How to Solve It

TRIZ (; Russian: теория решения изобретательских задач, romanized: teoriya resheniya izobretatelskikh zadach, lit. 'theory of inventive problem solving') is a methodology which combines an organized, systematic method of problem-solving with analysis and forecasting techniques derived from the study of patterns of invention in global patent literature. The development and improvement of products and technologies in accordance with TRIZ are guided by the laws of technical systems evolution.^{1 2} In English, TRIZ is typically rendered as the theory of inventive problem solving.^{3 4}

The development of TRIZ, led by Soviet inventor and science-fiction author Genrich Altshuller and his colleagues, began in 1946. TRIZ developed from a foundation of research into hundreds of thousands of inventions in many fields to produce an approach which defines patterns in inventive solutions and the characteristics of the problems which these inventions have overcome.⁵ The research produced three findings:

• Problems and solutions are repeated across industries and sciences.
• Patterns of technical evolution are replicated in industries and sciences.
• The innovations have scientific effects outside the field in which they were developed.

TRIZ applies these findings to create and improve products, services, and systems.⁶

History

TRIZ was developed by the Soviet inventor and science-fiction writer Genrich Altshuller and his associates. Altshuller began developing TRIZ in 1946 while working in the inventions inspection department of the Caspian Sea flotilla of the Soviet Navy. His role involved evaluating invention proposals, refining and documenting them, and preparing patent applications. Through this work, Altshuller recognised that many technical problems require inventive solutions because improving one parameter often leads to the deterioration of another, a situation he termed a technical contradiction.

Altshuller's work on what later became TRIZ was interrupted in 1950 by his arrest and a 25-year sentence to the Vorkuta Gulag. The arrest was partially triggered by letters that he and Refael Shapiro sent to Stalin, government ministers, and newspapers, criticising Soviet policy decisions they considered erroneous.⁷ Altshuller and Shapiro were released during the Khrushchev Thaw following Stalin’s death in 1953,⁸ and subsequently returned to Baku.

TRIZ flowchart

The first academic paper on TRIZ, On the psychology of inventive creation, was published in 1956 in the journal Issues in Psychology.⁹

By observing skilled inventors at work, Altshuller identified recurring patterns of creative thinking, which he used to develop a set of problem-solving tools and techniques. These included Smart Little People and Thinking in Time and Scale (also known as the Screens of Talented Thought).¹⁰

In 1986, Altshuller shifted his focus from technical problem-solving to the development of individual creativity. He adapted TRIZ for use by children, and this version was tested in several schools.¹¹ Following the end of the Cold War, emigrants from the former Soviet Union contributed to the international dissemination of TRIZ.¹²

Basic principles

The TRIZ method 40 principles of invention rendered schematically, stacked vertically in four-column, within single A3 paper sheet

TRIZ claims that by studying an individual parameter that is causing a problem (e.g., the mass of an object needs to be reduced), and the other parameters with which it conflicts (e.g., the lower mass would require thinner material, which is more likely to undergo catastrophic failure), solutions can be created. This is actually the simplification of some fact. Namely, the structured contents of each of the cells within the contradiction matrix, (i.e. cells fulfilled with ordered principles and identified by their order numbers) - the principles of inventions have been chosen, due to realized necessary statistical extensive studies.

Contradiction matrix

The 40 principles of invention are a suite of ideas that purport to aid in solving hard technical problems. The principles are based on TRIZ. One tool which evolved as an extension of TRIZ is a contradiction matrix, a structured and systematic representation of basic engineering parameters of objects, or systems, such as mass, length and manufacturing tolerances.¹³ The ideal final result (IFR) is the ultimate solution of a problem when the desired result is achieved by itself.¹⁴ Studies led by Altshuller led to this approach: according to Altshuller, every technical problem that requires a solution can be categorized in terms of what he called its main technical contradiction.

How to Solve It (1945) is a small volume by mathematician George Pólya, describing methods of problem solving.¹

This book has remained in print continually since 1945.

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

1. First, you have to understand the problem.²
2. After understanding, make a plan.³
3. Carry out the plan.⁴
4. Look back on your work.⁵ How could it be better?

If this technique fails, Pólya advises:⁶ "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions,⁷ depending on the situation, such as:

• What are you asked to find or show?⁸
• Can you restate the problem in your own words?
• Can you think of a picture or a diagram that might help you understand the problem?
• Is there enough information to enable you to find a solution?
• Do you understand all the words used in stating the problem?
• Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems.³ The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

• Guess and check ⁹
• Make an orderly list ¹⁰
• Eliminate possibilities ¹¹
• Use symmetry ¹²
• Consider special cases ¹³
• Use direct reasoning
• Solve an equation ¹⁴

Also suggested:

• Look for a pattern ¹⁵
• Draw a picture ¹⁶
• Solve a simpler problem ¹⁷
• Use a model ¹⁸
• Work backward ¹⁹
• Use a formula ²⁰
• Be creative ²¹
• Applying these rules to devise a plan takes your own skill and judgement.²²

Pólya lays a big emphasis on the teachers' behavior. A teacher should support students with devising their own plan with a question method that goes from the most general questions to more particular questions, with the goal that the last step to having a plan is made by the student. He maintains that just showing students a plan, no matter how good it is, does not help them.

Third principle: Carry out the plan

This step is usually easier than devising the plan.²³ In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work, discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals. ³

Fourth principle: Review/extend

Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what did not, and with thinking about other problems where this could be useful.^{24 25} Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

Heuristics

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

Influence

• The book has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.

• Marvin Minsky said in his paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems." ²⁷

• Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.²⁸

• Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.

• How to Solve it by Computer is a computer science book by R. G. Dromey.²⁹ It was inspired by Pólya's work.