Learn About Beautiful 3×3 Cube

3x3 Cube

Invention of Most Admiring 3×3 Cube

3×3 Cube invention, in March 1970, Larry D. Nichols invented the 2 × 2 × 2 puzzle “with rotating pieces in groups” 2 × 2 × 2 and filed a Canadian patent for it. Nichols’ cube was held together by magnets. Nichols was granted U.S. Patent 3,655,201 on April 11, 1972,

IN 1970, Frank Fox applied for a patent “fun device”, a type of sliding puzzle. He received his patent in the United Kingdom (1344259) on January 16, 1974.


Discovery of Fantastic 3×3 Cube

Cube was built as a teaching tool to help students to understand 3D objects, its current goal was to solve the structural problem of moving parts independently, without the whole mechanism disintegrating. He didn’t realize he had created a puzzle until the first time he shuffled his new cube and tried to restore it.

After international appearance, the Cube’s progress on Western toy store was briefly halted so that it could be manufactured to Western packaging. A Cube lighter was produced and Ideal decided to rename it.

3×3 Cube craze in 1980

Since most people could only solve one or two sides, several books were published. At one point in 1981, three of the ten best-selling books in the United States were 3×3 Cube resolution books, and the best-selling book of 1981 was James G. Nourse’s The Simple Solution to Rubik’s Cube, which sold over 6 millions of copies. .

In 1981, the Museum of Modern Art in exhibited a cube and at the 1982 World’s Fair in Knoxville, Tennessee, a six-foot cube was also exhibited.

In October 1982, the New York Times reported that “the craze was dead,”. However, in some Communist countries, such as China and the USSR, the craze started later and the demand was still high due to the shortage of cubes.

 Revival of 3×3 Cube in 21st century

The cubes continued to be marketed and sold in the 1980s and 1990s, but it wasn’t until the early 2000s that interest in the cube began to grow again. In the US, sales doubled between 2001 and 2003 and the Boston Globe noted that “it was getting nice to have a 3×3 Cube again.”

 The 2003 championship was the first speedcubing tournament since 1982. It took place in Toronto and saw the participation of 83 participants.


Taking advantage of the initial shortage of cubes, numerous imitations and variants emerged, many of which may have infringed one or more patents.

Mechanics of 3×3 Cube

A standard cube measures 5.6 centimeters on each side. The cube consists of 26 miniature cubes, also known as “cubelets”. Each of them includes a hidden inner extension that fits with the other cubes, allowing them to move to different locations. However, the central hub of each of the six fixed to the central mechanism. They provide the framework in which the other pieces fit and rotate.

 So there are 21 pieces: a single centerpiece made up of three axes that intersect and hold the six center squares in place but allow them to rotate, and 20 smaller plastic pieces that snap together to form the assembled puzzle.

Each of the six center pieces rotates on a screw (clamp) held by the center piece, a “3D cross”. A spring between each screw head and its corresponding part tensions the part inward so that, together, the entire assembly remains compact but can still be easily manipulated. The bolt can be tightened or loosened to change the “feel” of the hub.

The 3×3 Cube can be disassembled without too much difficulty, usually by rotating the top layer 45° and then pulling one of its edge cubes away from the other two layers. Consequently, it is a simple process to “resolve” a cube by disassembling and reassembling it in a resolved state.

There are six centerpieces showing a colorful face,

twelve corner pieces showing two colored faces and eight corner pieces showing three colored faces.

Each tile shows a unique color combination, but not all combinations are present (for example, if red and orange are on opposite sides of the solved 3×3 Cube, there is no border with red and orange sides).

 The position of these cubes to each other can be changed by rotating an outer third of the cube in 90 degree increments, but the position of the colored sides  cannot be changed; is fixed by the relative positions of the central squares

Mathematics in 3×3 Cube

The puzzle was having “more than 3,000,000,000 combinations, but only one solution”. Depending on how the combinations are counted, the actual number is significantly higher.

