Cubing challenge: peel the stickers off

[bcube]

This is the first cube.
I've made it in +/-30 minutes without so much hustle.

Cool. How about this one?



On a standard Rubik's Cube, the aim is to arrange the puzzle in such a way that every face has only one color; on the 9-color cube, the goal is opposite: to arrange it so that every face contains all nine colors exactly once.

 

[Tito_H]

Using the Gradient Cube concept, I get a 3x3 with 20 stickers as the minimum. Can this be reduced any further?

I don't think that a 3x3 can be reduced to less than 20 stickers.
The reason is that we can not tell what the colors are in the following 3 cases.

(2 EPs do not have stickers)


(2 CPs do not have stickers)


(Only 1 center has a sticker)

 

[jamol]

Cool. How about this one?

On a standard Rubik's Cube, the aim is to arrange the puzzle in such a way that every face has only one color; on the 9-color cube, the goal is opposite: to arrange it so that every face contains all nine colors exactly once.

Tried to solve it on my computer with mouse input but it's uncomfortable for me, so I solved on mobile with touch. Turning is hard on mobile too.

 

[bcube]

Tried to solve it on my computer with mouse input but it's uncomfortable, so I solved on mobile with touch. Turning is hard on mobile too.
View attachment 44683 View attachment 44684

Both PC and mobile inputs are comfortable for me (I prefer PC because I can use right mouse button (for cube rotation) there too which is kind of standard for the PC simulators I use).

Great to see you solved it, although I wanted Tito_H to come up with his own solution on the fly just as I did, not to reproduce already solved state.

 

[jamol]

Both PC and mobile inputs are comfortable for me (I prefer PC because I can use right mouse button (for cube rotation) there too which is kind of standard for the PC simulators I use).

I said turning hard on mobile but it maybe a bug or I did something wrong. On mobile if I turn(or rotate maybe) the whole cube and stop then continue turning, the cube turns like a glitch and the page itself starts moving. I have a video but couldn't attach it.

Great to see you solved it, although I wanted Tito_H to come up with his own solution on the fly just as I did, not to reproduce already solved state.

I learned this way of solving for testing when I made the cube. I needed a method to solve fast to test, and now I only use this method.
And I think cube has around 80 solutions, so Tito_H can come up with another solution.

 

[bcube]

I said turning hard on mobile but it maybe a bug or I did something wrong. On mobile if I turn(or rotate maybe) the whole cube and stop then continue turning, the cube turns like a glitch and the page itself starts moving. I have a video but couldn't attach it.

I can not reproduce the issue on any of my mobile devices, so based on your description I am not able to tell exactly what seems to be the problem on your end.

I learned this way of solving for testing when I made the cube. I needed a method to solve fast to test, and now I only use this method.
And I think cube has around 80 solutions, so Tito_H can come up with another solution.

Well, if you want to know how Tito_H feels right now, you could try solving the 16-color 4x4x4 cube I guess

 

[Tito_H]

Last night I was thinking the strategies & solution in my bed, and several ideas came up (while dreaming).
I don’t know if I have time today, but will reply asap.

 

[jamol]

I can not reproduce the issue on any of my mobile devices, so based on your description I am not able to tell exactly what seems to be the problem on your end.

Here's the video of problem I had a lot when turning the whole cube on mobile. The page itself starts moving and cube becomes kinda glitchy:

And I think I'm used to my own app, so maybe for this reason it's uncomfortable for me.

 

[bcube]

Here's the video of problem I had a lot when turning the whole cube on mobile. The page itself starts moving and cube becomes kinda glitchy:

And I think I'm used to my own app, so maybe for this reason it's uncomfortable for me.

I see. Thank you for reporting it and sending the video. Now I can reproduce the issue on mobile Chrome (which is not my default mobile browser).

If I remember correctly, this issue has something to do with auto-hide address & tab bars having different functionality across mobile browsers.

For instance, I can not reproduce the issue on any of these simulators if the address bar is already visible in mobile Chrome.

Let me check with the forum moderation/administration team if the issue can be easily fixed here. Apologies for (hopefully only temporary) inconvenience.

 

[Tito_H]

Last night I was thinking the strategies & solution in my bed, and several ideas came up (while dreaming).
I don’t know if I have time today, but will reply asap.

I completed the task in about 30 minutes. However, I spent several hours spinning the cube in my dream, so it probably took longer than that.

My strategies were as follows:
(Cube orientation: Yellow up, Blue front)

First, I divided the pieces into the following groups:

• Group A: Colors with only CP (green)
• Group B: Colors with only EP (red, magenta)
• Group C: Colors with a Center, CP, and EP (yellow, orange, blue, black, brown, white)

Next, I considered the placement of the pieces in each Group.

