Athefre
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- Jul 25, 2006
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Last year I presented the concept of recognizing steps using the minimum number of stickers, now called Straughan Recognition. Recently I was thinking about the application of the concept to the Mirror Cube and realized that the required number of stickers to recognize steps is fewer than that of Rubik’s Cube. This is because each “sticker” of each face of the Mirror Cube is unique, unlike the single color faces of Rubik’s Cube. I thought about how the Rubik's Cube color scheme could be altered to make each sticker unique and allow for this truly minimum sticker recognition and thought of several ideas. Once you’ve read through the examples and the step recognition reduction, maybe you could try making your own unique sticker cube and try solving as a fun challenge.
Gradient Cube: This is the cube that I have designed. Each piece is distinguished from all angles and there is no need to check additional information. This allows for perfectly optimized step recognition using the minimum necessary amount of information, which is described more later in this post. The potential negatives are its use by color blind solvers and in poor lighting conditions. But considering that each face only needs four unique stickers (because corners, edges, and centers are obviously different pieces), it is likely that there are colors that can be chosen that work well in all situations. A gradient cube is the color scheme idea that I had while considering recognition on the Mirror Cube. The standard pattern that I have designed is a spiral of matching corner and edge pairs going around each face. A physical version has been created after conversations with OreKehStrah, who tested various shades to find ones that are easily distinguishable.
Below is the original prototype stickered version by OreKehStrah:
There are many other possible ways to create a gradient cube, as we have tested in my Discord server. This is just the standard example. Even a 54 color cube could be used.
Super Cube: The super cube does something similar. A small bit of the side colors are shown at the edges of each sticker. This way there is no hidden information. This sticker scheme accomplishes the goal of distinguishing pieces from any angle. However, the issue is that the solver will still be checking the same amount of information as Rubik's Cube, because it still requires looking at the additional bits of color on each sticker. So the super cube isn't as efficient as the gradient style. The center pieces could be a single color if we want to remove the necessity of correctly orienting the centers.
Markers: Numbers, letters, or symbols could be placed over the current BOY color scheme to distinguish stickers. This contains the same advantages as gradient style, but may not look as nice. It’s possible that there could be a way to design it so that everything has an appealing look. I think it would be interesting to create a physical Speffz cube, to align with a BLD standard.
Mirror Cube: The Mirror Cube was previously mentioned and also contains the desired advantages. Differently shaped pieces or stickers allows for unique stickers among all pieces. Recognition may or may not be more difficult, depending on how well someone can learn the necessary piece shapes.
OreKehStrah has applied the minimum sticker concept to other WCA puzzles. Click below to check them out. If you want to try out a virtual version of a 3x3 sticker scheme, crystalcuber has designed a program where you can apply a custom scheme using a layout image.
Below are some examples of how steps can be recognized using the minimum number of stickers. It’s pretty amazing what becomes possible. The examples use the gradient cube, but the same thing applies to the other scheme types.
Solved state U face stickers
CLL / CMLL: Just 3 total stickers are all that is needed to recognize the final layer of corners. For example, the UFL, UFR, and UBR stickers uniquely identify each case. To maintain the standard 42 patterns, you choose a specific point among each 4 sticker orientation to check 3 stickers that belong on the U face. If you freely start at any random point and truly only look at just the three current U stickers at UFL, UFR, and UBR, this creates a more advanced version with 162 patterns.
Setup: U2 R U2 R’ U’ R U’ R’ (Permuted Sune)
PLL: The minimum number of stickers to recognize PLL is 5. PLL can be recognized using the UFL, UF, UFR, UR, and UBR stickers as an example. However, there is something much more interesting here. OLL also involves checking the U face stickers and PLL can be recognized using just U face stickers. This means that PLL can be very easily recognized and predicted during OLL. There is no need to check any side stickers.
Setup: R’ U L’ U2 R U’ R’ U2 R L U’ (J Perm)
ZBLL: ZBLL can be recognized using 5 total stickers. A standard example is the U face stickers that are currently within the UFL, UF, UFR, UR, and UBR pieces. Similar to CLL, a basic and advanced number of patterns also applies here.