Permutations of 3×3 Cube

The previous figure is limited to the permutations that can be obtained simply by rotating the sides of the 3×3 Cube. If we consider the permutations obtained by disassembling the cube, the number becomes twelve times greater:

There are about 519 quintillions of possible arrangements, but only one in twelve is actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge of the cube.

 Therefore, there are 12 possible sets of workable configurations, sometimes called “universes” or “orbits”, in which the cube can be positioned by disassembling and reassembling.

The preceding numbers assume that the center faces are in a fixed position. If we assume that we rotate the entire cube with a different permutation, each of the preceding numbers must be multiplied by 24. A chosen color can be on one of the six sides, and therefore one of the adjacent colors can be on one of the four positions; this determines the positions of all other colors.

Algorithms use in 3×3 Cube

Many algorithms are designed to solve cube only a small part of the cube without interfering with other parts which are already solved, so that they can be repeatedly applied to different parts of the cube until the puzzle is solved. For example, there are well-known algorithms for alternating three corners without changing the rest of the puzzle or reversing the orientation of one pair of edges, leaving the others intact.

Some algorithms have a desired effect on the 3×3 Cube (e.g. swapping two corners), but they can also have the side effect of changing other parts of the 3×3 Cube (like swapping some edges).

 These algorithms are often simpler than those without side effects and are employed at the beginning of the solution, when most of the puzzle has not yet been solved and the side effects are not important. Most are long and difficult to memorize. At the end of the solution more specific (and often more complicated) algorithms are used.

How to solve 3×3 Cube

Shift notation

Many fans of the 3×3 Cube use a notation developed by David Singmaster to indicate a sequence of moves, called “Singmaster notation”.algorithms to be written in such a way that they can be applied regardless of which side is designated at the top or how the colors are arranged in a given cube.

When a prime symbol (′) follows a letter, it indicates a counterclockwise turn of the face; while a letter without a prime symbol indicates a clockwise rotation. These instructions show you how you are looking at the specified face. A letter followed by a 2 (occasionally a superscript 2) indicates two turns, or a 180-degree turn. R is the right side clockwise, but R ′ is the right side counterclockwise.

 The letters x, y and z are used to indicate that the entire cube must be rotated around one of its axes, corresponding respectively to R, U and F. When x, y and z are turned on, it is an indication that the cube must be rotated in the opposite direction. When square, the cube must be rotated 180 degrees.

An alternative notation, the Wolstenholme notation, is designed to make it easier for beginners to memorize motion sequences. This notation uses the same letters for the faces, except it replaces U with T (top), so they are all consonants. 

The main difference is the use of the vowels O, A and I for clockwise, , which result in word-like sequences such as LOTA RATO LATA ROTI (equivalent to LU ′ R ′ UL ′ U ′ R U2 in Singmaster notation). Adding a C involves the rotation of the entire cube, so ROC is the clockwise rotation of the cube around its right face. The interlayer motions are indicated by adding an M to the corresponding face motion, so RIM means a 180 degree rotation of the interlayer adjacent to the R face.

Another notation appeared in the 1981 book The Simple Solution to 3×3 Cube. Singmaster notation was not well known at the time of publication. The faces were labeled Top (T), Bottom (B), Left (L), Right (R), Front (F) and Back (P), with + clockwise, – counterclockwise and 2 for 180 rotations degrees.

Optimal solutions of 3×3 Cube

A solution commonly used by speedcubers was developed by Jessica Fridrich. This method is called CFOP, which means “cross, F2L, OLL, PLL”. It is similar to the layer-by-layer method, but employs the use of a large number of algorithms, especially for orienting and permuting the last layer. 

The crossing is done first, followed by the corners of the first layer and the edges of the second layer at the same time, with each corner paired with a piece of the edge of the second layer, thus completing the first two layers (F2L). This is followed by last layer orientation and then last layer permutation (OLL and PLL respectively). Fridrich’s solution requires learning about 120 algorithms, but solves the Cube on average in just 55 moves.

