The trial and error method

From Wikipedia, the free encyclopedia

Trial and error is a fundamental method of problem-solving^[1] characterized by repeated, varied attempts which are continued until success,^[2] or until the practicer stops trying.

According to W.H. Thorpe, the term was devised by C. Lloyd Morgan (1852–1936) after trying out similar phrases «trial and failure» and «trial and practice».^[3] Under Morgan’s Canon, animal behaviour should be explained in the simplest possible way. Where behavior seems to imply higher mental processes, it might be explained by trial-and-error learning. An example is a skillful way in which his terrier Tony opened the garden gate, easily misunderstood as an insightful act by someone seeing the final behavior. Lloyd Morgan, however, had watched and recorded the series of approximations by which the dog had gradually learned the response, and could demonstrate that no insight was required to explain it.

Edward Lee Thorndike was the initiator of the theory of trial and error learning based on the findings he showed how to manage a trial-and-error experiment in the laboratory. In his famous experiment, a cat was placed in a series of puzzle boxes in order to study the law of effect in learning.^[4] He plotted to learn curves which recorded the timing for each trial. Thorndike’s key observation was that learning was promoted by positive results, which was later refined and extended by B. F. Skinner’s operant conditioning.

Trial and error is also a method of problem solving, repair, tuning, or obtaining knowledge. In the field of computer science, the method is called generate and test (Brute force). In elementary algebra, when solving equations, it is guess and check.

This approach can be seen as one of the two basic approaches to problem-solving, contrasted with an approach using insight and theory. However, there are intermediate methods which for example, use theory to guide the method, an approach known as guided empiricism.

This way of thinking has become a mainstay of Karl Popper’s critical rationalism.

Methodology[edit]

The trial and error approach is used most successfully with simple problems and in games, and it is often the last resort when no apparent rule applies. This does not mean that the approach is inherently careless, for an individual can be methodical in manipulating the variables in an attempt to sort through possibilities that could result in success. Nevertheless, this method is often used by people who have little knowledge in the problem area. The trial-and-error approach has been studied from its natural computational point of view ^[5]

Simplest applications[edit]

Ashby (1960, section 11/5) offers three simple strategies for dealing with the same basic exercise-problem, which have very different efficiencies. Suppose a collection of 1000 on/off switches have to be set to a particular combination by random-based testing, where each test is expected to take one second. [This is also discussed in Traill (1978–2006, section C1.2]. The strategies are:

• the perfectionist all-or-nothing method, with no attempt at holding partial successes. This would be expected to take more than 10^301 seconds, [i.e., 2^1000 seconds, or 3·5×(10^291) centuries]
• a serial-test of switches, holding on to the partial successes (assuming that these are manifest), which would take 500 seconds on average
• parallel-but-individual testing of all switches simultaneously, which would take only one second

Note the tacit assumption here that no intelligence or insight is brought to bear on the problem. However, the existence of different available strategies allows us to consider a separate («superior») domain of processing — a «meta-level» above the mechanics of switch handling — where the various available strategies can be randomly chosen. Once again this is «trial and error», but of a different type.

Hierarchies[edit]

Ashby’s book develops this «meta-level» idea, and extends it into a whole recursive sequence of levels, successively above each other in a systematic hierarchy. On this basis, he argues that human intelligence emerges from such organization: relying heavily on trial-and-error (at least initially at each new stage), but emerging with what we would call «intelligence» at the end of it all. Thus presumably the topmost level of the hierarchy (at any stage) will still depend on simple trial-and-error.

Traill (1978–2006) suggests that this Ashby-hierarchy probably coincides with Piaget’s well-known theory of developmental stages. [This work also discusses Ashby’s 1000-switch example; see §C1.2]. After all, it is part of Piagetian doctrine that children learn first by actively doing in a more-or-less random way, and then hopefully learn from the consequences — which all has a certain resemblance to Ashby’s random «trial-and-error».

Application[edit]

Traill (2008, espec. Table «S» on p.31) follows Jerne and Popper in seeing this strategy as probably underlying all knowledge-gathering systems — at least in their initial phase.

Four such systems are identified:

• Natural selection which «educates» the DNA of the species,
• The brain of the individual (just discussed);
• The «brain» of society-as-such (including the publicly held body of science); and
• The adaptive immune system.

Features[edit]

Trial and error has a number of features:

• solution-oriented: trial and error makes no attempt to discover why a solution works, merely that it is a solution.
• problem-specific: trial and error makes no attempt to generalize a solution to other problems.
• non-optimal: trial and error is generally an attempt to find a solution, not all solutions, and not the best solution.
• needs little knowledge: trials and error can proceed where there is little or no knowledge of the subject.

It is possible to use trial and error to find all solutions or the best solution, when a testably finite number of possible solutions exist. To find all solutions, one simply makes a note and continues, rather than ending the process, when a solution is found, until all solutions have been tried. To find the best solution, one finds all solutions by the method just described and then comparatively evaluates them based upon some predefined set of criteria, the existence of which is a condition for the possibility of finding a best solution. (Also, when only one solution can exist, as in assembling a jigsaw puzzle, then any solution found is the only solution and so is necessarily the best.)

