**Title**: How to Solve It: A New Aspect of Mathematical Method

**Authors**: George Pólya

**Status**: Book, published 1945. The first chapters are available online; the whole book is available from Princeton University Press [closed access]

There are several ways, in my experience, to get completely stuck in a research problem. I’ve been industriously stuck, spending hours optimizing code that does not accomplish the task I wrote it to accomplish. I’ve been idly stuck, so daunted by a problem that I put it off for days or weeks in favor of less important things. I’ve been stuck at the beginning, when I don’t understand the problem well enough to know where to start, or even what questions to ask to understand better. I’ve been stuck at the end, when perfectionism strikes and I so fear putting out a mediocre result that I delay putting out anything at all.

In all of these moments, especially the idle or ignorant ones when I’m somewhat ashamed to seek help getting un-stuck, I’ve wished that research came with an instruction manual: what do I do in this situation? The joy of research, ironically, is that there are no instructions for the particular problems any of us work on—our wonderful job is to figure them out. But there ought to be instructions for how to do a research problem *in* *general*.

Enter the subject of today’s astrobite: *How to Solve It*, which might be better subtitled “What to Do When You’re Stuck.” I’ve had it kicking around on my bookshelf for about ten years, but I only picked it up recently, and wow, I wish I had read it earlier. It’s written by a mathematician, George Pólya, as a guide to thinking about math problems, and he intended it primarily for students and teachers of mathematics. (Just check out how many math departments/professors discuss it on their websites!)

But it’s so much more generally useful than that! I recommend it to anyone who’s trying to figure anything out, which is to say, I recommend it to everybody.

**The Instructions**

Here they are! Here’s how to solve a problem:

*Understand the problem*. Break it down into its elements–given data, unknown, conditions–and figure out the relationships between those elements.*Devise a plan to solve it*. Think back to other problems you’ve solved before and decide whether you could apply similar reasoning here.*Carry out the plan*, carefully checking each step along the way.*Examine the solution*. Is it correct (provably, intuitively, both)? Could you have reached it another way, by another method, perhaps more elegantly? Does it illuminate other problems you’re working on? Could you devise new problems that this solution would, in turn, help you solve?

These steps are so basic as to seem trite (even if we know that carrying out the steps is not necessarily easy). But not many of us work through our research problems so systematically and concretely, and it’s refreshing to realize that we could! When we do get stuck, we’re not just floating, adrift, in the ether–we’re lost somewhere in that four-step plan, and to get un-stuck, all we have to do is figure out where we left the path and return to it.

Pólya first breaks down the four steps into many smaller pieces (summarized in a helpful chart on page xvi). “Understanding the problem,” for example, becomes a series of questions: “What is the unknown? What are the data? What is the condition? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down?” He offers examples, in the form of a dialogues between a fictional student and teacher about real mathematical problems; when the student flounders, the teacher leads her back to the path. We, the readers, imagine ourselves alternately as students and teachers: if we were stuck at this point in the problem, what push would we need to move forward? If someone offered us this hint, how would we use it to make progress?

Pólya devotes the second part of the book to a “short dictionary of heuristic,” by which he means, “the methods and rules of discovery and invention.” (This is slightly different from

“heuristic reasoning”, which means a problem-solving approach that helps us make short-term progress, but may not be optimal to find the final answer.) This dictionary is a long list of techniques that problem-solvers can use to get themselves un-stuck and make concrete headway. Its entries range from straightforward definitions to delightful meditations on the psychology of problem-solving; the entry for “Determination, hope, success,” for example, tells us:

“To solve a serious scientific problem, willpower is needed that can outlast years of toil and bitter disappointments…[A scientist] should have some hope to start with, and some success to go on. In scientific work, it is necessary to apportion wisely determination to outlook. You do not take up a problem, unless it has some interest; you settle down to work seriously if the problem seems instructive; you throw in your whole personality if there is great promise. If your purpose is set, you stick to it, but you do not make it unnecessarily difficult for yourself. You do not despise little successes, on the contrary, you seek them:

If you cannot solve the proposed problem try to solve first some related problem.” [Emphasis in original.]

I’ve found this advice to be immensely helpful. To be a researcher is to get stuck, but not to stay so. We can solve these problems of ours!