• I placed 2 CPs of the Group A on the diagonal position of the cube, since CP spans 3 faces of the cube, and these 3 faces should not be overlapped with other 3 faces.
• I placed the 6 EPs of the Group B on the edge common to the 2 sides of the cube, dividing the 6 faces into 3 consecutive two-sided sections. I first placed magenta on FL and red on RB positions, then decided the placement of the remaining 2 pieces in each section. For reference, this is the same position as the “six stars” on Master Yoda's cube.
• Group C has one Center, one CP, and one EP per color. The CP is placed in one of the four corners on the side opposite the Center. The EP's position is automatically determined once the positions of the Center and the CP are determined.

Finally, based on the above, I mentally considered the placement of all colors and came up with a pattern with no overlapping colors on any face.

After that, I completed Cross and F2L, then changed the position of the CPs and the EPs, using A and U perms respectively to complete LL.



What I found interesting was the shape that results when lines connect the same-colored parts of each group.

• Group A is the diagonal of the cube, so it is a line segment.
• Group B is an equilateral triangle, and two of the colors are "congruent shapes."
• Group C is a acute triangle, and all of the colors are "congruent shapes."

In other words, there are only three types of shapes made up of the same color inside the cube!

 

[Tito_H]

Since this is a combination of three simple shapes, I wondered if there were many possible completed patterns, so I decided to investigate how many there were.

The cube orientation is Yellow up, Blue front.

I will investigate the number of possible patterns by setting the Blue CP as UBR and the Blue EP as DL.



1. Consider the placement of the Brown CP which is on the opposite side of the Blue face. The Brown CP can be placed in one of the corners UFR, UFL, DFL, or DFR on the Blue face. However, if it is placed in UFR, the Brown and Blue EPs will overlap in DL, so UFR is not an option.

1-1. When placing the Brown CP in UFL:
Since two corners of the U face are filled, the third corner is filled with one of the Green CPs located diagonally across from the cube. The Green CP is convenient because it can fill two corners at once. This applies to either UFR&DBL or UBL&DFR.

1-1-1. When placing the Green CPs in UFR&DBL:
Since the three corners of the U face are filled, an Orange CP, the center color of the D face, is placed in the remaining corner, UBL, of the same face. Similarly, since the three corners of the L face are filled, a Black CP is placed in DFL. The White CP can go in either DFR or DBR, but if it is placed in DFR, the Black and White EPs will overlap in UB, so it is placed in DBR. Finally, the Yellow CP is placed in DFR.

Confirm that the Red and Magenta EPs are placed correctly and the process is complete. Note that the Red and Magenta colors can be swapped, so there are two possible patterns.


(Diagram with red and magenta swapped is omitted)

1-1-2. When placing the Green CPs in the UBL&DFR:
Since this is symmetrical to 1-1-1 (with respect to the vertical plane containing the Blue and Brown CPs), applying the same idea, the Orange CP will be placed in the UFR, the White CP in the DBR, the Black CP in the DFL, and the Yellow CP in the DBL.


(Diagram with red and magenta swapped is omitted)

1-2. When placing the Brown CP in the DFL:
Let's consider the placement of the Green CPs. The targets are UFR&DBL, UFL&DBR, or UBL&DFR.

1-2-1. When placing the Green CPs in UFR&DBL:
Consider the placement of an Orange CP. The targets are either UFL or UBL.

1-2-1-1. When placing the Orange CP in UFL:
Since the three corners of the L face are filled, a Black CP, the center color of the R face, is placed in the remaining corner UBL on the same face. Consider the placement of a White CP. The targets are either DFR or DBR.

1-2-1-1-1. When placing the White CP in DFR:
The Yellow CP is placed in DBR.


(Diagram with red and magenta swapped is omitted)

1-2-1-1-2. When placing the White CP in DBR:
The Yellow CP is placed in DFR.


(Diagram with red and magenta swapped is omitted)

1-2-1-2. When placing the Orange CP in UBL:
Since the three corners of the L face are filled, a Black CP, the center color of the R face, is placed in the remaining corner UFL of the same face. Consider the placement of a White CP. The targets are either DFR or DBR.

1-2-1-2-1. When placing the White CP in the DFR: The Yellow CP is placed in the DBR.


(Diagram with red and magenta swapped is omitted)

1-2-1-2-2. When placing the White CP in the DBR: The Yellow CP is placed in the DFR.


(Diagram with red and magenta swapped is omitted)

1-2-2. When placing the Green CPs in the UFL&DBR:
Consider the placement of the Orange CP. The targets are either UFR or UBL.

