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Benchmarking Beginner Algorithmsfor Rubik's cube




Supervisor: Michael SchliephakeExaminer: Örjan Ekeberg

AbstractOver the years different algorithms have been developedto step-by-step solve parts of the Rubik’s cube until fi-nally reaching the unique solution. This thesis explores twocommonly known beginner algorithms for solving Rubik’scube to find how they differ in solving speed and amountof moves. The algorithms were implemented and run on alarge amount of scrambled cubes to collect data. The re-sults showed that Layer-by-layer with daisy algorithm hada lower average amount of moves than the Dedmore al-gorithm. The main difference in amount of moves lies inthe steps that solve the last layer of the cube. The Layer-by-layer with daisy algorithm uses only one-seventh of thetime-consuming operations that Dedmore algorithm uses,which concludes that it is more suitable for speedcubing.


Över åren har ett antal olika algoritmer utvecklats för attsteg-för-steg lösa delar av Rubik’s kub för att till sist kom-ma fram till den unika lösningen. Denna rapport utforskartvå allmänt kända nybörjaralgoritmer för att lösa Rubik’skub, för att finna hur dem skiljer sig åt i tid samt antaloperationer för att nå lösningen. Algoritmerna implemen-terades och kördes på ett stort antal blandade kuber för attsamla data. Resultatet visar att Lager-för-lager med daisyalgoritmen hade ett lägre genomsnittligt antal förflyttning-ar jämfört med Dedmore algoritmen. Den största skillnadeni antalet förflyttningar ligger i stegen som löser sista lagretav kuben. Lager-för-lager med daisy algoritmen använderbara en sjundedel av de mest tidskrävande förflyttningarnajämfört med Dedmore algoritmen, slutsatsen av detta äratt Lager-för-lager med daisy algoritmen är bättre lämpadför lösning av kuben på tid.


0.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Introduction 31.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background 52.1 Rubik’s Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Competitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Method 133.1 Cube representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Scramble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Results 17

5 Discussion 215.1 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Conclusion 23

References 25

Bilagor 26

A Calculation of randomly rotated faces 27

B Data 29


0.1 TerminologyCubie one of the twenty-six small cubes that make up the Rubik’s cube.

Scramble perfoming some amount of random operations on a solved cube to reacha non-solved state.

Layer the nine cubies making up a side or the 8 cubies all around the cube ei-ther vertically or horizontally starting from a center-positioned cubie (middlelayer).

Operation rotating one layer of the cube.


Chapter 1


The Rubik’s cube is a 3-D combination puzzle, where each side of the cube is coveredwith nine squares in six possible colours: white, red, blue, orange, green and yellow.It was invented by the professor of architecture Ernő Rubik as a teaching tool tohelp his students to understand 3D objects. It was not until he scrambled his newcube and tried to restore it, that he realize his creation was a puzzle. Originallythe Rubik’s cube was called the Magic Cube and was licensed to be sold by theamerican toy company Ideal Toy Company in 1980. [5].

When solving the cube the idea is to start with a scrambled cube, meaning thatthe cubies are randomly positioned by executing random operations on the cube(fig 1.1a). The goal is to obtain the unique solution where each side of the cubeis covered with only one colour per side (fig 1.1b). Different algorithms have beendeveloped to solve subproblems one at the time with a series of operations to reachthe unique solution. Many algorithms are based on the idea to solve it one layer atthe time.

If you were to randomly rotate the faces in an attempt to solve the cube, thereis almost zero chance of achieving the solved state in your lifetime, because of allthe possible permutations of the cube. There are 4.3 ∗ 1019 (or 43 quintillion) [11]

(a) Scrambled cube (b) Solved cube

Figure 1.1



different states. Assuming you get to a unique state every second it would takemore than 130 billion years to test 10% of the cubes possible states [Appendix A].

There are two major ways to compete in solving the Rubik’s cube: the least amountof moves and solving the cube as fast as possible (speedcubing).