Setup: U2 R U2 R’ U’ R U’ R’ (Permuted Sune)
1LLL: 6 stickers is the minimum number to recognize 1LLL. Add in the UL edge to the ZBLL piece set to complete 1LLL recognition.
Setup: U2 r U2 R’ U’ R U’ r’ (Wide permuted Sune)
Edge orientation: When determining edge orientation during CFOP edge control, the Petrus edge orientation step, or in other methods, it can be difficult when stickers are out of view. The setup R U’ R’ U’ F’ U F is an example where two matching edge stickers are at UL and UB. But using a unique sticker cube this issue doesn’t exist.
Roux 4c: One of the minor annoyances with the Roux method is the final sub-step of permuting the middle slice edges. The 3-cycles that start with an M2 are impossible to distinguish when simply looking at them. Recognition methods, such as the one called DFDB, have been developed to handle this. With a unique sticker cube, there would be no need to have a recognition method and take up valuable mental processing power during previous sub-steps.
BLD: It is now possible to recognize corners and edges using just one sticker from each. This means that corners can always be memorized by looking at the U, F, and R faces without any tilt or rotation.
DR: This also benefits speed solving oriented DR methods or steps, such as MI2, SSC, Square-101, the separation variant of Mehta, or DR applied to Petrus like methods such as LEOR and CEOR. The DR state can be recognized entirely using the upper face stickers when corners are in the <U, R> permutation state.
Gradient Cube: This is the cube that I have designed. Each piece is distinguished from all angles and there is no need to check additional information. This allows for perfectly optimized step recognition using the minimum necessary amount of information, which is described more later in this post. The potential negatives are its use by color blind solvers and in poor lighting conditions. But considering that each face only needs four unique stickers (because corners, edges, and centers are obviously different pieces), it is likely that there are colors that can be chosen that work well in all situations. A gradient cube is the color scheme idea that I had while considering recognition on the Mirror Cube. The standard pattern that I have designed is a spiral of matching corner and edge pairs going around each face. A physical version has been created after conversations with OreKehStrah, who tested various shades to find ones that are easily distinguishable.
Below is the original prototype stickered version by OreKehStrah:
There are many other possible ways to create a gradient cube, as we have tested in my Discord server. This is just the standard example. Even a 54 color cube could be used.
Super Cube: The super cube does something similar. A small bit of the side colors are shown at the edges of each sticker. This way there is no hidden information. This sticker scheme accomplishes the goal of distinguishing pieces from any angle. However, the issue is that the solver will still be checking the same amount of information as Rubik's Cube, because it still requires looking at the additional bits of color on each sticker. So the super cube isn't as efficient as the gradient style. The center pieces could be a single color if we want to remove the necessity of correctly orienting the centers.
Markers: Numbers, letters, or symbols could be placed over the current BOY color scheme to distinguish stickers. This contains the same advantages as gradient style, but may not look as nice. It’s possible that there could be a way to design it so that everything has an appealing look. I think it would be interesting to create a physical Speffz cube, to align with a BLD standard.
Mirror Cube: The Mirror Cube was previously mentioned and also contains the desired advantages. Differently shaped pieces or stickers allows for unique stickers among all pieces. Recognition may or may not be more difficult, depending on how well someone can learn the necessary piece shapes.
OreKehStrah has applied the minimum sticker concept to other WCA puzzles. Click below to check them out. If you want to try out a virtual version of a 3x3 sticker scheme, crystalcuber has designed a program where you can apply a custom scheme using a layout image.
Recognition
Below are some examples of how steps can be recognized using the minimum number of stickers. It’s pretty amazing what becomes possible. The examples use the gradient cube, but the same thing applies to the other scheme types.
Solved state U face stickers
CLL / CMLL: Just 3 total stickers are all that is needed to recognize the final layer of corners. For example, the UFL, UFR, and UBR stickers uniquely identify each case. To maintain the standard 42 patterns, you choose a specific point among each 4 sticker orientation to check 3 stickers that belong on the U face. If you freely start at any random point and truly only look at just the three current U stickers at UFL, UFR, and UBR, this creates a more advanced version with 162 patterns.