Examples[edit]

Trial and error has traditionally been the main method of finding new drugs, such as antibiotics. Chemists simply try chemicals at random until they find one with the desired effect. In a more sophisticated version, chemists select a narrow range of chemicals it is thought may have some effect using a technique called structure–activity relationship. (The latter case can be alternatively considered as a changing of the problem rather than of the solution strategy: instead of «What chemical will work well as an antibiotic?» the problem in the sophisticated approach is «Which, if any, of the chemicals in this narrow range will work well as an antibiotic?») The method is used widely in many disciplines, such as polymer technology to find new polymer types or families.

Trial and error is also commonly seen in player responses to video games — when faced with an obstacle or boss, players often form a number of strategies to surpass the obstacle or defeat the boss, with each strategy being carried out before the player either succeeds or quits the game.

Sports teams also make use of trial and error to qualify for and/or progress through the playoffs and win the championship, attempting different strategies, plays, lineups and formations in hopes of defeating each and every opponent along the way to victory. This is especially crucial in playoff series in which multiple wins are required to advance, where a team that loses a game will have the opportunity to try new tactics to find a way to win, if they are not eliminated yet.

The scientific method can be regarded as containing an element of trial and error in its formulation and testing of hypotheses. Also compare genetic algorithms, simulated annealing and reinforcement learning – all varieties for search which apply the basic idea of trial and error.

Biological evolution can be considered as a form of trial and error.^[6] Random mutations and sexual genetic variations can be viewed as trials and poor reproductive fitness, or lack of improved fitness, as the error. Thus after a long time ‘knowledge’ of well-adapted genomes accumulates simply by virtue of them being able to reproduce.

Bogosort, a conceptual sorting algorithm (that is extremely inefficient and impractical), can be viewed as a trial and error approach to sorting a list. However, typical simple examples of bogosort do not track which orders of the list have been tried and may try the same order any number of times, which violates one of the basic principles of trial and error. Trial and error is actually more efficient and practical than bogosort; unlike bogosort, it is guaranteed to halt in finite time on a finite list, and might even be a reasonable way to sort extremely short lists under some conditions.

Jumping spiders of the genus Portia use trial and error to find new tactics against unfamiliar prey or in unusual situations, and remember the new tactics.^[7] Tests show that Portia fimbriata and Portia labiata can use trial and error in an artificial environment, where the spider’s objective is to cross a miniature lagoon that is too wide for a simple jump, and must either jump then swim or only swim.^[8][9]

See also[edit]

References[edit]

Further reading[edit]

• Ashby, W. R. (1960: Second Edition). Design for a Brain. Chapman & Hall: London.
• Traill, R.R. (1978–2006). Molecular explanation for intelligence…, Brunel University Thesis, HDL.handle.net
• Traill, R.R. (2008). Thinking by Molecule, Synapse, or both? — From Piaget’s Schema, to the Selecting/Editing of ncRNA. Ondwelle: Melbourne. Ondwelle.com — or French version Ondwelle.com.
• Zippelius, R. (1991). Die experimentierende Methode im Recht (Trial and error in Jurisprudence), Academy of Science, Mainz, ISBN 3-515-05901-6

Trial and error

Method of problem-solving / From Wikipedia, the free encyclopedia

Dear Wikiwand AI, let’s keep it short, summarize this topic like I’m. Ten years old or a College student

Trial and error is a fundamental method of problem-solving [1] characterized by repeated, varied attempts which are continued until success, [2] or until the practicer stops trying.

According to W.H. Thorpe, the term was devised by C. Lloyd Morgan (1852–1936) after trying out similar phrases «trial and failure» and «trial and practice». [3] Under Morgan’s Canon, animal behaviour should be explained in the simplest possible way. Where behavior seems to imply higher mental processes, it might be explained by trial-and-error learning. An example is a skillful way in which his terrier Tony opened the garden gate, easily misunderstood as an insightful act by someone seeing the final behavior. Lloyd Morgan, however, had watched and recorded the series of approximations by which the dog had gradually learned the response, and could demonstrate that no insight was required to explain it.

Edward Lee Thorndike was the initiator of the theory of trial and error learning based on the findings he showed how to manage a trial-and-error experiment in the laboratory. In his famous experiment, a cat was placed in a series of puzzle boxes in order to study the law of effect in learning. [4] He plotted to learn curves which recorded the timing for each trial. Thorndike’s key observation was that learning was promoted by positive results, which was later refined and extended by B. F. Skinner’s operant conditioning.

Trial and error is also a method of problem solving, repair, tuning, or obtaining knowledge. In the field of computer science, the method is called generate and test (Brute force). In elementary algebra, when solving equations, it is guess and check.

This approach can be seen as one of the two basic approaches to problem-solving, contrasted with an approach using insight and theory. However, there are intermediate methods which for example, use theory to guide the method, an approach known as guided empiricism.

This way of thinking has become a mainstay of Karl Popper’s critical rationalism.

Источник

Exceptions

Table of Contents

PHP has an exception model similar to that of other programming languages. An exception can be throw n, and caught (» catch ed») within PHP. Code may be surrounded in a try block, to facilitate the catching of potential exceptions. Each try must have at least one corresponding catch or finally block.

If an exception is thrown and its current function scope has no catch block, the exception will «bubble up» the call stack to the calling function until it finds a matching catch block. All finally blocks it encounters along the way will be executed. If the call stack is unwound all the way to the global scope without encountering a matching catch block, the program will terminate with a fatal error unless a global exception handler has been set.