1.1 ProblemThis thesis will explore two commonly known beginner algorithms for solving ofRubik’s cubes to find the differences. Both algorithms are based on the idea tosolve the cube one layer at the time, which is the easiest way for the human mind tosolve this kind of problem [8]. We will evaluate the algorithms operation varianceand the number of operations to reach a solved state, to find the most move-efficientof them both.

The thesis will not include any empirical study of humans solving the cube withdifferent beginner algorithms, and will instead rely on data gathered from compu-tations. The problem was stated as follows:

Which of the beginner algorithms would be more effective for speedcubing?

Which of the beginner algorithms solves the cube with the least amount ofmoves?

1.2 PurposeThe purpose of testing the algorithms is to find out which uses the least amount ofmoves and to find out which algorithm suits speedcubing best by analyzing the usageof different operations. The reason the operations shows if the algorithm is suitablefor speedcubing is because some of the operations are more time-consuming thanothers. The inexperienced user will find this as a guideline as to which algorithmto start his/her journey towards solving the Rubik’s cube.

1.3 StructureThe sections are devided as follows: The second section will introduce the reader toconcepts necessary to understand the algorithms implemented and benchmarked.The third section will explain the Algorithms used in this thesis together withexplanations of the representation and implementation of the cube. The fourthsection will be presenting the results and the fifth section contains a discussionregarding the results. Following that is a conclusions section that completes thecircle of the thesis, answering the problem statements. Lastly the references usedto this thesis will be listed followed by appendix containing computations and data.


Chapter 2


This section provides the reader with knowledge of: how the cube works, moveoperations, competitions and the algorithms that are in focus for this thesis.

2.1 Rubik’s Cube

Explanation of how the cube is constructed and the different notations for theoperations.

Description The cube consists of 26 cubies with three, two or one visible sidedepending on the type of cubie. There is one core piece consisting of three axesholding the center pieces together [1]. The corner and edge pieces (fig 2.1) are thecubies that are movable to different edge and corner positions, the center pieces (fig2.1) can only be moved according to the axis.

Operations An operation is a movement of a layer on the cube and is communi-cated by notations (abbreviations of operation name) in the Rubik’s cube commu-nity. This thesis uses the notations used by the World Cube Association (WCA)[12] explained below (all operations clockwise 90 degrees):F - Front faceB - Back faceR - Right faceL - Left faceU - Upper faceD - Bottom faceThe thesis also uses notations for turning the middle layer, defined as follows [4]:M - Middle layer vertical, in the same direction as LE - Middle layer (Equatorial) horizontal, in the same direction as DTo denote the anti-clockwise 90 degrees rotation just put a single citation mark (‘)after the letter. For example F’ - move front face anti-clockwise 90 degrees.



Figure 2.1: Cubie types

2.2 CompetitionsThere are two types of competitions for the cube that are relevant for this thesis.

Speedcubing When competing in an official event regulated by the WCA, thecompetitor has at maximum 15 seconds of inspection time of the cube before thesolve begins [13]. The time stops when the competitor have reached the uniquesolution.

Fewest moves The competitors have 60 minutes without any inspection time andthe competitor should also be able to hand in a written solution with the notationsused in the correct format [13].

2.3 AlgorithmsThe physical methods for solving the cube by hand matches the mathematical def-inition of an algorithm. Unlike many of the “near-optimal solvers” (some of whichsimulates several moves ahead to find the optimal move [7]), these algorithms canwith little effort be taught to humans. The algorithms are built on many heuristicsthat guarantees a solution, but the number of moves exceeds the proven maximumamount to solve any cube in any state (20) [9]. With the algorithms as heuristicscomes multiple ways to solve each step.

The algorithms of focus in this thesis are Layer-by-layer using daisy method andDedmore method which both are based on solving the cube layer-by-layer. Thismeans that the algorithms divides the cube into layers which makes it possible tosolve the layers in steps/subproblems without breaking the layers previously com-pleted.



