Any problem-solving activity includes problem defining, clarifying its causes, setting priorities, choosing alternatives for a better solution, and directly implementing this solution. The solve for x calculator allows you to enter your problem and solve the equation to see the result. Solve in one variable or many. Visit Mathway on the web. Download free on Google Play. Download free on iTunes. Download free on Amazon. Download free in Windows Store. Enter a problem.
Problem Solving
Solving a problem is reaching a goal state; there are many things that can stand in the way of solving a problem, but many strategies that can help.
Learning Objectives
Evaluate strategies for solving problems, and barriers to solving them
Key Takeaways
Key Points
- Problem solving is moving from a current state to a goal state, through barriers.
- Mental set is a barrier to problem solving; it is an unconscious tendency to approach a problem in a certain way. Functional fixedness is a subtype of mental set and refers to the inability to see an object’s potential uses aside from its prescribed uses.
- Unnecessary constraints are when people construct mental blocks to solving a problem.
- Irrelevant information is distracting information that is unnecessary to solving the problem but is framed as being part of the problem.
- A heuristic is a rule of thumb that is useful in problem solving but does not guarantee a correct solution; an algorithm is a set of steps that will lead to a solution.
Key Terms
- availability heuristic: When a person makes a judgment about the probability of an event based on the ease with which it comes to mind.
- functional fixedness: When the intended purpose of an object hinders a person’s ability to see its potential other uses.
- mental set: An unconscious tendency to approach a problem in a particular way.
- Problem: The difference between the current situation and a goal.
The human mind is a problem-solving machine. You may not realize it, but as you walk through the world, you are solving problems every second—everything from “I’m running late, what’s the quickest way for me to get to class?” to “Where did I leave my wallet?” to “What should my psychology paper be about?”
In psychology, “problem solving” refers to a way of reaching a goal from a present condition, where the present condition is either not directly moving toward the goal, is far from it, or needs more complex logic in order to find steps toward the goal. It is considered the most complex of all intellectual functions, since it is a higher-order cognitive process that requires the modulation and control of basic skills. There are considered to be two major domains in problem solving: mathematical problem solving, which involves problems capable of being represented by symbols, and personal problem solving, where some difficulty or barrier is encountered.
Barriers to Problem Solving
There are many common mental constructs that impede our ability to correctly solve problems in the most efficient manner possible.
Mental Set and Functional Fixedness
A mental set is an unconscious tendency to approach a problem in a particular way. Our mental sets are shaped by our past experiences and habits. For example, if the last time your computer froze you restarted it and it worked, that might be the only solution you can think of the next time it freezes.
Functional fixedness is a special type of mental set that occurs when the intended purpose of an object hinders a person’s ability to see its potential other uses. So for example, say you need to open a can of broth but you only have a hammer. You might not realize that you could use the pointy, two-pronged end of the hammer to puncture the top of the can, since you are so accustomed to using the hammer as simply a pounding tool.
Unnecessary Constraints
The dot problem: In the dot problem, described below, solvers must attempt to connect all nine dots with no more than four lines, without lifting their pen from the paper.
Choose 3 Waysproblem Solving Problems
This is a barrier that shows up in problem solving that causes people to unconsciously place boundaries on the task at hand. A famous example of this barrier to problem solving is the dot problem. In this problem, there are nine dots arranged in a 3 x 3 square. The solver is asked to draw no more than four lines, without lifting their pen or pencil from the paper, that connect all of the dots. What often happens is that the solver creates an assumption in their mind that they must connect the dots without letting the lines go outside the square of dots. The solvers are literally unable to think outside the box. Standardized procedures of this nature often involve mentally invented constraints of this kind.
The dot problem’s solution: Most solvers do not recognize that they can go outside the box and draw longer lines to connect the dots.
Irrelevant Information
Irrelevant information is information that is presented as part of a problem, but which is unrelated or unimportant to that problem and will not help solve it. Typically, it detracts from the problem-solving process, as it may seem pertinent and distract people from finding the most efficient solution. An example of a problem hindered by irrelevant information is this:
15% of people in Topeka have unlisted telephone numbers. You select 200 names at random from the Topeka phone book. How many of these people have unlisted phone numbers?