The thrown object must be an instanceof Throwable . Trying to throw an object that is not will result in a PHP Fatal Error.

As of PHP 8.0.0, the throw keyword is an expression and may be used in any expression context. In prior versions it was a statement and was required to be on its own line.

catch

A catch block defines how to respond to a thrown exception. A catch block defines one or more types of exception or error it can handle, and optionally a variable to which to assign the exception. (The variable was required prior to PHP 8.0.0.) The first catch block a thrown exception or error encounters that matches the type of the thrown object will handle the object.

Multiple catch blocks can be used to catch different classes of exceptions. Normal execution (when no exception is thrown within the try block) will continue after that last catch block defined in sequence. Exceptions can be throw n (or re-thrown) within a catch block. If not, execution will continue after the catch block that was triggered.

When an exception is thrown, code following the statement will not be executed, and PHP will attempt to find the first matching catch block. If an exception is not caught, a PHP Fatal Error will be issued with an » Uncaught Exception . » message, unless a handler has been defined with set_exception_handler() .

As of PHP 7.1.0, a catch block may specify multiple exceptions using the pipe ( | ) character. This is useful for when different exceptions from different class hierarchies are handled the same.

As of PHP 8.0.0, the variable name for a caught exception is optional. If not specified, the catch block will still execute but will not have access to the thrown object.

finally

A finally block may also be specified after or instead of catch blocks. Code within the finally block will always be executed after the try and catch blocks, regardless of whether an exception has been thrown, and before normal execution resumes.

One notable interaction is between the finally block and a return statement. If a return statement is encountered inside either the try or the catch blocks, the finally block will still be executed. Moreover, the return statement is evaluated when encountered, but the result will be returned after the finally block is executed. Additionally, if the finally block also contains a return statement, the value from the finally block is returned.

Global exception handler

If an exception is allowed to bubble up to the global scope, it may be caught by a global exception handler if set. The set_exception_handler() function can set a function that will be called in place of a catch block if no other block is invoked. The effect is essentially the same as if the entire program were wrapped in a try — catch block with that function as the catch .

Notes

Internal PHP functions mainly use Error reporting, only modern Object-oriented extensions use exceptions. However, errors can be easily translated to exceptions with ErrorException. This technique only works with non-fatal errors, however.

Example #1 Converting error reporting to exceptions

function exceptions_error_handler ( $severity , $message , $filename , $lineno ) <
throw new ErrorException ( $message , 0 , $severity , $filename , $lineno );
>

Examples

Example #2 Throwing an Exception

function inverse ( $x ) <
if (! $x ) <
throw new Exception ( ‘Division by zero.’ );
>
return 1 / $x ;
>

try <
echo inverse ( 5 ) . «n» ;
echo inverse ( 0 ) . «n» ;
> catch ( Exception $e ) <
echo ‘Caught exception: ‘ , $e -> getMessage (), «n» ;
>

// Continue execution
echo «Hello Worldn» ;
?>

The above example will output:

Example #3 Exception handling with a finally block

function inverse ( $x ) <
if (! $x ) <
throw new Exception ( ‘Division by zero.’ );
>
return 1 / $x ;
>

try <
echo inverse ( 5 ) . «n» ;
> catch ( Exception $e ) <
echo ‘Caught exception: ‘ , $e -> getMessage (), «n» ;
> finally <
echo «First finally.n» ;
>

try <
echo inverse ( 0 ) . «n» ;
> catch ( Exception $e ) <
echo ‘Caught exception: ‘ , $e -> getMessage (), «n» ;
> finally <
echo «Second finally.n» ;
>

// Continue execution
echo «Hello Worldn» ;
?>

The above example will output:

Example #4 Interaction between the finally block and return

function test () <
try <
throw new Exception ( ‘foo’ );
> catch ( Exception $e ) <
return ‘catch’ ;
> finally <
return ‘finally’ ;
>
>

The above example will output:

Example #5 Nested Exception

class Test <
public function testing () <
try <
try <
throw new MyException ( ‘foo!’ );
> catch ( MyException $e ) <
// rethrow it
throw $e ;
>
> catch ( Exception $e ) <
var_dump ( $e -> getMessage ());
>
>
>

$foo = new Test ;
$foo -> testing ();

The above example will output:

Example #6 Multi catch exception handling

class MyOtherException extends Exception

class Test <
public function testing () <
try <
throw new MyException ();
> catch ( MyException | MyOtherException $e ) <
var_dump ( get_class ( $e ));
>
>
>

$foo = new Test ;
$foo -> testing ();

The above example will output:

Example #7 Omitting the caught variable

Only permitted in PHP 8.0.0 and later.

class SpecificException extends Exception <>

function test () <
throw new SpecificException ( ‘Oopsie’ );
>

try <
test ();
> catch ( SpecificException ) <
print «A SpecificException was thrown, but we don’t care about the details.» ;
>
?>

Example #8 Throw as an expression

Only permitted in PHP 8.0.0 and later.