Setup: U2 R U2 R’ U’ R U’ R’ (Permuted Sune)
PLL: The minimum number of stickers to recognize PLL is 5. PLL can be recognized using the UFL, UF, UFR, UR, and UBR stickers as an example. However, there is something much more interesting here. OLL also involves checking the U face stickers and PLL can be recognized using just U face stickers. This means that PLL can be very easily recognized and predicted during OLL. There is no need to check any side stickers.
Setup: R’ U L’ U2 R U’ R’ U2 R L U’ (J Perm)
ZBLL: ZBLL can be recognized using 5 total stickers. A standard example is the U face stickers that are currently within the UFL, UF, UFR, UR, and UBR pieces. Similar to CLL, a basic and advanced number of patterns also applies here.
Setup: U2 R U2 R’ U’ R U’ R’ (Permuted Sune)
1LLL: 6 stickers is the minimum number to recognize 1LLL. Add in the UL edge to the ZBLL piece set to complete 1LLL recognition.
Setup: U2 r U2 R’ U’ R U’ r’ (Wide permuted Sune)
Edge orientation: When determining edge orientation during CFOP edge control, the Petrus edge orientation step, or in other methods, it can be difficult when stickers are out of view. The setup R U’ R’ U’ F’ U F is an example where two matching edge stickers are at UL and UB. But using a unique sticker cube this issue doesn’t exist.
Roux 4c: One of the minor annoyances with the Roux method is the final sub-step of permuting the middle slice edges. The 3-cycles that start with an M2 are impossible to distinguish when simply looking at them. Recognition methods, such as the one called DFDB, have been developed to handle this. With a unique sticker cube, there would be no need to have a recognition method and take up valuable mental processing power during previous sub-steps.
BLD: It is now possible to recognize corners and edges using just one sticker from each. This means that corners can always be memorized by looking at the U, F, and R faces without any tilt or rotation.
DR: This also benefits speed solving oriented DR methods or steps, such as MI2, SSC, Square-101, the separation variant of Mehta, or DR applied to Petrus like methods such as LEOR and CEOR. The DR state can be recognized entirely using the upper face stickers when corners are in the <U, R> permutation state.
With a unique sticker scheme, every corner, every edge, and every center is truly unique from all angles. With the current scheme, matching stickers in combination with additional stickers that are out of view creates a doppelganger situation where it is often difficult to determine which piece is which.
There is something else really cool. Think about the fact that many tutorials display PLL, ZBLL, 1LLL, and other steps with arrow lines placed on the U face showing the cycles or swaps that will occur. With a unique sticker puzzle, recognition now more closely aligns with that because it is possible to recognize those steps using just U face stickers.
The overall idea of unique stickers works on other puzzles as well. If each sticker of other puzzles was made to be unique, the recognition minimums would be reduced.
Depending on the chosen unique sticker scheme, solving may be more or less difficult than the BOY scheme. A super cube, potentially with a single color center, would be the easiest transition, but doesn't truly reduce the amount of information required to be checked. The other schemes, such as the gradient cube, are more difficult to transition to but do truly reduce the required amount of information for step recognition. Whether faster or slower, the overall realizations are still very interesting. The realizations being that our current color scheme (unintentionally) limits our information within a solve, and that a unique sticker scheme unlocks new minimums and abilities relating to recognition.
There is something else really cool. Think about the fact that many tutorials display PLL, ZBLL, 1LLL, and other steps with arrow lines placed on the U face showing the cycles or swaps that will occur. With a unique sticker puzzle, recognition now more closely aligns with that because it is possible to recognize those steps using just U face stickers.
The overall idea of unique stickers works on other puzzles as well. If each sticker of other puzzles was made to be unique, the recognition minimums would be reduced.
Depending on the chosen unique sticker scheme, solving may be more or less difficult than the BOY scheme. A super cube, potentially with a single color center, would be the easiest transition, but doesn't truly reduce the amount of information required to be checked. The other schemes, such as the gradient cube, are more difficult to transition to but do truly reduce the required amount of information for step recognition. Whether faster or slower, the overall realizations are still very interesting. The realizations being that our current color scheme (unintentionally) limits our information within a solve, and that a unique sticker scheme unlocks new minimums and abilities relating to recognition.
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