Our whole life is solving puzzles.
I found myself this year occupying an office with an extra desk that was disturbingly tidy for the first half of the fall semester, when I happened upon a jigsaw puzzle featuring an image of Edward Gorey’s “An Exhibition” and decided it would be the perfect way to add some clutter to my office. And so, during the months of November and December, Gorey’s drawing emerged on my desk as the 1000 pieces of the puzzle were slowly but steadily assembled during midday breaks in my mathematical work.
I enjoyed jigsaw puzzles as a child, but this was my first time completing a puzzle in over a decade, and my first time seriously working on a puzzle since starting to do research in mathematics. And as I progressed through its solution, I was struck by how the experience of solving it mirrored the experience of doing math research. I’ll be the first to admit that the observation that doing math is like solving puzzles is not particularly deep or original, but what increasingly stood out to me was how unexpectedly rich and multifaceted the similarities between the acts of research and puzzle-solving turned out to be.
So today, we are going to explore some of these similarities, with the aim of perhaps helping to begin to illuminate what it actually feels like, at least at an abstract level, to do math research. For although a relatively small percentage of the population has actively engaged in mathematical research, the act of solving jigsaw puzzles is a more universal experience, and I expect many people will be familiar with its joys and frustrations.
Solving a jigsaw puzzle and completing a mathematical research project are both processes, with distinct phases, and these phases will furnish a useful framework for our comparison. With this in mind, let us begin at the beginning.
1. Assembling the edge
When starting a puzzle, there are certain easily identifiable pieces about which one knows some additional information. These are, of course, the edge pieces, which are straight on one (or, in rarer but even more informative cases, two) of their sides. And so, the most common and most reliable strategy for beginning a jigsaw puzzle is first to isolate the edge pieces and then to fit all of them together to construct the “frame” of the puzzle. The reasons for this being an effective strategy are many, but chief among them are surely:
- It is easy to identify edge pieces.
- It is relatively easy to fit them together, since for two edge pieces there are at most two possible ways in which they can be joined, whereas for two interior pieces there are up to sixteen possible ways in which they can be joined.
- Once constructed, the edge of the puzzle will help give structure to the more chaotic interior, providing scaffolding off of which one can steadily build into the center of the puzzle.
The counterpart to assembling the edge pieces in mathematical research is, I claim, doing background reading, getting yourself up to speed with regards to the current state of knowledge about the problem you are trying to solve. While it is certainly conceivable that one could solve a problem with little or no knowledge of previous work, in practice it is immensely helpful and time-saving to be familiar with what other researchers have already discovered about the problem, with techniques that have been developed to tackle this or similar problems, and with ways in which these techniques have succeeded or failed in the past. In somewhat strained analogy with the above, let me isolate three reasons why this is effective:
- Given modern bibliographic and communication tools, it is relatively easy to identify relevant books or papers in the existing literature (though it is not always as easy as one might hope).
- Reading the background literature should be an “easier” or at least more consistently productive process than jumping immediately in to the research, in large part because it will not require the novel insights that the eventual solution to your problem will involve. (This is, in part, assuming that the background literature is available to you and reasonably well-written, neither of which is an entirely safe assumption.)
- After becoming familiar with previous work, you will have a solid foundation that should provide some firm jumping-off points to begin your research.
2. Experimental tinkering
Once you have completed the edge of your puzzle, it is probably wise to take a break to prepare yourself for the more daunting prospect of working your way into its interior. At the beginning, this process is likely be slow, frustrating, overwhelming, and bewildering, made especially difficult by the fact that, although the puzzle pieces fit together to form a coherent image, individually they are abstract blobs of color and texture, with little indication as to the form they will eventually be a part of. This phenomenon is particularly well depicted by the labyrinthine opening sentence (containing not one nor two but three colons!) of Georges Perec’s (yes, that Georges Perec) masterpiece novel, Life: A User’s Manual, in which jigsaw puzzles feature prominently:
To begin with, the art of jigsaw puzzles seems of little substance, easily exhausted, wholly dealt with by a basic introduction to Gestalt: the perceived object — we may be dealing with a perceptual act, the acquisition of a skill, a physiological system, or, as in the present case, a wooden jigsaw puzzle — is not a sum of elements to be distinguished from each other and analysed discretely, but a pattern, that is to say a form, a structure: the element’s existence does not precede the existence of the whole, it comes neither before nor after it, for the parts do not determine the pattern, but the pattern determines the parts: knowledge of the pattern and of its laws, of the set and its structure, could not possibly be derived from discrete knowledge of the elements that compose it. That means that you can look at a piece of a puzzle for three whole days, you can believe that you know all there is to know about its colouring and shape, and be no further on than when you started.