class SpecificException extends Exception <>

function test () <
do_something_risky () or throw new Exception ( ‘It did not work’ );
>

try <
test ();
> catch ( Exception $e ) <
print $e -> getMessage ();
>
?>

User Contributed Notes 13 notes

If you intend on creating a lot of custom exceptions, you may find this code useful. I’ve created an interface and an abstract exception class that ensures that all parts of the built-in Exception class are preserved in child classes. It also properly pushes all information back to the parent constructor ensuring that nothing is lost. This allows you to quickly create new exceptions on the fly. It also overrides the default __toString method with a more thorough one.

interface IException
<
/* Protected methods inherited from Exception class */
public function getMessage (); // Exception message
public function getCode (); // User-defined Exception code
public function getFile (); // Source filename
public function getLine (); // Source line
public function getTrace (); // An array of the backtrace()
public function getTraceAsString (); // Formated string of trace

/* Overrideable methods inherited from Exception class */
public function __toString (); // formated string for display
public function __construct ( $message = null , $code = 0 );
>

abstract class CustomException extends Exception implements IException
<
protected $message = ‘Unknown exception’ ; // Exception message
private $string ; // Unknown
protected $code = 0 ; // User-defined exception code
protected $file ; // Source filename of exception
protected $line ; // Source line of exception
private $trace ; // Unknown

public function __construct ( $message = null , $code = 0 )
<
if (! $message ) <
throw new $this ( ‘Unknown ‘ . get_class ( $this ));
>
parent :: __construct ( $message , $code );
>

public function __toString ()
<
return get_class ( $this ) . » ‘ < $this ->message > ‘ in < $this ->file > ( < $this ->line > )n»
. » < $this ->getTraceAsString ()> » ;
>
>
?>

Now you can create new exceptions in one line:

class TestException extends CustomException <>
?>

Here’s a test that shows that all information is properly preserved throughout the backtrace.

function exceptionTest ()
<
try <
throw new TestException ();
>
catch ( TestException $e ) <
echo «Caught TestException (‘ < $e ->getMessage ()> ‘)n < $e >n» ;
>
catch ( Exception $e ) <
echo «Caught Exception (‘ < $e ->getMessage ()> ‘)n < $e >n» ;
>
>

echo » ;
?>

Here’s a sample output:

Источник

Trial and error

From Wikipedia, the free encyclopedia

Trial and error is a fundamental method of problem-solving [1] characterized by repeated, varied attempts which are continued until success, [2] or until the practicer stops trying.

According to W.H. Thorpe, the term was devised by C. Lloyd Morgan (1852–1936) after trying out similar phrases «trial and failure» and «trial and practice». [3] Under Morgan’s Canon, animal behaviour should be explained in the simplest possible way. Where behavior seems to imply higher mental processes, it might be explained by trial-and-error learning. An example is a skillful way in which his terrier Tony opened the garden gate, easily misunderstood as an insightful act by someone seeing the final behavior. Lloyd Morgan, however, had watched and recorded the series of approximations by which the dog had gradually learned the response, and could demonstrate that no insight was required to explain it.

Edward Lee Thorndike was the initiator of the theory of trial and error learning based on the findings he showed how to manage a trial-and-error experiment in the laboratory. In his famous experiment, a cat was placed in a series of puzzle boxes in order to study the law of effect in learning. [4] He plotted to learn curves which recorded the timing for each trial. Thorndike’s key observation was that learning was promoted by positive results, which was later refined and extended by B. F. Skinner’s operant conditioning.

Trial and error is also a method of problem solving, repair, tuning, or obtaining knowledge. In the field of computer science, the method is called generate and test (Brute force). In elementary algebra, when solving equations, it is guess and check.

This approach can be seen as one of the two basic approaches to problem-solving, contrasted with an approach using insight and theory. However, there are intermediate methods which for example, use theory to guide the method, an approach known as guided empiricism.

This way of thinking has become a mainstay of Karl Popper’s critical rationalism.

Источник

JavaScript Errors

Throw, and Try. Catch. Finally

The try statement defines a code block to run (to try).

The catch statement defines a code block to handle any error.

The finally statement defines a code block to run regardless of the result.

The throw statement defines a custom error.

Errors Will Happen!

When executing JavaScript code, different errors can occur.

Errors can be coding errors made by the programmer, errors due to wrong input, and other unforeseeable things.

Example

In this example we misspelled «alert» as «adddlert» to deliberately produce an error:

JavaScript catches adddlert as an error, and executes the catch code to handle it.

JavaScript try and catch

The try statement allows you to define a block of code to be tested for errors while it is being executed.

The catch statement allows you to define a block of code to be executed, if an error occurs in the try block.

The JavaScript statements try and catch come in pairs:

JavaScript Throws Errors

When an error occurs, JavaScript will normally stop and generate an error message.

The technical term for this is: JavaScript will throw an exception (throw an error).

JavaScript will actually create an Error object with two properties: name and message.

The throw Statement

The throw statement allows you to create a custom error.

Technically you can throw an exception (throw an error).

The exception can be a JavaScript String , a Number , a Boolean or an Object :

If you use throw together with try and catch , you can control program flow and generate custom error messages.

Input Validation Example

This example examines input. If the value is wrong, an exception (err) is thrown.

The exception (err) is caught by the catch statement and a custom error message is displayed:

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Trial and error, or trial by error or try an error, is a general method of problem solving, fixing things, or for obtaining knowledge.
«Learning doesn’t happen from failure itself but rather from analyzing the failure, making a change, and then trying again.»^[1]

In the field of computer science, the method is called generate and test. In elementary algebra, when solving equations, it is «guess and check».