1-2-2-1. When placing the Orange CP in the UFR:
Three corners of the U face are filled, and the remaining corner of the same face, the UBL, cannot accommodate the Yellow and White CPs, so a Black CP is placed there. The White CP cannot be placed in the DBL, so it is placed in the DFR. Finally, the Yellow CP is placed in the DBL.

However, as shown in the figure below, the Red and Magenta EPs cannot be placed correctly, so this is excluded.



1-2-2-2. When placing the Orange CP in the UBL:
This is symmetrical to 1-2-2-1 (with respect to the vertical face containing the Blue, Brown, and Green CPs), so applying the same logic, the White CP is placed in the UFR, the Black CP in the DBL, and the Yellow CP in the DFR.


(Diagram with red and magenta swapped is omitted)

1-2-3. When placing the Green CPs in UBL&DFR:
This is symmetrical to 1-2-1 (with respect to the vertical plane containing the Blue and Brown CPs), and there are 8 patterns. Details omitted.

Consider the placement of the Orange CP. The targets are UFR and UFL.

1-2-3-1. When placing an Orange CP in UFR:
Since the three corners of the R face are filled, the White CP, the center color of the L face, is placed in the remaining corner, DBR, of the same face. The Yellow CP cannot be placed in UFL, so it is placed in DBL. Finally, the Black CP is placed in UFL.


(Diagram with Red and Magenta swapped is omitted)

1-2-3-2. When placing the Orange CP in UFL:
Since the three corners of the L face are filled, the Black CP, the center color of the R face, is placed in the remaining corner, DBL, of the same face. The Yellow CP cannot be placed in UFR, so it is placed in DBR. Finally, the White CP is placed in the UFR.

However, as shown in the figure below, the Red and Magenta EPs cannot be placed correctly, so they are excluded.



1-3. When placing the Brown CP in DFR:
This is symmetrical to 1-1 (with respect to the plane containing the Blue center, CP, and EP), and there are following 4 patterns. Details omitted.


(Diagram with Red and Magenta swapped is omitted)


(Diagram with Red and Magenta swapped is omitted)

The above are the results of examining the Blue CP as UBR and the Blue EP as DL, and there are 26 20 patterns in total.

The Blue CP can also be placed in UBL, DBL, or DBR, but if the cube is rotated by z, z2, or z3, respectively, it will be in the same position as when the Blue CP is UBR. Therefore, it will be as follows.

If the Blue CP is UBL: Instead of Yellow, Black, Orange, and White, it will be White, Yellow, Black, and Orange, respectively, for a total of 26 20 patterns.

If the Blue CP is DBL: Instead of Yellow, Black, Orange, and White, it will be Orange, White, Yellow, and Black, respectively, for a total of 26 20 patterns.

If the Blue CP is DBR: Instead of Yellow, Black, Orange, and White, it will be Black, Orange, White, and Yellow, respectively, for a total of 26 20 patterns.

As a conclusion, the total number of possible patterns is 26x 4 = 104. 20 x 4 = 80.

 

[bcube]

Here's the video of problem I had a lot when turning the whole cube on mobile. The page itself starts moving and cube becomes kinda glitchy:

Please check again. Everything should be back to normal.

I completed the task in about 30 minutes

Very nice! It took me a couple of hours to come up with my own solution (details in next post) for the 9-color cube.

What I found interesting was the shape that results when lines connect the same-colored parts of each group.

• Group A is the diagonal of the cube, so it is a line segment.
• Group B is an equilateral triangle, and two of the colors are "congruent shapes."
• Group C is a acute triangle, and all of the colors are "congruent shapes."

In other words, there are only three types of shapes made up of the same color inside the cube!

Although I see where are you getting with this, I personally like to think of it in terms of planes - all 3 groups form a plane (as you were saying), groups A and B are cutting the cube in 2 equal parts, and if I am visualizing group C correctly, it is cutting the cube in 2 equal parts too.

 

[PapaSmurf]

I just tried the puzzle that was generated in the OP and it was great! ZZ with keyhole seems to be the way to go. I did find that I got a sort of parity state with my EO, but maybe that was my fault all along.

 

[LukeKing67]

Here's the video of problem I had a lot when turning the whole cube on mobile. The page itself starts moving and cube becomes kinda glitchy:

And I think I'm used to my own app, so maybe for this reason it's uncomfortable for me.

Is uncomfortable for me to. I'm on mobile.

 

[bcube]

Since this is a combination of three simple shapes, I wondered if there were many possible completed patterns, so I decided to investigate how many there were.

Nice analysis in that post. I kind of did the same analysis when I tried to solve the 9-color cube with one difference: I didn´t come up with a solution before even solving the puzzle, I was figuring out (gradually eliminating) the solution while solving the puzzle. The consequence is that my solution will lead to a different solved state each time.

