Brainteasers and market sizing

Let's set expectations first, because most students get this backwards. Brainteasers are a small minority of what you'll face in an IB interview. When one shows up, it's usually an interviewer with ten minutes to kill, not a make-or-break gate. Do not spend a weekend memorizing riddles off some list. Spend it on the three statements, valuation, and your story. Those are what actually get you the offer.

That said, there's one "brainteaser" you should be ready for every single time: market sizing. It's the estimation question, and it's the one bankers actually like, because it looks like the job. How big is the market for a product? How many of something fit in a space? You will not have a calculator, and there is no right answer sitting in the interviewer's notes. They're watching how you think.

What they're actually grading

Here's the part beginners miss. The number you land on almost doesn't matter. The interviewer wants to see a structured, out-loud reasoning process with assumptions they can follow and poke at. A candidate who says "about 8 million" with no path gets nothing. A candidate who builds to "roughly 8 million" through four clean, stated steps gets the check mark, even if the true figure is 5 million or 12 million.

The method: top-down, round numbers, out loud

There are two ways to attack these. Top-down starts from a big population and narrows with percentages. Bottom-up starts from a single unit and scales up. For most interview prompts, top-down is faster and cleaner. Pick one, say which you're using, and go.

The sequence that works every time:

1. Restate the question and define the scope. "How many people in the US, per year? Just retail, not commercial? Okay."
2. Start from a number you can anchor to. US population is about 330 million. Round it to 300 million so the arithmetic stays clean. Say that out loud: "I'll use 300 million to keep the math simple."
3. Segment down with round percentages. Cut the population by the filters that matter (age, ownership, usage frequency). Use 10%, 25%, 50%, halves and quarters. Never 37%.
4. Multiply through, then sanity-check the magnitude. Does the answer feel right? If you get "2 billion coffees a day in the US," you know you slipped a zero.

A worked example: golf balls in a classroom

The classic packing question. Let's do it the way you'd say it.

"I'll estimate the room's volume, then a golf ball's volume, then divide, then knock off some for wasted space between the balls."

Say the room is 10 meters by 10 meters by 3 meters. That's 300 cubic meters. A golf ball is roughly 4 cm across, so treat it as a small cube about 4 cm on a side to keep it simple (packing spheres wastes space anyway, so a cube approximation is fair). Convert: 4 cm is 0.04 m, and $0.04^3$ is a small number, so let me switch units instead to avoid the decimals.

Work in centimeters. The room is $1000 \times 1000 \times 300 = 3 \times 10^{8}$ cubic cm. Each ball-cube is $4 \times 4 \times 4 = 64$ cubic cm, call it 50 to round. Then:

$\frac{3 \times 10^{8}}{50} = 6 \times 10^{6}$

So about 6 million balls if the packing were perfect. Balls don't pack perfectly, so shave off roughly a third for the gaps:

$6{,}000{,}000 \times \tfrac{2}{3} \approx 4{,}000{,}000$

Call it 4 million golf balls. Notice what happened: every number got rounded, the arithmetic stayed whole, and the reasoning was audible the entire way. That's the deliverable.

A second example: sizing a product market

Say they ask, "How big is the annual US market for running shoes, in dollars?" Top-down:

| Step | Assumption | Running figure | |------|-----------|----------------| | US population | Round it | 300 million | | Share who run regularly | ~1 in 6, call it 15% | 45 million runners | | Pairs per runner per year | Round to 1 | 45 million pairs | | Price per pair | Round to $100 | $4.5 billion |

So roughly $4 to $5 billion a year. Then the sanity check out loud: "That feels reasonable for a specialty athletic category in a country this size." If they push, you flex an assumption: "If casual buyers push it to 1 in 4, the market's closer to $7 billion." That flexibility, the ability to move a lever and see the answer move, is what separates a strong candidate.

The rest of the brainteaser zoo

You may catch a logic or probability question at more quant-leaning groups: coin flips, dice, a bag of colored marbles, a simple expected-value bet. The method is the same as market sizing. Structure it aloud, state what you're assuming, and reason step by step. For anything with odds, name the total outcomes and the favorable ones and take the ratio. Nobody's asking you to be a competition mathematician. They want to see you stay organized under a little pressure.

And the raw mental math underneath all of this is trainable. Fast percentages, halving and doubling, rough multiplication. That fluency shows up everywhere in the job, not just in brainteasers, so it's worth reps. It's the same muscle you use in the paper LBO, where you grow EBITDA and pay down debt in your head, and in a back-of-envelope DCF sanity check.