This approach can be seen as one of the two basic approaches to problem solving and is contrasted with an approach using insight and theory.

Process

Bricolage —
In trial and error, one selects a possible answer, applies it to the problem and, if it is not successful, selects (or generates) another possibility that is subsequently tried. The process ends when a possibility yields a solution.

In some versions of trial and error, the option that is a priori viewed as the most likely one should be tried first, followed by the next most likely, and so on until a solution is found, or all the options are exhausted. In other versions, options are simply tried at random.

Methodology

This approach is more successful with simple problems and in games, and is often resorted to when no apparent rule applies. This does not mean that the approach need be careless, for an individual can be methodical in manipulating the variables in an attempt to sort through possibilities that may result in success. Nevertheless, this method is often used by people who have little knowledge in the problem area.

Simplest applications

Ashby (1960, section 11/5) offers three simple strategies for dealing with the same basic exercise-problem; and they have very different efficiencies:
Suppose there are 1000 on/off switches which have to be set to a particular combination by random-based testing, each test to take one second. [This is also discussed in Traill (1978/2006, section C1.2]. The strategies are:

• the perfectionist all-or-nothing method, with no attempt at holding partial successes. This would be expected to take more than 10^301 seconds, [i.e. 2^1000 seconds, or 3·5×(10^291) centuries!];
• a serial-test of switches, holding on to the partial successes (assuming that these are manifest) would take 500 seconds; while
• a parallel-but-individual testing of all switches simultaneously would take only one second.

Note the tacit assumption here that no intelligence or insight is brought to bear on the problem. However, the existence of different available strategies allows us to consider a separate («superior») domain of processing — a «meta-level» above the mechanics of switch handling — where the various available strategies can be randomly chosen. Once again this is «trial and error», but of a different type. This leads us to:

Trial-and-error Hierarchies

Ashby’s book develops this «meta-level» idea, and extends it into a whole recursive sequence of levels, successively above each other in a systematic hierarchy. On this basis he argues that human intelligence emerges from such organization: relying heavily on trial-and-error (at least initially at each new stage), but emerging with what we would call «intelligence» at the end of it all. Thus presumably the topmost level of the hierarchy (at any stage) will still depend on simple trial-and-error.

Traill (1978/2006) suggests that this Ashby-hierarchy probably coincides with Piaget’s well-known theory of developmental stages. [This work also discusses Ashby’s 1000-switch example; see §C1.2]. After all, it is part of Piagetian doctrine that children learn by first actively doing in a more-or-less random way, and then hopefully learn from the consequences — which all has a certain resemblance to Ashby’s random «trial-and-error».

The basic strategy in many fields?

Traill (2008, espec. Table «S» on p.31) follows Jerne and Popper in seeing this strategy as probably underlying all knowledge-gathering systems — at least in their initial phase.

Four such systems are identified:

• Darwinian evolution which «educates» the DNA of the species!
• The brain of the individual (just discussed);
• The «brain» of society-as-such (including the publicly-held body of science); and
• The immune system.

An ambiguity: Can we have «intention» during a «trial»

In the Ashby-and-Cybernetics tradition, the word «trial» usually implies random-or-arbitrary, without any deliberate choice.
However amongst non-cyberneticians, «trial» will often imply a deliberate subjective act by some adult human agent; (e.g. in a court-room, or laboratory). So that has sometimes led to confusion.

Of course the situation becomes even more confusing if one accepts Ashby’s hierarchical explanation of intelligence, and its implied ability to be deliberate and to creatively design — all based ultimately on non-deliberate actions! The lesson here seems to be that one must simply be careful to clarify the meaning of one’s own words, and indeed the words of others. [Incidentally it seems that consciousness is not an essential ingredient for intelligence as discussed above.]

Features

Trial and error has a number of features:

• solution-oriented: trial and error makes no attempt to discover why a solution works, merely that it is a solution.
• problem-specific: trial and error makes no attempt to generalise a solution to other problems.
• non-optimal: trial and error is generally an attempt to find a solution, not all solutions, and not the best solution.
• needs little knowledge: trials and error can proceed where there is little or no knowledge of the subject.

It is possible to use trial and error to find all solutions or the best solution, when a testably finite number of possible solutions exist. To find all solutions, one simply makes a note and continues, rather than ending the process, when a solution is found, until all solutions have been tried. To find the best solution, one finds all solutions by the method just described and then comparatively evaluates them based upon some predefined set of criteria, the existence of which is a condition for the possibility of finding a best solution. (Also, when only one solution can exist, as in assembling a jigsaw puzzle, then any solution found is the only solution and so is necessarily the best.)

Examples

Trial and error has traditionally been the main method of finding new drugs, such as antibiotics. Chemists simply try chemicals at random until they find one with the desired effect. In a more sophisticated version, chemists select a narrow range of chemicals it is thought may have some effect. (The latter case can be alternatively considered as a changing of the problem rather than of the solution strategy: instead of «What chemical will work well as an antibiotic?» the problem in the sophisticated approach is «Which, if any, of the chemicals in this narrow range will work well as an antibiotic?») The method is used widely in many disciplines, such as polymer technology to find new polymer types or families.

