Today we’ll look at a question from a student who was troubled by the amount of guessing needed to solve certain problems. This leads to an interesting survey of different kinds of guessing, and ways to develop that skill.
When do we need to guess in math?
The question is from 2017:
More Methodical Than Guessing It seems like guessing is involved in many operations that we do in maths. For example, in finding square roots, we make guesses as to what they could be. In factorisation, we think of numbers that can multiply to be another, in a guessing kind of way. Another example is in solving the general cubic equation. One important step is to simply rearrange an algebraic expression to another, more complex one -- just because it works. (How do we know we should re-arrange it that way and not in any other way?) While I only have a high school level of mathematical experience, I love learning about more advanced concepts. But all this guessing is difficult for me to master, especially in complex problems. In middle school, I was very good in geometry, but bad in algebra where, for example, we factorised expressions in a guessing kind of way. All of which leads me to two questions. Are there better options for these mathematical operations? Can we classify mathematical operations into the categories of "guess it first" on the one hand; and on the other hand, "Do steps 1, 2, 3, ... and then you will get the solution directly"?
Interesting question! Are there “guess operations” and “systematic operations”? I answered:
Hi, Egan. There are a number of different things that can be said about this. My first thought is that we need to distinguish between an "operation" (in the sense of what we want to accomplish; for example, find the square root) and an "algorithm" or process or method to perform the operation or otherwise reach the goal (do this, then that, ...). There are usually multiple ways to accomplish a task. Some may involve guessing, while others may not; some may be more efficient than others, while others may be easier to explain. Sometimes guessing turns out to be the most efficient method, even though non-guessing methods are available. Guessing is not inherent to the operation, itself -- just a feature of a particular method you may choose to use. So we can't sort "operations" according to whether they involve guessing.
So we may choose to solve a problem by guessing even though we don’t have to; on the other hand, some things can’t (reasonably) be solved by guessing. We’ll see some examples later.
Second, "guessing" can take several very different forms. You are familiar with algorithms of the "guess and check" variety, where you make a guess, see whether it works, and then make a new guess based on the outcome. Sometimes the "guess" will almost always work, and the "check" is mostly a matter of continuing on with your work. (I am thinking of long division; longhand square roots are similar, once you have the necessary experience.)
In these examples, we don’t merely make a wild guess, but rather estimate intelligently, and the check is inherent in the work. (If we get too large a product, we know exactly what to do to fix it.)
Other processes, such as some factoring techniques, can be done by a quick "guess" (insight) that turns out to be correct, or by systematically trying all possible answers (e.g., all pairs of factors of a given number), with the conclusion being either the answer or the knowledge that there is no answer. (Pure "guess and check" can never determine whether you have just missed an answer.) Such a procedure may be done blindly (just plodding through the list) or intelligently (using clues to know what to skip).
I’ll be demonstrating this.
Still other "guess-like" processes are not really guesses at all, but successive approximations to the correct answer, which may not be possible to find exactly. (Here I am thinking of the "divide and average" method for finding a square root.)
Here we are actually following a routine procedure, starting with a reasonable “guess”, that promises to improve the “guess” at every step.
Examples of guessing … or not
Let's consider some examples. I've mentioned two different methods for finding a square root, and there are a couple others. This illustrates my first point. One method ("longhand") involves guessing for each digit (which should rarely be far off). Another ("divide and average") starts with a "guess," but goes on from there to repeatedly improve the guess, with no more guessing needed: Square/cube roots without a calculator http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html
I showed these techniques and others in my post Evaluating Square Roots by Hand.
Long division as normally done involves guessing (I'd rather call it estimating) quotient digits. But you could do it with no guessing at all: Just make a list of multiples of the divisor, and you can look up the closest multiple to the dividend. (We guess to save time, not because it is needed!) Or, you could do what a computer or calculator does: convert to binary, divide in binary (which requires no guessing), and convert back. The method with guessing is easier and faster for humans, but not the only way. See this page: Long Division, Egyptian Division, Guessing http://mathforum.org/library/drmath/view/58858.html
Egyptian division is equivalent to dividing in binary; there, each quotient digit is either 0 or 1, and it is always obvious which to use.