The answer, of course, is none of them: if they are in the phone book, they do not have unlisted numbers. But the extraneous information at the beginning of the problem makes many people believe they have to perform a mathematical calculation of some sort. This is the trouble that irrelevant information can cause.
Problem-Solving Strategies
There are many strategies that can make solving a problem easier and more efficient. Two of them, algorithms and heuristics, are of particularly great psychological importance.
Heuristic
A heuristic is a rule of thumb, a strategy, or a mental shortcut that generally works for solving a problem (particularly decision-making problems). It is a practical method, one that is not a hundred percent guaranteed to be optimal or even successful, but is sufficient for the immediate goal. The advantage of heuristics is that they often reduce the time and cognitive load required to solve a problem; the disadvantage is that they cannot always be relied on to solve the problem—just most of the time.
Algorithm
An algorithm is a series of sets of steps for solving a problem. Unlike a heuristic, you are guaranteed to get the correct solution to the problem; however, an algorithm may not necessarily be the most efficient way of solving the problem. Additionally, you need to know the algorithm (i.e., the complete set of steps), which is not usually realistic for the problems of daily life.
The difference between an algorithm and a heuristic can be summed up in the example of trying to find a Starbucks (or some other national chain) in a city. An algorithm would be a series of steps: “Walk in an increasingly large grid pattern around the city blocks until you find a Starbucks or you have looked at every street.” But a heuristic could simply be, “Well, usually they’re at busy intersections; I’ll just walk to the nearest busy intersection.”
Other Strategies
There are many other ways of solving a problem. The most effective depends on the type of problem and the resources at hand.
Choose 3 Waysproblem Solving Worksheets
- Abstraction: solving the problem in a model of the system before applying it to the real system.
- Analogy: using a solution for a similar problem.
- Brainstorming: suggesting a large number of solutions and developing them until the best is found.
- Divide and conquer: breaking down a large, complex problem into smaller, solvable problems.
- Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption.
- Lateral thinking: approaching solutions indirectly and creatively.
- Means-ends analysis: choosing an action at each step to move closer to the goal.
- Morphological analysis: assessing the output and interactions of an entire system.
- Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it.
- Reduction: transforming the problem into another problem for which solutions exist.
- Root-cause analysis: identifying the cause of a problem.
- Trial and error: testing possible solutions until the right one is found.
Because of the popularity and relative simplicity of the 3x3x3 cube, there are a large number of methods out there. The most common and fast will be explored in this article.
Basic Methods
Basic methods require very few algorithms and little practice, but they also have a high move count and are very difficult to get fast times with. The fastest times that have been recorded with these methods are around 30 seconds. These methods are good beginner's methods, meaning they are good for teaching people who have never solved a Rubik's Cube before.
- In the Layer-By-Layer method, the solver finishes the layers one at a time. This is one of the most common method for new cubers to discover on their own. To become faster at LBL better techniques such as Keyhole or F2L will speed up the first layers, and learning more algorithms for the last layer will allow it to be solved in fewer steps using the 4 Look Last Layer.
- In the Corners First style, the corners are all solved, and then the edges and centers are filled in. This is a very easy method to learn, and it has many variations.
- In the block building methods the first parts of the solve is done by setting up blocks of some form, a 2x2x3 as in the Petrus method or F2B (first two blocks) as in the Roux method.
Intermediate Methods
Intermediate methods are very popular because they are quite fast without requiring extensive knowledge or memorization. They have a moderate move count (although they can be heavily optimized) and are good for cubers who have mastered a basic method.
- The CFOP Method (Fridrich) is the most popular speedcubing method. First the bottom layer edges are solved, then the first two layers are filled in either using intuition or algorithms, and finally the top layer is solved in two steps: OLL then PLL.
- The Roux Method is a hybrid of block-building and corners first methods. Two opposite 1x2x3 blocks are made, then the last four corners are solved (CMLL). The remaining six edges are solved intuitively.
- The ZZ Method is a relatively modern method where the edges are all oriented in the first step, allowing for more fingertricks and zero cube rotations. Last layer edges are pre-oriented, allowing for many options including 1LLL.