A well known method was developed by Lars Petrus. In this method, a 2 × 2 × 2 section is solved first, followed by a 2 × 2 × 3, then the wrong edges are solved using a three-movement algorithm, which eliminates the need for a possible 32-move algorithm afterwards. 

The principle behind this is that, layer by layer, you must constantly break and repair the completed layer or layers; the 2 × 2 × 2 and 2 × 2 × 3 sections allow you to record three or two layers (respectively) without hampering your progress. One of the advantages of this method is that it tends to provide solutions in fewer steps. For this reason, the method is also popular for races with the fewest moves.

The Roux method, developed by Gilles Roux, is similar to the Petrus method in that it is based on building blocks rather than layers, but derives from corner-first methods. In Roux, a 3 × 2 × 1 block is solved, followed by another 3 × 2 × 1 on the opposite side. After that, the top level corners are resolved. The cube can then be solved using just the U-layer and M-slice motions.

Methods for beginners

Most beginner solving methods involve solving the 3×3 Cube one layer at a time, using algorithms that preserve what has already been solved. The simpler layer-by-layer methods require only 3-8 algorithms.

In 1981, 13-year-old Patrick Bossert developed a solution for solving the cube, along with graphical notation, designed to be easily understood by beginners.

In 1997, Denny Dedmore published a solution described using diagrammatic icons representing the moves to be made, instead of the usual notation.

Philip Marshall’s The Ultimate Solution to Rubik’s Cube takes a different approach, averaging just 65 twists but requiring only two algorithms to be memorized. The cross is resolved first, followed by the remaining edges, then by the five corners, and finally by the last three corners.

Blindfolded resolution of 3×3 Cube

Blindfolded multiple resolution, or “multi-blind”, in which the competitor solves any number of blindfolded cubes in sequence

Solve the 3×3 Cube using only one hand

Solve the cube with the fewest possible moves

In blindfolded solving, the contestant first studies the encrypted cube (i.e., looking at it normally not blind), then is blindfolded before starting to turn the faces of the cube. The time logged for this event includes the time it took to store the cube and the time it took to manipulate it.

In Multiple Blindfolded, all cubes are stored and then all cubes are solved once blindfolded; therefore, the main challenge is to memorize many separate 3×3 Cube, usually ten or more. 

The event is scored not based on time, but based on the number of points earned after the one hour limit with the number of unsolved cubes after the end of the attempt, where the greater the number of points, the better. If several competitors get the same number of points, the rankings are evaluated based on the total time of the at3x3 Cube

Master Cube 4×4 CUBE

4x4 cube

What is 4×4 cube?

4×4 cube also known as Master Cube is a 4×4×4 version of  Cube. It was released in 1981. Invented by Péter Sebestény, This was almost called Sebestény’s Cube until a last-minute decision changed the puzzle’s name to attract fans of the original Rubik’s Cube.  like a 3×3 cube, it has no fixed center. the central facets are free to move into positions many different.

Technique To Solve 4×4 cube?

The methods for solving the 3×3×3 cube work for the edges and corners of the 4×4 cube, as long as you have correctly identified the relative positions of the colors, as the center facets can no longer be used for identification.

The puzzle consists of 56 miniature cubes unique on the surface. They consist of 24 centers of one color each, 24 edges of two colors each, and 8 corners of three colors each. The 4×4 cube can be disassembled without too much difficulty, usually by turning one side at a 30° angle and pushing one end up until it comes free.

The original mechanism designed by Sebestény uses a fluted ball to hold the centerpieces in place. The edge pieces are held in place by the centers and the corners are held in place by the edges, just like the original cube. There are three perpendicular grooves. Each slot is wide enough to allow a row of centerpieces to pass through. The shape of the sphere prevents the centerpieces of the other row from slipping, ensuring that the ball is aligned with the outside of the cube. Rotating one of the middle layers moves only that layer or even the ball.