Layer-by-layer using daisy algorithm Due to this algorithm has no name, itwill from here be referred to as Layer-by-layer with daisy (or Lbl with daisy forshort). The daisy-part of the name refers to the algorithm to solve the white cross.This by first making a cross with white edges and a yellow center and then turnthe white edges over to the opposite side, completing the white cross in the bottom.This algorithm focuses on an edge first solution. The steps can be found in moredetail at [6] and the Daisy method can be found at [2].

1. White cross

The goal here is to achieve a white cross, so that the white center-piece is alignedwith its 4 white edge-pieces in the bottom. For it to be a completed step, the secondcolor of the white edge-pieces must also align with it’s center-piece counterpart asshown in fig 2.2a.This is done using daisy’s method, flipping the edges one at thetime to make sure the edge-pieces on the vertical sides are aligned with the center-pieces on the same sides.

2. White corners

With the white cross done the next step is to complete the first layer by positioningthe white corner-pieces correct between the cross edges (fig 2.2b). This is achievedwith three different operation-combinatio[2].ns depending on the colour positions,all focusing on the upper-front-left corner of the cube.

3. Middle layer edges

The next step is to solve the middle layer by moving down a cubie from the middleof the third layer to the correct position on the second layer (fig 2.3a).

4. Yellow cross

With the second layer complete (fig 2.3b) it is time to work on the yellow cross.This is achieved by applying one of two operation-combinations or both dependingon how many white pieces that are correct positioned (fig 2.4).

(a) The white cross done (b) First layer complete

Figure 2.2



5. Yellow corners

Positioning the yellow corners by applying different operations to move the edgesdepending och how many corners that are in position already.

6. Last layer permutation

Now when the corners of the last layer are in position so that the top is all yellow,the only thing left to do is to positioning the pieces of the last layer so they matchwith the colours of the vertical sides. This is achieved by applying three differentoperation-combinations depending on if its edges or corners that should switchplaces.

(a) Technique for each side (b) Middle layer complete

Figure 2.3

Figure 2.4: Different states to achieve the yellow cross



Dedmore algorithm The Dedmore algorithm differs from Lbl with daisy in fo-cusing on a corners-first solution [3]. It was one of the first solution guides to theRubik’s cube [10]. The following steps can be found in more detail at the websitecreated by Denny Dedmore himself [3].

1. Top corners (the X)

Start off by finding a starting corner. For example aim to solve the blue top facefirst. Rotate the cube to obtain the blue side of the corner in top position. Thenmove the blue center-piece into position without moving the corner. When this isdone, rotate the whole cube to put the corner in the top left on the front side, asin the fig 2.5a.

Search for the next corner that should be positioned in the top right front side.Match the position of this corner to one of 6 scenarios and execute different operation-combinations to get it in position like the fig 2.5b. Rotate the cube and continuewith the next corner. The goal of this step is to form a ‘X’ on the top face as in fig2.5c.

2. Top edges

The goal of this step is to get the edges in place. The strategy is to get them inplace one by one, by matching the cube to one of five scenarios and then executean operation-combination for solving that scenario. When this step is finished thecube looks like fig 2.6a.

3. Middle layer

The next step is to solve the middle layer by moving down a cubie from the middleof the third layer to the correct position on the second layer (fig 2.3a).

(a) First corner in position (b) Second corner in posi-tion (c) Top edges in position

Figure 2.5



4. Bottom corners

Here the cube is turned upside down and the goal is again to build a (green) ‘X’with the corners(fig 2.6b). This is done by regarding the 4 corner cubies as 2 pairsand moving each pair into correct position. The correct pair can be either in eachother’s position, or diagonally from each other. Both requiring different operation-combinations to solve.