The scientific method can be regarded as containing an element of trial and error in its formulation and testing of hypotheses. Also compare genetic algorithms, simulated annealing and reinforcement learning — all varieties for search which apply the basic idea of trial and error.

Biological evolution is also a form of trial and error. Random mutations and sexual genetic variations can be viewed as trials and poor reproductive fitness, or lack of improved fitness, as the error. Thus after a long time ‘knowledge’ of well-adapted genomes accumulates simply by virtue of them being able to reproduce.

Bogosort, a conceptual sorting algorithm (that is extremely inefficient and impractical), can be viewed as a trial and error approach to sorting a list. However, typical simple examples of bogosort do not track which orders of the list have been tried and may try the same order any number of times, which violates one of the basic principles of trial and error. Trial and error is actually more efficient and practical than bogosort; unlike bogosort, it is guaranteed to halt in finite time on a finite list, and might even be a reasonable way to sort extremely short lists under some conditions.

Issues with trial and error

Trial and error is usually a last resort for a particular problem, as there are a number of problems with it. For one, trial and error is tedious and monotonous. Also, it is very time-consuming; chemical engineers must sift through millions of various potential chemicals before they find one that works. Fortunately, computers are best suited for trial and error; they do not succumb to the boredom that humans do, and can potentially do thousands of trial-and-error segments in the blink of an eye.

References

• Ashby, W. R. (1960: Second Edition). Design for a Brain. Chapman & Hall: London.
• Traill, R.R. (1978/2006). Molecular explanation for intelligence…, Brunel University Thesis, HDL.handle.net
• Traill, R.R. (2008). Thinking by Molecule, Synapse, or both? — From Piaget’s Schema, to the Selecting/Editing of ncRNA. Ondwelle: Melbourne. Ondwelle.com — or French version Ondwelle.com.

See also

• Brute force attack
• Brute-force search
• Empiricism

References

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Factoring Trinomials by Trial and Error – Method & Examples

Are you still struggling with the topic of factoring trinomials in Algebra? Well, no worries, because you are at the right place.

This article will introduce you to one of the simplest methods of factoring trinomials known as trial and error.

As the name suggests, trial and error factoring entails trying all possible factors until you find the right one.

Trial and error factoring is regarded as one of the best methods of factoring trinomials. It encourages students to develop their mathematical intuition and thus increase their conceptual understanding of the topic.

How to Unfoil trinomials?

Suppose we want to unfoil the general equation of a trinomial ax² + bx + c where a ≠ 1. Here are the steps to follow:

• Insert the factors of ax²in the 1^st positions of the two sets of brackets that represent the factors.
• Also, insert the possible factors of c into the 2^nd positions of brackets.
• Identify both the inner and outer products of the two sets of brackets.
• Keep on trying different factors until the sum of the two factors is equal to “bx.”

NOTE:

• If c is positive, both factors will have the same sign as “b”.
• If c is negative, one factor will have a negative sign.
• Never put in the same parentheses’ numbers with a common factor.

Trial and error factoring

Trial and error factoring, which is also referred to as reverse foil or unfoiling, is a method of factoring trinomials built upon different techniques such as foil, factoring by grouping, and some other concepts of factoring trinomials with a leading coefficient of 1.

Example 1

Use trial and error factoring to solve 6x² – 25x + 24

Solution

Paired factors of 6x² are x (6x) or 2x (3x), therefore our parentheses will be;

(x – ?) (6x – ?) or (2x – ?) (3x – ?)

Replace “bx” with possible paired factors of c. Try all paired factors of 24 that will produce -25 The possible choices are (1 & 24, 2 & 12, 3 & 8, 4 & 6). Therefore, the correct factoring is;

6x² – 25x + 24 ⟹ (2x – 3) (3x –

Example 2

Factor x² – 5x + 6

Solution

The factors of the first term x², are x and x. Therefore, insert x in the first position of each parentheses.

x² – 5x + 6 = (x – ?) (x – ?)

Since last term is 6, therefore the possible choices of factors are:

(x + 1) (x + 6)
(x – 1) (x – 6)
(x + 3) (x + 2)
(x – 3) (x – 2)

The correct pair which gives -5x as the middle term is (x – 3) (x – 2). Hence,

(x – 3) (x – 2) is the answer.

Example 3

Factor x² – 7x + 10

Solution

Insert the factors of the first term in the first position of each parentheses.

⟹ (x -?) (x -?)

Try the possible pair of factors of the 10;

⟹ (-5) + (-2) = -7

Now replace the question marks in the parentheses with these two factors

⟹ (x -5) (x -2)

Hence, the correct factoring of x² – 7x + 10 is (x -5) (x -2)

Example 4

Factor 4x² – 5x – 6

Solution

(2x -?) (2x +?) and (4x -?) (x +?)

Try the possible pair of factors;

6 x² − 2x – 151 & 6, 2 & 3, 3 & 2, 6 & 1

Since the correct pair 3 and 2, therefore, (4x – 3) (x + 2) is our answer.

Example 5

Factor the trinomial x² − 2x – 15

Solution

Insert x in the first position of each parentheses.

(x -?) (x +?)

Find two numbers whose product and sum are -15 and -2, respectively. By trial and error, the possible combinations are:

15 and -1;

-1 and 15;

5 and -3;

-5 and 3;

Our correct combination is – 5 and 3. Therefore;

x² − 2x – 15 ⟹ (x -5) (x +3)

How to factor trinomials by grouping?