The Eastsheen version of the 4×4 cube

The Eastsheen version of the cube, which is slightly less than 6 cm from one edge, has a completely different mechanism. Its mechanism is very similar to Eastsheen’s version of the teacher’s cube rather than the ball mechanism. There are 42 pieces of which 36 are movable and six are fixes This design is stronger than the original and also allows you to use screws to tighten or loosen the hub. The central shaft is specially shaped to prevent it from misaligning with the outside of the hub

4x4 cube

Solution of 4×4 cube

There are 24 angle brackets with two colored sides each and eight angle brackets with three colors. Each corner piece of edges shows a unique color combination. The position of these cubes relative to each other can be changed by rotating the layers of the cube, but the position of the colored sides relative to each other in the complete form of the puzzle cannot be changed because it is fixed by the relative positions of the central squares and the distribution of color combinations on edges and corners.

The edge pairs are often called double-edged suitcases. For the newer cubes, the sticker colors are red opposite orange, yellow opposite white, and green opposite blue. However, there are also cubes with alternating color arrangements (yellow opposite green, blue opposite white, and red opposite orange).

The Eastsheen version 4×4 cube

The Eastsheen version has purple (as opposed to red) instead of orange. There are 8 corners, 24 edges, and 24 centers. Any permutation of angles is possible, including odd permutations. Seven corners are rotated independently and the orientation of the eighth depends on the other seven corners. There are 24 centers, which can be organized into 24! different ways. It is raised to the sixth power because there are six colors.

An odd permutation of the angles implies an odd permutation of the centers and vice versa; however, the odd and even permutations of the centers are indistinguishable due to the identical appearance of the pieces. There are several ways to make centerpieces distinguishable, which would make a unique central permutation visible.

The 24 edges cannot be reversed because the internal shape of the parts is asymmetrical. The corresponding edges are distinguishable as they are mirror images of each other Assuming that the cube has no fixed orientation in space and that the permutations resulting from rotating the cube without twisting it are identical, the number of permutations is reduced by a factor of 24.

This is happening all 24 possible positions of the first angles are equivalent due to lack of fixed centers. This factor does not appear in the computation of the permutations of N × N × N cubes where N is odd, since these puzzles have fixed centers that identify the spatial orientation of the cube. There are several methods that can be used to solve a 4×4 cube. One such method is the reduction method, so-called because it effectively reduces 4×4×4 to 3×3×3. Cubers first group the centerpieces of common color, then pair the edges that shows the same two colors. Once this is done, by rotating only the outer layers of the cube it is possible to solve it as a 3×3×3 cube.

Yau method 4×4 cube

Another known method is nothing but the Yau method, on the name of Robert Yau. The Yau method is similar to the reduction method and is the most common method used by speedcubers. Yau’s methods begin by solving two centers on opposite sides. Three cross-deductions are then resolved. Then the remaining four centers are resolved. Subsequently, any remaining edges are resolved. This boils down to a 3x3x3 cube.

A method that is almost similar to the Yau method is known as Hoya. It was invented by Jong-Ho Jeong. It is as similar as Yau, but the order is different. It begins with the resolution of all centers except 2 adjacent centers. Then a cross is formed at the bottom, solving the last two centers. After that, it is identical to Yau, refines the edges, and solves the cube as a 3×3.

Parity Errors in 4x4x4 cube

Some positions that cannot be resolved on a standard 3×3×3 cube can be reached. There are two possible problems not found in 3×3×3. The first is two inverted edges on one edge, with the result that the colors of this edge do not match the rest of the cubes on both sides.

Note that these two edges are swapped. The second is two pairs of edges that are swapped with each other (PLL parity), two angles can be swapped depending on the situation and/or method: These situations are known as parity errors. These positions can still be resolved; however, special algorithms must be applied to correct errors.