When the corner pairs are in position, the next part is to make sure the cubiesare facing the correct way. This is done by matching the cube to one of sevendifferent scenarios, and perform an operation-combination. This might have to berepeated up to 3 times for the step to be completed.

5. Bottom edges

In this step, at least one edge is already in the correct place (the color might beswitched as in fig 2.6c). Find that edge and put it on the front side. Then posi-tion every other edge correctly by executing an operation-combination up to 2 times.

When every edge are correctly positioned, there are 2 possible states left. Eachrequires a different sequence of operations to solve. The ‘fish’ pattern (fig 2.7a) andthe ‘H’ pattern (fig 2.7b). Execute the sequence for the pattern and the cube issolved.

(a) Top edges in position (b) The green X (c) One edge correctlyplaced

Figure 2.6



(a) Fish pattern (b) H pattern

Figure 2.7


Chapter 3


For this thesis the two methods used were implementation of algorithms and anal-ysis of the data representation. An implementation was motivated considering thatimplementations allowed analysis and collection of data from multiple algorithmruns. The analysis was motivated considering that a comparison of two algorithmshad to be made in order to answer the problem-statements.

Both of the algorithms were implemented in Java and with the same built-in func-tions and data structures to rule out differences and obtain a realistic comparison.The algorithms were run a number of times on different cubes to obtain a good setof data for the comparison. The data was obtained during the runs by gatheringinformation such as: number of moves used, statistics of which moves used andnumber of times one algorithm used less moves than the other. For each algorithmthe test data was evaluated by use of different operations, number of operationsused and time-consuming operations.

3.1 Cube representationThe cube was represented as six separate sides with nine positions per side, each ofwhich has a color (see fig 3.1). There exists no relation between positions in differ-ent sides; unlike the physical cube, where for example an edge has two colors. Thismeans any correctly performed operation needed to move the colours to the correctside and location. The only way to keep the rules of the physical cube (correctcolour-combination of a cubie) was to enforce them during the operations.

For example performing the ‘F’ operation (flip the front layer 90 degrees clockwise)on the cube shown in fig 3.1 would result in these internal assignments:

Side tempTop = new Side(top);Side tempFront = new Side(front);top.c7 = left.c9;top.c8 = left.c6;



top.c9 = left.c3;left.c3 = bot.c1;left.c6 = bot.c2;left.c9 = bot.c3;bot.c1 = right.c7;bot.c2 = right.c4;bot.c3 = right.c1;right.c1 = tempTop.c7;right.c4 = tempTop.c8;right.c7 = tempTop.c9;front.c1 = tempFront.c7;front.c2 = tempFront.c4;front.c3 = tempFront.c1;front.c4 = tempFront.c8;front.c6 = tempFront.c2;front.c7 = tempFront.c9;front.c8 = tempFront.c6;front.c9 = tempFront.c3;

Time-consuming operations The authors own experience (as novices) of theoperations indicates that the operations E and M are most time-consuming becausethey both work with the middle-layer (vertically and horizontally). Due to the factthat no empirical studies were conducted, an assumption was made that operationsworking with the middle layer took a beginner twice as long as ‘normal’ operationsto perform.

3.2 Scramble

The idea of the scrambler is to always start with a solved cube. The scramble issimulated by making a large amount of operations in pseudo-random sequence [Al-gorithm 1]. The scrambled cube is saved to a list awaiting the two solvers to pickand solve it.



Input: A cube, Number of operations to be madeOutput: A scrambled cubeLet rand be a pseudo-random generator.for Number of op to be made do

int opNr = rand.nextInt(Tot num of op)switch opNr do

All operations represented by a number, perform the operation withthe number opNr


Algorithm 1: Scramble

3.3 ImplementationBoth algorithms contains several steps and all of them uses the same structure.Starting off by looking if the goal of the step is already achieved. While the goalis not achieved: put the cube in a (by the algorithm) desired position and thenperform operations in an order given by the algorithm to reach the solution statefor the step [Algorithm 2].

