The Art of Guessing (And Then Checking)
Good problem solvers often make smart guesses and refine them.
There's a dirty secret in mathematics: guessing works.
Not wild guessing. Not "close your eyes and hope" guessing. But thoughtful, strategic, informed guessing—followed by checking, adjusting, and guessing again—is one of the most powerful problem-solving techniques there is.
Somewhere along the way, many students learned that guessing is cheating. Real mathematicians, they imagine, see through problems instantly, deploying elegant formulas like magic spells. The truth is messier. Real mathematicians guess constantly. They estimate, approximate, triangulate, and iterate. They ask "what if?" and then find out.
This post is a rehabilitation of the guess. Used well, it's not a crutch—it's a compass.
Safe Guessing for Kids
Children are natural guessers. Ask a five-year-old how many sweets are in a jar, and they'll shout a number without hesitation. The trick isn't to suppress this instinct—it's to channel it.
Guess and check is a legitimate strategy for early math problems. Consider this: "I'm thinking of a number. When I double it and add 3, I get 11. What's my number?"
A child might guess 5. Double it: 10. Add 3: 13. Too high! Try 4. Double it: 8. Add 3: 11. Perfect.
No algebra required. Just logic, feedback, and adjustment.
The key is teaching kids to use their wrong guesses. A wrong guess isn't failure—it's information. "Too high" tells you something. "Too low" tells you something. Each guess narrows the search space until the answer appears.
This builds two crucial skills:
- Number sense: developing intuition about how big or small answers should be
- Resilience: learning that wrong attempts are steps toward right answers, not evidence of inability
Games like "higher or lower" and "20 questions" are guess-and-check trainers in disguise. So is any puzzle where you try something, see what happens, and adjust.
Narrowing Down Options for Teens
As problems get harder, guessing gets smarter.
Consider solving equations. Yes, there are formal methods—but sometimes a quick estimate gets you there faster, especially for checking your work. If you're solving a quadratic and your answer is x = 47, but the problem is about the dimensions of a small garden, something's wrong. Your number sense should raise a flag before you even check the algebra.
Bracketing is a powerful refinement of guess-and-check. You make two guesses that you know are on opposite sides of the answer, then systematically narrow the gap.
Example: Find √50 to one decimal place.
- 7² = 49 (too low)
- 8² = 64 (too high)
- So √50 is between 7 and 8.
- 7.1² = 50.41 (slightly high)
- 7.0² = 49 (slightly low)
- So √50 ≈ 7.1
This is essentially how calculators find square roots—just faster and with more decimal places. The method is called bisection, and it works for any problem where you can tell whether your guess is too high or too low.
Trial and improvement shows up in science too. Chemists adjust concentrations, engineers tweak parameters, economists calibrate models. The formal methods get the spotlight, but behind the scenes, smart guessing does heavy lifting.
Approximations and Bounds for Adults
In the real world, exact answers are often impossible—or irrelevant. What matters is getting close enough, fast enough.
Fermi estimation is the art of making educated guesses about quantities you can't measure directly. How many piano tuners are in London? How much does the atmosphere weigh? How many golf balls fit in a school bus?
These aren't trick questions. They're exercises in decomposition. You break a hard question into easier parts, estimate each part, and combine. You might be off by a factor of two, but you won't be off by a factor of a thousand. That's often good enough.
Bounding takes this further. Instead of guessing a single number, you establish a range: "The answer is definitely more than X and definitely less than Y." This is how professionals handle uncertainty. A project manager might not know exactly how long a task will take, but they can often say "at least two weeks, at most six." That range is actionable.
Sanity checks are bounds in disguise. Before submitting a financial model, an analyst asks: "Does this output make sense? Is the profit margin plausible? Is the growth rate physically possible?" These aren't rigorous proofs—they're guesses informed by experience. And they catch errors that formal methods miss.
The progression from kid to professional isn't from guessing to not-guessing. It's from wild guesses to calibrated ones, from single guesses to strategic refinement, from "I don't know" to "I know it's between here and here."
Try These
Ready to train your estimation instincts? Here are three puzzles that reward smart guessing and systematic refinement.
Puzzle 1: The Mystery Number (Small Number Guess)
I'm thinking of a whole number between 1 and 50.
Here are three clues: - When you divide it by 4, the remainder is 3. - When you divide it by 5, the remainder is 1. - It's greater than 30.
What's my number?
Hint: Start by listing numbers that fit the first clue (3, 7, 11, 15...). Then narrow down using the second clue. Finally, apply the third.
Puzzle 2: The Balanced Equation (Equation/Unknown)
Find a positive whole number that satisfies this equation:
n³ − 5n = 220
You could try to factor or use the cubic formula—or you could make smart guesses.
Hint: Since n³ grows fast, test small values. If n = 5, then n³ = 125, and 125 − 25 = 100 (too low). If n = 7, then n³ = 343, and 343 − 35 = 308 (too high). You're getting close...
Puzzle 3: The Stadium Estimate (Real-World Approximation)
You're asked to estimate how many plastic cups are used at a major football stadium during a single match.
The stadium seats 60,000 people. Estimate the total cups used, stating your assumptions.
Then calculate reasonable lower and upper bounds for your estimate.
Hint: Think about what fraction of attendees buy drinks, how many drinks each, and whether any cups are reused. There's no single "right" answer—but some estimates are more defensible than others.
Final Thought
Guessing isn't the opposite of rigour. It's the beginning of it.
Every formal method was once a hunch. Every proof started as "I wonder if..." The mathematician's job isn't to avoid guessing—it's to guess well, check honestly, and refine relentlessly.
So the next time you face a problem that seems impenetrable, try something. Make a guess. See what happens. If it's wrong, you've learned something. If it's right, you've solved it.
Either way, you're doing mathematics.
What's your favourite estimation trick? Ever made a guess that turned out to be surprisingly accurate—or spectacularly wrong? Share your stories in the comments!