I did the corners-first solution, my idea being that if corners are solvable, the edges would be solvable as well. When the white center is facing down, the white corner belongs to the top layer. Green corner belongs to the top layer as well. Similarly, orange and blue corners belong to the top layer as well, so I made these corners solvable by placing them "correctly". Next I made the corners in the bottom layer solvable, so that all corners are in a solvable state.

Next I tried to solve, one by one, the edges from your group C (edge-center-corner group). I indeed encoutered a situation in which 2 edges required to be placed in one place, realizing I need to find another solvable state for corners - so I cycled the corners (green corner not included in this cycle) in one layer so that the 8 corners would be still solvable relative to the centers. Eventually, that lead to the corners and all except red and magenta edges being solvable.

As my last step, I tried to solve the group B pieces (red and magenta edges). I indeed encoutered a situation in which these edges are not solvable, realizing I need to make an E or E´ move. Only one of these moves will make the corners still solvable, so I went with that move. Next I solved the edges from group C again, and finally, group B pieces would be solvable, thus solving the whole puzzle.

As a conclusion, the total number of possible patterns is 26x 4 = 104.

Jaap is stating there are 80 solutions (see the section "Nine-Colour Cube"). Is it possible that you counted the same state multiple times? For example 3 states with cycled red edges should be counted as 1 state because all 3 edges are indistinguishable.

 

[bcube]

Is uncomfortable for me to. I'm on mobile.

If that is the case, go here and view its source code (typing "view-source:" in front of URL works for me). Refresh the page with source code. Now you should see a page starting with:

<script type="text/javascript">
/**
 * @author Josef Jelinek
 * @version 3.5b
 * Conversion to JavaScript by Michael Feather
 */

Go back to this page, refresh it, and you should not see any glitches anymore when rotating the 3x3x3 cube or twisting its layer.

 

[jamol]

Please check again. Everything should be back to normal.

I checked it on mobile and yes the problem is fixed.
And I noticed something else but don't know should I tell about it because it's a minor issue.

 

[Tito_H]

Jaap is stating there are 80 solutions (see the section "Nine-Colour Cube"). Is it possible that you counted the same state multiple times? For example 3 states with cycled red edges should be counted as 1 state because all 3 edges are indistinguishable.

Last night I noticed my error on 1-2-3.

I will revise later that section as follows, and the total number.

_____________________________________
1-2-3. When placing the Green CPs in UBL/DFR:
Consider the placement of the Orange CP. The targets are UFR and UFL.

1-2-3-1. When placing an Orange CP in UFR:
Since the three corners of the R face are filled, the White CP, the center color of the L face, is placed in the remaining corner, DBR, of the same face. The Yellow CP cannot be placed in UFL, so it is placed in DBL. Finally, the Black CP is placed in UFL.


(Diagram with Red and Magenta swapped is omitted)

1-2-3-2. When placing the Orange CP in UFL:
Since the three corners of the L face are filled, the Black CP, the center color of the R face, is placed in the remaining corner, DBL, of the same face. The Yellow CP cannot be placed in UFR, so it is placed in DBR. Finally, the White CP is placed in the UFR.

However, as shown in the figure below, the Red and Magenta EPs cannot be placed correctly, so they are excluded.


_____________________________________

 

[bcube]

I checked it on mobile and yes the problem is fixed.

Glad to hear that. Thanks for checking.

And I noticed something else but don't know should I tell about it because it's a minor issue.

I am all ears. If you want to keep discussing publicly, I suggest switching to the AnimCubeJS thread where the discussion would be more appropriate I think, If you want to keep the conversation private, you can send me a direct message. Don´t be shy and tell me .

 

[Tito_H]

Last night I noticed my error on 1-2-3.

I will revise later that section as follows, and the total number.

_____________________________________
1-2-3. When placing the Green CPs in UBL/DFR:
Consider the placement of the Orange CP. The targets are UFR and UFL.

1-2-3-1. When placing an Orange CP in UFR:
Since the three corners of the R face are filled, the White CP, the center color of the L face, is placed in the remaining corner, DBR, of the same face. The Yellow CP cannot be placed in UFL, so it is placed in DBL. Finally, the Black CP is placed in UFL.


(Diagram with Red and Magenta swapped is omitted)

1-2-3-2. When placing the Orange CP in UFL:
Since the three corners of the L face are filled, the Black CP, the center color of the R face, is placed in the remaining corner, DBL, of the same face. The Yellow CP cannot be placed in UFR, so it is placed in DBR. Finally, the White CP is placed in the UFR.

However, as shown in the figure below, the Red and Magenta EPs cannot be placed correctly, so they are excluded.


_____________________________________

I duly corrected 1-2-3 section, and fortunately the total number of the possible solution became 80, as indicated by Jaap