We can also factor trinomials by using a method of grouping. Let’s walk through the following steps to factor ax² + bx + c where a ≠1:

• Find the product of the leading coefficient “a” and the constant “c.”

⟹ a * c = ac

• Look for the factors of the “ac” that add to coefficient “b.”
• Rewrite bx as a sum or difference of the factors of ac that add to b.
• Now factor by grouping.

Example 6

Factor the trinomial 5x² + 16x + 3 by grouping.

Solution

Find the product of the leading coefficient and the last term.

⟹ 5 *3 = 15

Perform trial and error to find pair factors of 15 whose sum is the middle term (16). The correct pair is 1 and 15.

Rewrite the equation by replacing the middle term 16x by x and 15x.

5x² + 16x + 3⟹5x² + 15x + x + 3

Now, factor out by grouping

5x² + 15x + x + 3 ⟹ 5x (x + 3) + 1(x + 3)

⟹ (5x +1) (x + 3)

Example 7

Factor 2x² – 5x – 12 by grouping.

Solution

2x² – 5x – 12

= 2x² + 3x – 8x – 12

= x (2x + 3) – 4(2x + 3)

= (2x + 3) (x – 4)

Example 8

Factor 6x² + x – 2

Solution

Multiply the leading coefficient a and the constant c.

⟹ 6 * -2 = -12

Find two numbers whose product and sum are -12 and 1 respectively.

⟹ – 3 * 4

⟹ -3 + 4 = 1

Rewrite the equation by replacing the middle term -5x by -3x and 4x

⟹ 6x² -3x + 4x -2

Finally, factor out by grouping

⟹ 3x (2x –  1) + 2(2x – 1)

⟹ (3x + 2) (2x – 1)

Example 9

Factor 6y² + 11y + 4.

Solution

6y² + 11y + 4 ⟹ 6y² + 3y + y + 4

⟹ (6y² + 3y) + (8y + 4)

⟹ 3y (2y + 1) + 4(2y + 1)

= (2y + 1) (3y + 4)

Trial and error, or trial by error, is a general method of problem solving for obtaining knowledge, both propositional knowledge and know-how. In the field of computer science, the method is called generate and test. In elementary algebra, when solving equations, it is «guess and check».

This approach can be seen as one of the two basic approaches to problem solving and is contrasted with an approach using insight and theory.

Process

Bricolage -In trial and error, one selects a possible answer, applies it to the problem and, if it is not successful, selects (or generates) another possibility that is subsequently tried. The process ends when a possibility yields a solution.

In some versions of trial and error, the option that is a priori viewed as the most likely one should be tried first, followed by the next most likely, and so on until a solution is found, or all the options are exhausted. In other versions, options are simply tried at random.

Methodology

This approach is more successful with simple problems and in games, and is often resorted to when no apparent rule applies. This does not mean that the approach need be careless, for an individual can be methodical in manipulating the variables in an attempt to sort through possibilities that may result in success. Nevertheless, this method is often used by people who have little knowledge in the problem area.

Features

Trial and error has a number of features:
* solution-oriented: trial and error makes no attempt to discover «why» a solution works, merely that it «is» a solution.
* problem-specific: trial and error makes no attempt to generalise a solution to other problems.
* non-optimal: trial and error is an attempt to find «a» solution, not «all» solutions, and not the «best» solution.
* needs little knowledge: trials and error can proceed where there is little or no knowledge of the subject.

Examples

Trial and error has traditionally been the main method of finding new drugs, such as antibiotics. Chemists simply try chemicals at random until they find one with the desired effect.

The scientific method can be regarded as containing an element of trial and error in its formulation and testing of hypotheses. Also compare genetic algorithms, simulated annealing and reinforcement learning — all varieties for search which apply the basic idea of trial and error.

Biological evolution is also a form of trial and error. Random mutations and sexual genetic variations can be viewed as trials and poor reproductive fitness as the error. Thus after a long time ‘knowledge’ of well-adapted genomes accumulates simply by virtue of them being «able» to reproduce.

Bogosort can be viewed as a trial and error approach to sorting a list.

In mathematics, the method of trial and error can be used to solve formulae — it is a slower, less precise method than algebra, but is easier to understand.

ee also

* Brute force attack
* Brute-force search
* Empiricism

Notes

The Use of Trial and Error To Solve Problems

Some complex problems can be solved by a technique that is called trial and error. Trial and error is typically good for problems where you have multiple chances to get the correct solution. However, this is not a good technique for problems that don’t give you multiple chances to find a solution.

Trial and error is also an excellent tool for inventors. The inventor will first imagine a device they would like to invent, and then they may go through the trial and error process to find the best ways of inventing the device. While trial and error is an extremely powerful tool that can be used to solve problems, it also has some weaknesses. I mentioned this briefly in the first paragraph. Trial and error is inefficient in situations where making an error can lead to serious injury or death. A good example of this would be the attempts that were made by aircraft engineers to break the sound barrier. Though they succeeded, many pilots died because they would use the trial and error process. In the situation, and error would often lead to the plane crashing.

Author Description

Analytical Reasoning: Trial and Error Method

Learn the right approach to handle Analytical Reasoning questions with the help of Hit & Trial method.