Georges Perec, Life: A User’s Manual
In practice, this initial phase of exploration into the center of the puzzle consists of repeatedly finding two pieces that have similar colors or textures and trying to fit them together in any conceivable configuration. Most of the time, this will fail. On the rare occasion when it succeeds, the two pieces will melt into one another, becoming a single larger piece, although, at least initially, this larger piece will still remain an abstract blob of color and texture, giving little indication of its eventual identity.
A useful strategy in this early stage of puzzle assembly is to try to directly attach interior pieces to pieces in the completed edge. This allows you to use to your advantage the facts that the completed edge already provides some nontrivial information about the structure of the interior of the puzzle, at least providing partitions into discrete regions of similar colors and textures, and that the edge pieces have fewer degrees of freedom than unconnected interior pieces, so progress will be quicker at the edge. Of course, once an interior piece is attached to the edge, it becomes part of the edge itself, to be further built upon.
The initial stages of mathematical research are equally slow, frustrating, overwhelming, and bewildering. As above, it is prudent to build off of your background knowledge, taking advantage of the little bit of structure it already provides. Make small tweaks to existing techniques to see if they lead anywhere interesting. Try to connect different bits of existing information to understand how they form complementary parts of a coherent whole. Again, most of these initial attempts at creating new knowledge fail. But every now and then something works, and a tiny breakthrough is achieved. And while these initial achievements don’t yet give you an idea of the final shape of the eventual solution, they gradually add both to existing store of background knowledge and to your own personal store of experience.
3. Making Connections
As the small groups of connected pieces constructed in Phase 2 begin to grow, something curious happens. The abstract patterns of the individual pieces begin coalescing into recognizable forms. A formerly mystifying red ribbon reveals itself to be a bicycle frame. A black and orange pattern becomes a person’s clothing. A black smudge, when placed in context, is instantly recognizable as said person’s shoe. And then, sometimes, something truly satisfying happens: two of these groups become rotated in just the right way and moved to just the right spots within the frame of the puzzle, and, in a moment of great serendipity, they are found to fit precisely together, in a way that was entirely unforeseen but in hindsight appears obvious and inevitable. This coupling creates a larger, even more structured group that gives the solver information not just about its constituents but about the pieces that will eventually fill in the remaining gaps. In this way, the landscape of the puzzle is gradually revealed so that, even if not all of the pieces are in place, the solver knows in a broad sense what the final picture will be.
Mathematical research progresses in much the same way. The disparate facts built up in Phase 2 begin to fit together to form a coherent narrative. Bits of knowledge from two seemingly distant domain reveal themselves to be two sides of the same coin, shedding light not only on these specific bits but on a whole constellation of ideas. The correct questions to ask begin to emerge and, later on, you see outlines of their solutions.
It is in this third phase of puzzle solving and mathematical research that the real magic happens, where new discoveries are made, where fleeting moments of pure joy are unearthed. And it is these moments that make the admittedly rather toilsome other phases worth it.
4. Filling in the Gaps
When all of the major features in the puzzle have been fixed in place, when the solver understands the final image, some gaps will remain to be filled: stray undiscovered pieces in the midst of otherwise completed forms, or small expanses of blue sky or green grass. This phase again becomes somewhat tedious and involves none of the discovery of Phase Three, but its finality lends it a satisfaction not found in any of the early solving phases. There is also pleasure in the increasing ease with which you progress through this phase. As the stocks of remaining pieces and open spots simultaneously dwindle, the pieces are slotted in with accelerating rapidity until, suddenly, the image is complete.
In mathematics, too, there is still work to be done after Phase Three. Technical details must be worked out. Proofs must be checked. The paper must be written. And though a mathematical project does not have as obvious an ending as a jigsaw puzzle, there must come a point when you say, “It is done,” and send it to a journal for publication.
There are two additional connections that I want to draw between puzzle solving and mathematical research that don’t quite fit within the above “phases” framework. The first involves the gradual development of intuition and expertise.
The early phases of solving a puzzle consist of a great deal of trial and error. Any two pieces with similar patterns are tried together, in every conceivable configuration, in a slow and rather tedious process. As the puzzle-solving progresses, though, you begin to be able quickly to see, through increased familiarity with subtleties in gradation or piece shape or other more ineffable factors, which pieces are more likely to fit together and which you needn’t bother trying, cutting down greatly on the amount of trial and error needed and considerably speeding up the process. You also begin to see more quickly through the abstraction of the individual pieces, identifying them for what they are even before connecting them with their neighbors.