And when I first taught my oldest son to do long division, he made a table of multiples of the divisor, as I suggested here, so that he never had to guess.
Factoring can be done using methods that require various amounts of guessing. Looking in our archives for examples that discuss guessing, I found one that shows a traditional trial and error method with lots of guessing: Factoring Trinomials: 9x^2 - 42x + 49 http://mathforum.org/library/drmath/view/61570.html There, Doctor Ian refers to another page in which he shows how to factor by completing the square, or using the quadratic formula, each of which entirely bypasses guessing (and, unlike the other method, even works when the factors do not involve integers): Factoring Quadratics Without Guessing http://mathforum.org/library/drmath/view/60700.html There is also a commonly-taught method of "ac-grouping," which uses a simpler kind of guessing: Factorization by Decomposition and the Distributive http://mathforum.org/library/drmath/view/77809.html But when I saw the title of the first of these, I immediately saw that 9x^2 - 42x + 49 could be factored using only a single "guess": recognizing that the first and last terms are both squares, we can guess that it MIGHT be a perfect square, of the form (a - b)^2 = a^2 - 2ab + b^2. In that case, the factorization has to be (3x - 7)^2. Then all I have to do is check that by multiplication, and I can determine that it's correct. (If the guess had been wrong, I'd have switched to one of the other methods.)
This last example involves “guessing that” it is a perfect square, not “guessing a number” — a very different kind of guess.
Do you see my point here? There are MANY ways to factor a quadratic, which can range from no guesses, to one guess, to potentially hundreds of guesses. The optimal strategy is to first "guess" what might be the most efficient method for a particular problem, and then do it. The methods with fewer guesses involve various levels of complexity and risk, as a trade-off to the challenge of guessing.
My standard advice for solving a quadratic equation is to stare at it for ten seconds and decide whether you want to go to the trouble of the usual method (that is, whether guessing pairs of numbers will be efficient), and if not, “Use the formula, Luke!”
Solving cubic equations
Your other example was the cubic equation. I suppose you have read something like this: Cubic and quartic equations http://mathforum.org/dr.math/faq/faq.cubic.equations.html There is actually a formula for this that requires no guessing. Far beyond the scope of high-school algebra, it takes up half a page, at least. I have never tried to master it. Alternatively, you can follow the procedure discussed on our page, which does not involve guessing, but is an orderly process.
On the other hand, we often solve cubic equations by the Rational Root Theorem, which involves listing possible rational numbers that could work; this is yet another sort of guessing, where we first “guess” (hope) that there is a rational root, then “guess” each of a restricted number of possibilities, and then, if that didn’t work, use an approximation method.
Guessing a method
Still referring to the cubic,
You also allude to the role of guessing in not just the process, itself, but rather in how that process was devised. You may be right that this involved some guessing (somewhat like my perfect square guess above, where I tried something that seemed useful, and it worked). That's a very different kind of guessing, which arises in any non-routine problem-solving. I compare this sort of thinking to finding your way to a goal through a forest you have never seen before. You have to develop an intuition about what might work, based on experience using the mathematical "tools" you have learned. There is often no way around that; that's why we call such a problem "non-routine." And that kind of math may be the most useful in real life: when you learn how to solve problems you have never seen before, you are ready for the real world, where nothing is ever quite what you have been taught! Willingness to try and fail and try something else -- to persevere -- is the most important thing you can learn.
Here, with experience we develop a sense of what may work to solve a problem, perhaps by trying to rework a problem into a form to which we know we can apply familiar methods. In other words, we just try to make the problem look more familiar.
Learning to guess well
As far as learning to do the guessing part, all I can say (apart from the fact that you can often work around it) is that as you gain experience in any field, you develop a feel for how things work; and that sense helps you make the right guesses much of the time. How that is done depends very much on the particular field you are trying to master. Perhaps we could discuss that in detail for a particular problem, or kind of problem. Show us one that interests you and how you currently try to solve it, and we can suggest ways to do it better.
Egan wrote back with an encouraging compliment:
Thank you very much, sir, for your detailed answer! I have been using this site for about eight years, and I always recommend it to my friends (especially those with more of a mathematical background), as this is so much better than Quora or Mathematics Stack Exchange. :)
We are quite different from those sites, but I like to think we offer something better on occasion.