Here's how to find the smallest integer when three consecutive integers sum to 72.

Consecutive numbers and a clean trick

Ever notice how some math questions feel like neat little puzzles hiding in plain sight? Here’s a tidy example that lands with surprising ease: three consecutive integers add up to 72. What’s the smallest one? The choices you’d see are something like 21, 22, 23, or 24. The answer is 23, and the path there is quick, reliable, and wonderfully explainable.

Let’s slow down a notch and savor the idea behind it. Consecutive integers are numbers that follow one another in order—like 23, 24, 25. They’re not random; they line up in a line, which means their sum has a simple relationship to the middle number. That relationship is the heart of the trick.

Set it up: three in a row

A natural way to stand in for three consecutive numbers is to write them as n, n+1, and n+2. Here, n is the smallest. This makes the algebra clean and transparent.

So, if you know their sum is 72, you write:

n + (n+1) + (n+2) = 72

Now you’re staring at a straightforward equation. Combine like terms:

3n + 3 = 72

From here, it’s a tiny step to isolation:

3n = 69

n = 23

That’s the moment the puzzle clicks—the smallest of the three is 23. The others, by the way, are 24 and 25, and together they sum to 72. Simple, tidy, and repeatable.

A quick sanity check you can actually perform in your head

Let me explain why this works in a quick, mental-friendly way. The middle number in any trio of consecutive integers is the one right between the smallest and largest. In our setup, that middle number is n+1. The sum of the three numbers is three times that middle value, because you’re adding the middle one once and then mirroring it on either side. Put another way: three consecutive integers have a total of 3 times the middle number.

With a sum of 72, the middle number must be 72 divided by 3, which is 24. If the middle is 24, the smallest is 23 and the largest is 25. It all fits together in a neat little triangle of numbers, like three chairs in a row at the same height.

A friendly generalization for future puzzles

Once you see the pattern, you can handle almost any similar situation without breaking a sweat. Here’s the quick version:

• For three consecutive integers, the sum S equals 3 times the middle number. So the middle is S/3.

• The smallest is middle minus 1, and the largest is middle plus 1.

Drawing it out helps, too. If you write the three numbers as x-1, x, x+1, their sum is (x-1) + x + (x+1) = 3x. If you know the sum, you just divide by 3 to find x, the middle, and from there you can pin down the ends. It’s like a tiny blueprint you can reuse for any trio of consecutive integers.

Why this kind of thinking matters beyond one problem

You might wonder, why bother with this little exercise? The beauty is in the mental model. Mathematics isn’t just about plugging numbers into a formula; it’s about recognizing patterns and translating a real-world situation into a tidy model. Consecutive integers appear in many contexts—groupings, sequences, even simple scheduling. Seeing the middle as a pivot point gives you a reliable anchor. It’s a small but powerful mindset that carries over to more complex topics, whether you’re graphing, solving inequalities, or untangling word problems that involve order and spacing.

A few more reflections that make the method feel intuitive

• The middle matters: When you’re dealing with an odd count of items that are evenly spaced, the middle item becomes a natural anchor for totals, averages, and symmetry. That’s exactly what we’re exploiting here.

• Pairing and balance: If the sum is a fixed number, the three numbers must balance around the middle. You can picture it as a seesaw—the middle sits right at the fulcrum, and the ends balance out on either side.

• Quick checks with a calculator, not a must: If you’re allowed to glance at a calculator, you’ll see the middle value pop out as S/3. If you’re calculating by hand, you can still get there with a couple of tidy steps.

A few little detours that still circle back

While we’re on the topic, you can apply a similar line of thinking to longer runs of consecutive numbers. Suppose there are five consecutive integers summing to a certain value. The same “middle is the pivot” idea extends, but the arithmetic shifts a bit. The sum would be five times the middle number, so the middle becomes the sum divided by five, and the smallest and largest stretch out by two steps on either side. The rhythm is the same; it just scales up with the count.

If you’re into exploring with tools, try sketching a quick diagram on paper or firing up Desmos to plot consecutive integers and their sums. It’s satisfying to watch the numbers line up as you tweak the target sum. Real-world math often rewards this kind of visual check, not because you need it to solve the problem, but because it reinforces your intuition.

A tiny set of practice prompts you can try (no pressure, just curiosity)

• Three consecutive integers sum to 99. What are they?

• Five consecutive integers sum to 275. What’s the middle number?

• If the sum of four consecutive integers is 52, what’s the smallest one?

• Hmmm, does the same approach work if the numbers aren’t strictly consecutive but are in an increasing sequence with a fixed gap? See how it changes the latitude of the problem.

When the answer feels a little obvious, that’s a good sign you’ve internalized the pattern

There’s something satisfying about a clean solution that doesn’t require hunting through a maze of steps. The moment you realize the middle number is the key, you’ve learned a skill that translates well beyond this one question. It’s not about memorizing a trick; it’s about recognizing a structure and letting it point you to the solution.

A note on the language of math—and why words matter

In problems like this, the words do a lot of the heavy lifting. Consecutive numbers, sum, smallest, middle—these phrases aren’t just labels. They guide your reasoning. When you see “consecutive,” you know the numbers sit in a row with a steady march of +1. When you see “sum,” you know you’re stacking those numbers together. Paying attention to these cues sharpens your ability to translate a written puzzle into a clean set of equations.

A closing thought

Math often hides its best lessons in the simplest setups. The trio n, n+1, n+2 is one of those friendly anchors that makes the whole world feel legible. The trick is to treat the middle as a compass; from there, the smallest becomes a natural neighbor, and the total invites you to verify with a quick check. The next time you stumble on a problem that mentions consecutive integers, you’ll hear that quiet voice in your head—middle first, then the ends—and you’ll move with confidence.

If this kind of reasoning resonates, you’re not alone. It’s the practical kind of math that shows up in everyday problem solving—no dramatic leaps, just clear steps and a moment of tidy math harmony. And that’s a pretty satisfying place to land, don’t you think?