I was able to witness the effects of this intuition recently. About a year ago, I visited some friends and spent about 15 minutes working on a jigsaw puzzle with their 5-year-old son. At the time, the puzzle was well into what would be labeled “Phase 2” in our framework. Over the course of these 15 minutes, I think I succeeded in connecting precisely one piece after a large amount of trial and error. The five-year-old, meanwhile, seemed to be making much fewer connection attempts but, in the same amount of time, connected about ten pieces. This disparity is not because he is better at puzzles or has faster visual cognition or has better geometric visualization than me, though all of those things are possible. Rather, I think it is because he had been working on this puzzle for enough time that he developed a feel for which types of pieces are likely to work in certain places in the puzzle.
Intuition works similarly in mathematics and continues to be developed across different projects and throughout many years. Every young researcher has had the experience of telling a more senior figure in the field about a recent result they have proven only to have the senior mathematician immediately give a simpler proof of a more general version of the result or make sweeping, initially surprising connections between the new result and existing mathematical facts. This can be initially disconcerting to the young mathematician, since this result that they have spent months or years of hard work on has been absorbed and improved upon by the senior mathematician in a matter of minutes, but in fact it should not take anything away from their accomplishment. After all, they proved this result, while the senior mathematician did not. What the senior mathematician brings is an extensive knowledge of the landscape in which the result sits that lets them quickly assimilate it in their mental space. And through experiences like this, the younger mathematician begins to build up their intuition as well.
The second and final additional connection I want to draw concerns what happens when the project is over.
After the last puzzle piece is placed and you step away to admire your work, the lines between the puzzle pieces begin to mentally dissolve, and the puzzle simply becomes the image printed on its face. All of the difficult toil of the previous hours and days gets washed away. Visitors who look at the completed puzzle do not see all of the ways in which the pieces didn’t fit together. The final solution looks obvious and inevitable. Of course that is how one should place the pieces; how else would you do it?
A completed mathematical project similarly takes on a sheen of inevitability. In your published account of your work, all of the dead ends and initial mistakes are excised from history, and the path from the beginning to the end is laid out in a nice orderly manner rather than the winding maze in which it was initially traversed. After you’ve had a bit of time to get perspective on your work (sometimes only a matter of hours), it begins to seem trivial and obvious. Of course this is how the problem was solved; how else would you do it?
This phenomenon (and some others we have discussed) is well-described by passages in Life: A User’s Manual (for puzzles) and Birth of a Theorem: A Mathematical Adventure, by Cedric Villani (for mathematics). Let’s look first at the passage on puzzles, which in the book comes shortly after the passage excerpted above:
The pieces are readable, take on a sense, only when assembled; in isolation, a puzzle piece means nothing — just an impossible question, an opaque challenge. But as soon as you have succeeded, after minutes of trial and error, or after a prodigious half-second flash of inspiration, in fitting it into one of its neighbours, the piece disappears, ceases to exist as a piece. The intense difficulty preceding this link-up — which the English word puzzle indicates so well — not only loses its raison d’être, it seems never to have had any reason, so obvious does the solution appear. The two pieces so miraculously conjoined are henceforth one, which in its turn will be a source of error, hesitation, dismay, and expectation.
Life: A User’s Manual
Next, Villani, a distinguished French mathematician, thinks about the process of mathematical research while walking through a dark pedestrian tunnel at night:
This gloomy tunnel is a little like the one you pass through when you begin work on a new mathematical problem […] Total obscurity. Bilbo in Gollum’s tunnel.
A mathematician’s first steps into unknown territory constitute the first phase of a familiar cycle.
After the darkness comes a faint, faint glimmer of light, just enough to make you think that something is there, almost within reach, waiting to be discovered…. Then, after the faint, faint glimmer, if all goes well, you unravel the thread — and suddenly it’s broad daylight! You’re full of confidence, you want to tell anyone who will listen about what you’ve found.
And then, after day has broken, after the sun has climbed high into the sky, a phase of depression inevitably follows. You lose all faith in the importance of what you’ve achieved. Any idiot could have done what you’ve done, go find yourself a more worthwhile problem and make something of your life. Thus the cycle of mathematical research…
Cedric Villani, Birth of a Theorem: A Mathematical Adventure
I finally want to end with one key difference between puzzle-solving and mathematical research. Jigsaw puzzles are bounded, existing entirely within their frame. When you are done with a puzzle, you are done, and the next puzzle you complete will have nothing to do with it except its form. Each mathematical project, on the other hand, is part of the same landscape. When you zoom out and adjust your perspective, it becomes a puzzle piece of its own, to be connected in surprising ways to other projects, forming ever larger and more detailed pieces that combine to form the infinitely rewarding and exquisitely detailed background image of the puzzle of mathematics.
Cover Image: Illustration from The Doubtful Guest by Edward Gorey