Combinatorial Thinking

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What is Combinatorial Thinking?

Combinatorial Thinking is the practice of building a new option from elements you can already name, then discarding the joins that fail a real constraint.

To create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority. Invention is discernment, choice.

That is Henri Poincaré in Science and Method (1908), in the chapter on mathematical creation. Anyone can pair known pieces. The pairings run on without end, and most of them are empty. The skill is the short list you are willing to keep. The everyday picture is a fridge before dinner. Rice, eggs, and a lime are already there, so you write two or three plates you can actually cook, and then you taste. Most of those plates fail once they hit the pan, and the plate that gets eaten is the combination that survived. A gift bought from what is already in the house, or a Saturday plan built from people who are free, works the same way. Creative Thinking is the wider job: generate, judge, and improve until an idea is both original and useful. Combinatorial Thinking is one engine inside that job. The new idea arrives as a join, and the join still has to pass the tests. Analogical Thinking maps a relation from a familiar domain onto a new one. Combinatorial Thinking puts parts side by side and asks whether the assembly holds.

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Combinatorial Thinking in 3 Depths

• Beginner: A new option is usually old parts in a new join, and the cue is leftovers, a budget gift, or two tools you already know how to use.
• Practitioner: Write the inventory first—parts, constraints, and half-solutions—then cross two pairs, including one less obvious pair, and cut anything that breaks a real limit. Rewrite the survivor with whatever the first join revealed.
• Advanced: A wider search raises the cost of selection, because distant pairs often lift surprise and bruise usefulness on the first pass. The structural bet is a second draft built on the first join, under a rule that throws most combinations away.

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Origin

Combinatorial Thinking enters the public record as two habits that share one claim: choose among combinations, and treat a real change as a new combination of things that already exist. In 1908, Henri Poincaré published Science et méthode (Science and Method). The chapter on mathematical creation recounts his work on Fuchsian functions. For fifteen days he tried to prove that no such functions existed. One evening he drank black coffee, could not sleep, and felt ideas collide until pairs interlocked into what he called a stable combination. By morning he had a class of those functions, drawn from the hypergeometric series, and he needed only a few hours to write the results. His point was the filter that follows the collision. Preliminary conscious work sets some elements in motion, and the fertile combination is the rare one that reveals a kinship between facts that had looked unrelated. George Bruce Halsted’s English translation, in The Foundations of Science, carries the line that invention is discernment, choice. In 1911, Joseph Schumpeter published Theorie der wirtschaftlichen Entwicklung. The 1934 English edition, The Theory of Economic Development, defines development as the carrying out of new combinations. He lists five: a new good or a new quality of a good, a new method of production, a new market, a new source of supply, or a new organization of an industry. He calls that carrying-out enterprise, and he calls the people who do it entrepreneurs. In his terms, a small improvement of the old circular flow counts as growth, and a new combination breaks that flow. In 1945, Jacques Hadamard published The Psychology of Invention in the Mathematical Field and printed Albert Einstein’s reply to his questions. Einstein wrote that words played little role in his thought. The elements were signs and images he could combine. Combinatory play, he said, seems to be the essential feature in productive thought, before any logical construction in words that other people can follow. In 1962, Sarnoff Mednick gave the habit a laboratory definition in Psychological Review. He defined the creative process as forming associative elements into new combinations that meet a requirement or prove useful in some way. The more mutually remote the elements, the more creative the process or the solution. He turned the definition into the Remote Associates Test: three words, and a fourth word that links them. In 1990, and again in the 2004 edition of The Creative Mind, Margaret Boden split creativity into three roads. Combinational creativity is the making of unfamiliar combinations of familiar ideas: a poetic image, a collage, an analogy. She argues that the join needs a rich store of knowledge, and an intelligible path between the pieces. She compares a purely random process to shaking marbles in a bag, and she treats that shake as the less usual route. A combination is kept when an intelligible path runs between the ideas. Exploratory creativity moves inside a style of thought you already have, and transformational creativity changes the rules of that style. Boden holds that most psychological definitions cover only the first road. Dean Keith Simonton, in a 2021 paper in Frontiers in Psychology, disagrees with that boundary. He argues that discoveries Boden would call exploratory or transformational still combine representations. On his account, a creative combination is unlikely at the start, useful at the end, and surprising given what the maker already knew. He notes that a similar trio—new, useful, and nonobvious—is what the United States patent system asks of an invention. The disagreement is live. Use Boden when you need to see that some breakthroughs change the rules, and use Simonton when you need a filter for the combination itself.

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Key Points

Combinatorial Thinking pays when the pieces are already in the room and the missing move is a join you can test. It stalls when the list of joins has no exit.