- The Petrus Method The cuber creates a 2x2x2 block, expands it to a 2x2x3 block, orients all of the remaining edges, then finishes the first two layers. The last layer is finished in three steps, although many Petrus users elect to finish the last layer with OLL and PLL instead.
- The Waterman Method is a method based on corners first methods. First cuber solves the first layer, then does the opposite layer's corners (CLL), and finally solves the remaining edges in three steps.
- Columns First Methods are a group of methods that in some way build four columns of three pieces each, orients the remaining edges, and permutes the remaining edges.
Advanced Methods
An advanced method requires extensive memorization, understanding of intuitive blockbuilding, or both. Few people have ever mastered these methods, but they are theoretically capable of extremely low move counts and times.
- The Heise Method is a very tricky block-building method that requires no algorithms at all. Four 1x2x2 blocks are created, and then paired up to finish all of the F2L minus two pieces, while also orienting the last layer edges. Then two corner-edge pairs are made and all of the edges are solved along with two corners. Finally the three remaining corners are finished with a commutator.
- The Snyder Method is a very hard to learn method where the cube is solved in stages, using algorithms that simultaneously permute and orient throughout. The F2L is solved using a variety of block building techniques for convenience, and a large number of algorithms are used to solve the last layer in 1 or 2 looks. Generally, the last layer is divided into two stages: 1) solve all edges + one corner, 2) solve remaining corners. And where this 2-look method requires more than 4 turns compared to a direct solve, the solution reverts over to a 1-look direct solve, or an approximation.
- The Human Thistlethwaite method is a human-usable version of the Thistlethwaite algorithm that reduces the cube to subgroups, and ends by solving the cube with only 180 degree face turns.
- SSC (Shadowslice Snow Columns) is an interesting method and has been variously described as a variation on Orient First, an advanced Columns first or an improved Belt Method. In truth, it is all and none of those. Perhaps the most accurate and descriptive name is Human Kociemba Algorithm in a similar way to Human Thistlethwaite described above as the first 'phase' reduces to oriented corners and edges and the second 'phase solves' completely. It has only a handful of algorithms- (<35 in full, <12 min depending on the variant) with a large amount being done intuitively. It is also notable as using pseudo-blocks and pseudo-pairs.
Partial Methods
There are a couple of methods which only provide a way to solve a part of the cube; a cuber using one of these methods would be able to choose another method to do the other parts of the cube with. Sometimes these partial methods improve the move count of a normal method, and sometimes they just provide a different style that some people may prefer.
- The MGLS Method is one way to solve the last layer plus a corner/edge slot. There are three steps: insert the F2L edge and orient the LL edges, insert the F2L corner and orient the LL corners, and then PLL.
- The ZB Method solves the last layer plus a corner/edge slot in just two steps, but requires hundreds of lengthy algorithms. First the cuber finishes the F2L while orienting the LL edges, and then the cuber solves the entire LL in one step.
- Vandenbergh-Harris (VH) is a method to solve the last F2L slot and orient last layer edges by connecting the last pair in U, then using a single case of ZBLS/ZBF2L. It's a good stepping stone for learning ZBLS.
- Winter Variation (WV) is a method to solve the last F2L slot while orienting corners of the last layer in a single step.
- CLL/ELL is an alternate way to finish the last layer with about the same speed and number of algorithms as OLL/PLL. There are two steps: solve the corners in one step (CLL), and solve the edges in one step (ELL). If you already have the edges oriented, you can modify the method to use CLL algorithms which keep the edges oriented, and then finish the cube with an edges-only PLL.
- M-CELL is one way to solve the 3x3 cube after solving an F2L-2 (Cross-1 and F2L-1). It has an algorithm count of 96, all for 3TCLL. See the original post here
- My World is another way to solve F2L-2 assuming the edges are already oriented. During third slot 3 LL edges are solved relative to each other reducing the number of algs needed for 1LLSLL. Note: Unlike M-CELL both the last slot and last layer is solved with 1 algorithm. It is also a 5-Look Solve Method or 5LS. Meaning the entire cube is solved with only 5 looks. The method has only ~3.8k algs; less than full 1LLL. See the original post here. For the SS Forum post look here.
External links
- Speedsolving.com: Method Survey
- Video on 3x3 speedsolving methods intro/comparison by CUBE PORTAL on youtube