To avoid the parity errors described above some different solutions are made. For example, resolving corners and edges first and centers last would avoid these parity errors. Once the rest of the cube is resolved, any permutation of the centerpieces can be resolved.

Note that it is possible to apparently swap a pair of face centers by alternating 3 face centers, two of which are visually identical. PLL parity occurs on all cubes with an even number of edges from 4x4x4 onwards. However, it does not occur on cubes with an odd number of edges, such as 3x3x3 and 5x5x5. This is because the latter have fixed centerpieces and the former does not. Direct 4×4×4 resolution is rare but possible with methods like K4. This combines a variety of techniques and relies heavily on alternators for the final steps.

Learn How to Solve a 4×4 in 10 Minutes (Full Yau Method Tutorial)

Pyraminx Cube – An Amazing 3×3 Pyramid Cube


Pyraminx Cube: The Pyraminx was first made by Mèffert in the year 1970. He did nothing with his design until 1981 when he first took it to Hong Kong for production. Uwe likes to say that if it hadn’t been for Ernő Rubik’s invention of the cube, his Pyraminx would never have been produced.

Pyraminx is a straight-shaped puzzle with 4 axial pieces, 6 edge pieces, and 4 trivial points. It can be twisted along your cuts to market your pieces. Axial parts are octahedral, and can only rotate around the axis to which they are attached The six edges can be placed in 6!

because they can be twisted independently of all the other pieces, making them trivial to put them in the resolved position. Meffert also produces a similar puzzle called Tetraminx, which is the same as Pyraminx, except that the mundane edges are removed, turning the puzzle into a truncated tetrahedron.

Pyraminx stirred

The purpose of Pyraminx is to mix colors and then return them to their original settings.

The 4  tips can be easily rotated to align with the axial piece to which they respectively match, and the axial pieces can also be easily rotated so that their colors align with each other. This leaves only the 6 pieces on the board as a real challenge to the puzzle. They can be solved by repeatedly applying two 4-twist sequences, which are mirror versions of each other. These sequences switch 3 pieces of the board at a time and change their orientation differently, so a combination of the two sequences is enough to solve the puzzle. However, more efficient solutions (requiring fewer total twists) are generally available (see below).

The twist of an axial part is independent of the other three, as in the case of the tips. The six edges can be placed in many positions and inverted 25 ways, taking parity into account. Multiplying by a factor of 38 for the axial parts gives 75,582,720 possible positions. However, setting the mundane hints to the correct positions reduces the possibilities to 933,120, which is also the number of possible patterns in Tetraminx. Defining the axial pieces also reduces the number to just 11,520, making this a very easy puzzle to solve.

How To Solve Pyraminx cube


There are 2 general methods for solving Pyraminx, V-First – where you solve one level except one edge before finishing the puzzle, and Top-First – where you solve the top of the puzzle before solving the rest. None of the approaches to solving the Pyraminx cube is a great approach than the other one, Moreover, the V-First method I like to adapt when solving as it is much more easier to understand and intuitive rather than the Top-First approach. Therefore, in this guide, I will provide tips on more advanced V-Firsts solution techniques (as well as more general advice).

This guide is a tip, tricks, and advice that will improve your Pyraminx play and solution. I will mainly deal with L4E resolution, as by reading this guide you should already have a good understanding of LBL resolution.

L4E (last 4 edges) is an advanced “V” method mainly used for sub-5 and higher resolution. Instead of creating a complete layer as you would with the LBL (Layer By Layer) method; L4E consists of creating a “V” (level minus one edge) before using an intuitive (or learned) algorithm to solve the remaining 4 edges, hence the method name. This method has much more “speed potential” than the LBL method, such as looking for just a “V” during an inspection rather than an entire layer; you are able to look ahead and influence the L4E case, offering more opportunities for a more fluid and advanced solution.