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Applications

Use Combinatorial Thinking when a plan has to be assembled from parts, people, or rules that already exist. Bring a rule for throwing joins away, or the list will outgrow the afternoon.

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Case Study

The public measurement of Combinatorial Thinking is a 2015 study in Cognition by Joel Chan and Christian Schunn of the University of Pittsburgh. They followed ideas on OpenIDEO, a site where people propose answers to social and environmental challenges. The archive held 2,341 concepts and 4,557 inspirations, posted on 12 problems by 1,190 contributors. The problems included how to manage electronic waste, and how to raise the number of registered bone-marrow donors. Contributors cited the inspirations or earlier concepts they built on. That citation trail let the authors draw a genealogy: which idea used which source, and which source had itself been built on something further back. The bet they tested is Mednick’s, carried into a public archive: pair more distant sources, and the concept should come out more creative. Creative, in this study, meant shortlisted by a panel of the stakeholders who posted the problem and a team of OpenIDEO designers. The OpenIDEO team told the authors that the panel weighed novelty and usefulness together, and read usefulness as potential impact. On the 456 concepts with enough direct sources, distance left the shortlist where it was. A model that already included feedback and the other controls fitted no better once distance was added (p = 0.92). Six of the twelve challenges leaned positive, and six leaned negative. In that subsample, about 16 percent of concepts were shortlisted, and direct distance did not sort them. One step later, the pattern shifted. On 522 concepts, Chan and Schunn measured distance among indirect sources, the sources behind the sources a concept cited. A move of about one standard deviation toward more distant indirect sources, 0.10 on their scale, raised the fitted chance of a shortlist from about 13 percent to about 19 percent. The other predictors stayed at their averages. That model comparison was only marginal (p = 0.07), and the band of plausible values still included a very small drop. Every one of the twelve challenges leaned positive, and the chance of that sweep arising by luck was very small (p = 0.0005). An outside check kept the shortlist honest. Independent experts rated 318 concepts from five challenges, about four judges per concept, on novelty, impact, and feasibility. Impact predicted the shortlist, and novelty’s link was positive but too weak to trust on its own. The panel was a creativity score in the authors’ sense, new and useful, with the weight sitting on impact. Combinatorial Thinking explains the lag, and it explains why the shift is modest. A far join is a seed. The lift showed up in concepts built on earlier concepts that had already drawn on a distant combination. The study is correlational, so it does not show that asking people to combine distant ideas caused those shortlists. Chan and Schunn also limit the finding to how ideas develop, and they are silent on whether a shortlisted concept later worked in the street, the clinic, or the budget.

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Boundaries and Failure Modes

Combinatorial Thinking fails when nothing is thrown away, and when the first distant join is treated as the finished answer. Poincaré’s first boundary is the explosion. Known elements can be paired without end, and almost all of those pairs are empty. A workshop that fills a wall and keeps every sticky has left the method. The exit is a constraint you can fail: time, money, a regulation, or a person who must actually use the result. When the field is large enough to need a full grid, Morphological Analysis is the instrument that marks contradictions in pairs. First Principles Thinking makes a different prior cut, stripping a problem to what remains true and then rebuilding. Combinatorial Thinking starts from parts you have decided to keep. The second boundary is the first draft of a distant pair. Chan and Schunn report that the direct effect of far combinations, in their OpenIDEO data, had a mean of about zero. They also note that two earlier lab studies linked more distant combinations to lower-quality ideas: Mobley, Doares, and Mumford in 1992, and Baughman and Mumford in 1995. In the 1995 study, 155 undergraduates solved 12 category-combination problems. Naming features and mapping them across categories lifted originality, and it left quality where it was. How related the categories were did move quality. Distance can buy a stranger first draft and a worse fit, so the second pass is part of the method. The common misuse is the slogan join. Putting two fashionable words in one sentence skips the inventory, the job, and the rewrite. A second misuse treats school combinatorics, the count of ways to choose k items from n, as this habit. Counting arrangements is a mathematics topic, and selecting a useful join is a decision. A third misuse treats the habit as the whole of creativity. Boden’s exploratory and transformational roads change what a style of thought can even express, and Simonton replies that those roads still recombine representations. Keep both warnings: a combination can be the mechanism, and it can still be too shallow for the breakthrough you need.

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Common Misconceptions

Three confusions travel with the name. A mashup gets mistaken for the skill, distance gets mistaken for quality, and the word gets mistaken for either arithmetic or genius.

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Related Concepts

These neighboring pages cover the wider creative loop, the mapped analogy, the full grid, and the cut that follows a burst of options.

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One-Line Takeaway