Venture Capital puzzles, solved step by step

Puzzles: 100
Traced to a firm: 7
Topics: 12
Hard: 30

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042A mental maths set from a first-round interview: work out 49 x 51, 998 x 1,002 and 12.5% of Rs 3,680 crore, each inside ten seconds.Mental maths and speed testsWarm upGeneral AtlanticNew York · 2026
Try it first
What is 998 x 1,002?
Show the worked solution
2,499, then 9,99,996, then Rs 460 crore. The first two use the same shortcut: numbers equally spaced either side of a round number multiply to its square minus the gap squared, so 49 x 51 is 2,500 minus 1, and 998 x 1,002 is 10,00,000 minus 4. The third is a fraction in disguise: 12.5% is one eighth, so halve Rs 3,680 crore three times, to 1,840, 920 and 460.
Why does 49 x 51 come out one short of 2,500?
Picture a square garden 50 metres a side. Take a one-metre strip off the bottom, 50 square metres, and lay it along the right-hand side. It only fits for 49 metres, so one square metre is left over. A rectangle 49 by 51 is a 50 by 50 square with one corner square missing, so the product is 2,500 minus 1, or 2,499. The same picture works for any pair spread evenly around a round number: the product is the middle number squared minus the gap squared.
The relationship
$(a - b)(a + b) = a^2 - b^2 \qquad 49 \times 51 = 50^2 - 1^2 = 2{,}499 \qquad 998 \times 1{,}002 = 1{,}000^2 - 2^2 = 9{,}99{,}996$
a the round number in the middle, 50 or 1,000
b how far each factor sits from it, 1 or 2
a squared minus b squared the product, always slightly below the square
What it says in wordsTwo numbers spread evenly either side of a round number multiply to that number squared, less the gap squared.
49 x 51 is a 50 by 50 square with one unit square missing, so it equals 2,499, and 12.5% of Rs 3,680 crore is one of eight equal blocks, Rs 460 crore, found by halving three times.
How do you take 12.5% of anything in a few seconds?
Recognise the fraction first. 12.5% is one eighth, and dividing by eight is the same as halving three times, which is easier to do aloud than any long division. Rs 3,680 crore halves to 1,840, then 920, then 460. Keep a short list of these anchors ready: 12.5% is an eighth, 37.5% three eighths, 16.7% a sixth, 6.25% a sixteenth. Interviewers who give percentage questions usually pick one of them, because the speed of recognising the fraction is what they are testing.
What is the interviewer actually checking?
Speed sets are less about arithmetic than about whether you look for structure before you calculate. Saying the shortcut out loud, square minus one, or halve three times, shows a method that will also work on the next question. Two habits help. Check the size of your answer against an anchor: 49 x 51 must sit just under 2,500. And say the number back with its units, Rs 460 crore, not just 460. The limit of these tricks is that they only work on numbers chosen to suit them; for awkward numbers, round and say so.
Where candidates lose it
The common loss on the products is grinding through long multiplication and running out of time, or writing 10,00,004 because the sign of the correction is guessed. The product of numbers either side of a round number is always below its square.
On the percentage, candidates multiply 3,680 by 0.125 digit by digit and fumble. Name the fraction, one eighth, and halve three times; the clock is part of the question.
What the interviewer asks next
Work out 97 x 103 and 4.95 x 5.05 the same way.
What is 37.5% of Rs 2,400 crore?
Estimate 1,999 squared in your head.
Asked at General Atlantic, Generalist, New York, 2026 (Wall Street Oasis): The first round was behavioral with mental math at the end.
065An online aptitude test has 50 questions in 12 minutes, five options each, and no penalty for wrong answers. You can answer about 30 questions carefully in the time, at 85% accuracy. Should you guess the other 20, and what is your expected score?Mental maths and speed testsWarm upVista Equity PartnersAustin · 2022
Try it first
What is your expected score if you guess the last 20?
Show the worked solution
Yes, guess every remaining question; the expected score rises from 25.5 to 29.5. Your 30 careful answers at 85% are worth 25.5 points. A blind guess on a five-option question is right one time in five, and a wrong answer costs nothing, so 20 guesses add 20 x 0.2, or 4 expected points. A blank is a sure zero. The only real decision is leaving yourself the last 30 seconds to fill them in.
Why is a blind guess worth anything at all?
A free raffle ticket is worth taking even if the odds are poor, because it costs nothing. When wrong answers carry no penalty, every guess has a positive expected value, one point times the chance of being right, and a blank has an expected value of exactly zero. With five options that chance is 1 in 5, so each guess is worth 0.2 points and twenty are worth 4. On a 50-question test, 4 points is often the gap between two scoring bands.
Thirty careful answers at 85% give 25.5 expected points, and filling the remaining 20 with blind guesses adds 20 x 0.2 = 4 more, lifting the expected score to 29.5 at no cost because wrong answers are not penalised.
The relationship
$E[\text{score}] = 30 \times 0.85 + 20 \times \tfrac{1}{5} = 25.5 + 4 = 29.5$
30 x 0.85 careful answers times your accuracy
20 x 1/5 blind guesses times the chance a random pick is right
What it says in wordsAdd what the careful answers are worth to what the guesses are worth; blanks add nothing.
What changes if wrong answers are penalised, and how do you manage the clock?
Suppose a wrong answer cost a quarter of a point. A blind guess would then be worth 0.2 minus 0.8 x 0.25, which is exactly zero. The rule is to guess whenever the chance of being right times the reward beats the chance of being wrong times the penalty, and eliminating even one option tips a penalised guess back into positive value. On the clock: 12 minutes for 50 questions is 14.4 seconds each. Skip any question that will take more than about 30 seconds, come back if time allows, and stop answering carefully with half a minute left to fill every blank.
Two honest notes. The 4 points are an average: 20 guesses have a standard deviation of about 1.8 points, and there is roughly a 1% chance none of them land. And the no-penalty rule is the premise of this question; tests differ, so check the instructions of the one you are sitting before deciding.
Where candidates lose it
The trap is leaving blanks out of a sense that guessing is unserious. On a no-penalty test that throws away free expected points, and the people who score well on these tests always fill every answer.
The second loss is spending too long on hard questions early. At 14.4 seconds a question, one stubborn item can cost you three easy ones; skip, mark and return.
What the interviewer asks next
If a wrong answer cost a quarter point, would you still guess?
You can eliminate one option on each of the 20. What is the guess value now?
Would you rather answer 35 at 75% accuracy or 30 at 85%?
Asked at Vista Equity Partners, Technology, Media and Telecom, Austin, 2022 (Wall Street Oasis): Application requires a CCAT, 50 questions in 12 minutes.

a	the round number in the middle, 50 or 1,000
b	how far each factor sits from it, 1 or 2
a squared minus b squared	the product, always slightly below the square

30 x 0.85	careful answers times your accuracy
20 x 1/5	blind guesses times the chance a random pick is right

Try it first

Why does 49 x 51 come out one short of 2,500?

How do you take 12.5% of anything in a few seconds?

What is the interviewer actually checking?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a blind guess worth anything at all?

What changes if wrong answers are penalised, and how do you manage the clock?

Where candidates lose it

What the interviewer asks next