of such edge pieces and 2 corners having the same colour, so these can be brought together.
Next, the 6 remaining sets of edge-corner-center pieces can be grouped together to form L-shaped patterns, each time covering a center, edge and corner piece.
It is probably easier to show on a real cube, but below is such a pattern.
Attachment:
9 colour cube, alternative patterrn.jpg
9 colour cube, alternative patterrn.jpg [ 54.76 KiB | Viewed 2493 times ]


In case you have other proposals for additional challenges, possibly also including different colour sets, Uwe will be very interested.
Please let him (and/or me) know.


Geert


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 Post subject: Re: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sun Jan 22, 2012 3:03 pm 
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Hey, how about my 8-color SG Cube? :lol:


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 Post subject: Re: The 9 colour Ball Sudoku Cube puzzle
PostPosted: Sun Feb 19, 2012 7:16 am 
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Joined: Tue Sep 06, 2011 4:57 am
Location: Japan
I made same puzzle, color is transposing to number.
Image

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