Complexity The steps of both algorithms used the same general idea with thesame data-structures in the implementation (see [Algorithm 2]). Using the same

Figure 3.1: Side numberings



data-structure simplifies the comparison of the algorithms due to no big differencesin implementations. Both algorithms run in linear time: the loops that break whena step is complete have a constant maximum number of iterations, because eachstep will focus on a finite part of the cube in form of a layer, resulting in a finiteamount of permutations to try.

Input: A cubeOutput: A cube that has a step solvedif stepIsDone(cube) then

return cubeendwhile not(stepIsDone(cube)) do

if readyForOperations(cube) thenPerform the operations necessary to complete the step


while not(readyForOperations(cube)) doOperations to get the cube in a state to be able to perform thesteps operations


endAlgorithm 2: The general idea of algorithm steps


Chapter 4


In this section the results are displayed as a comparison of the algorithms amountof moves used, statistics for specific operation usage and number of times the al-gorithm used least amount of moves. The amount of moves in each step of bothalgorithms are also presented.

Fig 4.1 represents the sum of executions that used the same amount of moves,where the x-axis is amount of moves and the y-axis number of runs. As shown bythe highest pillars for both algorithms, the Lbl with daisy algorithm had a loweraverage (170) amount of moves than the Dedmore algorithm (189). Turning thewhole cube in any direction was not counted for as a move.

Fig 4.2 represents a comparison of the number of runs the algorithms used leastamount of moves, with the y-axis as number of runs. This shows that over tenthousand scrambled cubes the Lbl with daisy algorithm won 71% of the time, theygot equal amount of moves 1% of the time and Dedmore algorithm had the fewestin comparison 28% of the time.

Fig 4.3 represents the average use of operations for both algorithms over all runswith the x-axis as the average number of times the operations where used.The data for both fig 4.2 and fig 4.3 can be found in detail in Appendix B.

Fig 4.4 and fig 4.5 represents the amount of moves used in each step for bothalgorithms. By comparing fig 4.4 and 4.5 it is clear that the biggest difference inamount of moves are between the middle layer step and the last steps of the twoalgorithms.

Fig 4.6 visualizes the average total time to solve the cubes and how much of thetime that was spent on time-consuming operations (included in total time).



Figure 4.1

Figure 4.2


Figure 4.3: Average use of operations

Figure 4.4: Number of moves per step



Figure 4.5: Number of moves per step

Figure 4.6: Time-consuming operations and total time


Chapter 5


In this section we discuss the results in form of comparisons and reflect on whythese results were given. We also discuss any errors that could have had an impacton the final results.

5.1 ComparisonAs we can see in the results fig 4.1, Lbl with daisy algorithm has the lowest amountof moves the majority of the times. Lbl with daisy algorithm also has the lowestminimum and maximum amount of moves with 95 and 253 respectively. The cor-responding values for the Dedmore algorithm are 96 and 292 as can be seen in fig4.1 and in Appendix B. This shows that Lbl wins in lowest average, best case andworst case and the result of this is shown in fig 4.2.

The algorithms often assumed the correct cubies to be on a certain side of thecube, in reality this simply means rotating the cube to a desired position. The im-plemented versions prioritized execution of operations instead, to get the cube intoposition, which led to an increase in the amount of total operations. This might ex-plain the frequent uses of a few operations, for example U, Ui and F, shown in fig 4.3.

According to the result graphs fig 4.4 and fig 4.5 the part of the algorithms with thebiggest differences in amount of moves are the steps solving the middle layer andlast layer. But with the middle layer using the same technique in both algorithmsthe focus is turned to the last layer steps, to find differences affecting the numberof moves. The steps in focus are yellow cross (average 6 moves) and yellow corners(average 13 moves) in Lbl with daisy algorithm and bottom corners (average 42moves) in Dedmore algorithm. The cube has two layers completely solved beforethese steps are performed. These steps are comparable because they are solving thesame layer of the cube, which limits the choice of operation-combinations to usewithout destroying parts of the first two layers that are completed.