Questions in the analytical reasoning section test your ability to analyze the situation, reason the cause/ effect and then find a solution. While questions in most of the competitive exams have options with them, but sometimes, we are not presented with options. A lot of effort is required to identify and reach the right option. What works best in such Analytical Reasoning questions is what we call as a trial and error strategy.

What is Trial and Error Strategy?

Try something (Guess), if it does not work, guess again. Over time, our guesses become more educated — and we tend to require fewer trials to reach our solutions.

Tips to remember:

• First of all you need to read the directions of a particular Data Sufficiency question very carefully as examiner can change the directions and even after solving all the questions correctly you mark the wrong answers.
• You need to remember the steps involved in solving a particular Data Sufficiency question and follow them in this particular order: Check A (i.e. the first statement), then Check B (i.e. the second statement) and lastly, if required, combine the two statements to get the answer.
• Do not make any assumptions while solving Data Sufficiency questions.

Solved Questions:

Question: 44 people live in eight houses (3 on each side of the Albert Square). Each house has a different number of people living in it. Each line of three houses has 15 people living in it. How many people live in each house?

Solution: Simplifying, we have to get 4 sets of numbers, each of which add up to 15. So start guessing.

На основании Вашего запроса эти примеры могут содержать грубую лексику.

На основании Вашего запроса эти примеры могут содержать разговорную лексику.

методом проб и ошибок

метода проб и ошибок

методу проб и ошибок

Предложения

Therefore, nothing prevents these people from using the proven and reliable trial and error method to identify true aspirations.

Следовательно, ничто не мешает применить проверенный и надежный «метод проб и ошибок», чтобы выявить истинные склонности.

The trial and error method seems appropriate in that case.

According to scientific data, the trial and error method actually implies 99.99% mistakes.

The trial and error method is responsible for the absence of good criteria in evaluating new technological ideas.

Метод проб и ошибок несет ответственность и за отсутствие критериев оценки новых технических идей.

Construction of new mental actions by an organism occasionally appeared to be successful in the behaviorism selection («trial and error method«).

Построение организмом новых психических действий объяснялось в русле бихевиоризма отбором случайно оказавшихся успешными («метод проб и ошибок»).

If not, then you have to play by the trial and error method.

Altogether, the best procedure for finding the perfect solution is the trial and error method.

Другими словами, наилучший способ найти идеальное решение — это метод проб и ошибок.

Using the trial and error method, you test one arrangement after another and check their stability.

Используя метод проб и ошибок, Вы одно за другим тестируете эти расположения и проверяете их стабильность.

And the first of them is the notorious trial and error method for each of us.

И первый из их числа — печально известный каждому из нас метод проб и ошибок.

So they remember things better if they use the trial and error method, since they have to make certain associations that require more work within the brain.

Поэтому они лучше запоминают вещи, если используют метод проб и ошибок, поскольку им приходится создавать определенные ассоциации, которые требуют больше работы в мозге.

Key words: knowledge, the science, the common-life mind, the intellectual mind, poverty, social security, trial and error method.

Ключевые слова: знание, наука, общежизненный ум, интеллегентский ум, бедность, социальное обеспечение, метод проб и ошибок.

Since ancient times, man has used the trial and error method as the only available way of making decisions on all aspects of his life.

С древнейших времен человек использует Метод проб и ошибок как единственный имеющийся в его арсенале способ принятия решений по любым аспектам его жизни и деятельности.

Resume: A reasonable choice would be a trial and error method, that is, learning partners better through joint projects.

Резюме: Для Экономического пояса Шелкового пути разумным выбором является метод проб и ошибок, то есть узнавание партнеров посредством совместных проектов.

While a trial and error method may be the best option for selecting the right web hosting company for your needs, it would certainly take time, effort, and money.

Хотя метод проб и ошибок может быть лучшим вариантом для выбора подходящей веб-хостинговой компании для ваших нужд, это, безусловно, потребует времени, усилий и денег.

A brute-force attack is a trial and error method used by hackers to guess credentials or encrypted data such as login, passwords or encryption keys, through exhaustive effort (using brute force) with the hope of eventually guessing correctly.

Атака методом «грубой силы» — это метод проб и ошибок, используемый хакерами для подбора учетных данных или зашифрованных данных, таких как логин, пароли или ключи шифрования, посредством исчерпывающих усилий (с использованием грубой силы) с надеждой в конечном итоге угадать правильно.

Each time to find the right one is a special art, and each time it requires the intuition of the researcher, as well as the trial and error method.

Каждый раз подобрать нужную — это особое искусство, и каждый раз — это интуиция исследователя, а также метод проб и ошибок.

Careful consideration of the activities of Khrushchev, Kosygin, Andropov, Gorbachev, and Yakovlev allows us to make a conclusion that they put into practice the trial and error method.

Внимательное рассмотрение деятельности Хрущёва, Косыгина, Андропова, Горбачёва и Яковлева позволяет сделать вывод, что они реализовывали на практике метод проб и ошибок.

Trial and error method is not for you?

Trial and error method, based on the accumulation of life experiences to find effective behaviors

метод проб и ошибок, основанный на накоплении жизненного опыта и поиске эффективных моделей поведения

Use the good old trial and error method.

Ничего не найдено для этого значения.

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