The reason why the two steps in Lbl with daisy (total average 19 moves) uses lessmoves than Dedmores step are because of the difference in strategy in the steps anda possible error described below. The step Yellow cross has three possible start-ing positions, which needs two different operation-combinations or both of themcombined, and each operation-combination consists of 6 moves. The step Yellowcorners has the same structure as yellow cross with three cases. Two of these casesare operation-combinations which are needed to be able to perform the solving casedepending on the permutation of the cube. Each case uses at a minimum 8 movesand the solving case can use 16 moves in the worst scenario.

Analyzing Dedmores bottom corner step shows that it works with two corners at thetime, positioning them by using two cases of operation-combination for the first pair(11 moves each) and one case if needed for the second pair (11 moves). Now in thefinishing part of the step the cube can be matched with one of seven possible statesand when matched an operation-combination (10 moves) needs to be performed tosolve the step. That operation-combination should (according to the descriptionof the step) be performed at a maximum of three times before the step is finished,which turned out to be incorrect. We have found that it is possible that the finishingpart of the step needs more than three performed operation-combinations, makingthe amount of moves increase to over 30 in worst case, for just this part. Accordingto the description of the step the max amount of moves should be 52 and leastamount of moves should be 21, making the least amount of moves for Dedmore’sbottom corner step higher than the average for Lbl with daisys steps.

The data in fig 4.3 and Appendix B shows that Dedmore used both E, Ei andM, Mi more than Lbl with daisy. With these results in mind we investigate thegraph 4.6 which shows the amount of time-units used by the time-consuming op-erations compared to the total time. This should, according to the assumptionsmade, imply that Dedmore uses more time than Lbl with daisy algorithm becauseof the time-consuming operations used.

5.2 ErrorsConverting the algorithms effeciently from description of physical solving of the cubeto the code solver proved to be challenging. Ambigous and unfinished descriptionof steps in the algorithms led to an extra amount of operations on the cubes. Thishad an impact on the final result.


Chapter 6


The measurements shows that the Layer-by-layer using daisy algorithm solved thecube with the least amount of moves in more than 70% of the 10,000 executionsmade. It also shows that the same algorithm had the lowest moves in minima andmaxima on the tested cubes. This concludes that the Layer-by-layer using daisyalgorithm is the better choice of algorithm regarding the least amount of moves.

Time measurements shows that Layer-by-layer using daisy algorithm is more suit-able for speedcubing than Dedmore. This conclusion is based on the statistics ofoperation-usage, showing that Dedmore used these time-consuming operations 7times more frequently than Lbl with daisy algorithm. This is also supported bythe assumption made on the time-differences between time-consuming and normaloperations.



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Appendix A

Calculation of randomly rotated faces

Assumption: Every second you get to a new state.There are 4.3∗1019 different states (rounded down) [11]. 10% of the different states:4.3 ∗ 1019 ∗ 0.1 = 4.3 ∗ 1018

4.3 ∗ 1018 seconds to years = 136.4 ∗ 109 years.


Appendix B


Average use of operations.Operations Lbl DedmoreD 1.0479 11.4515E 0.1197 6.0673F 36.4233 10.7832B 3.1820 0.0648Ri 15.0851 16.8868L 2.9329 8.7679M 0.000 6.1943Ui 18.1466 37.7807U 36.0456 24.2579Bi 3.1820 0.1031Mi 0.000 6.1943Li 2.9329 8.7187R 25.0051 12.7996Ei 4.5945 13.3863Fi 15.1346 9.2321Di 6.4249 16.5976

Number of times the algorithms had least amount of moves.Lbl Dedmore Equals7098 2819 83

Min and max movesLbl with daisy algorithm Dedmore

Average amount of moves 170 189Min amount of moves 95 96Max amount of moves 253 292