Investment Banking puzzles, solved step by step

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All topicsProbability7Mental maths and counting11Growth and compounding9Valuation riddles11Logic and brainteasers11Rates, risk and options9Estimation and market sizing7Accounting riddles9Expected value and games7Enterprise value and dilution6Deal maths6DCF and cost of capital7
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  1. 003A pond has one lily pad on day 1. The lily pads double every day and the pond is completely covered on day 30. On what day was the pond half covered?Growth and compoundingWarm upHarris WilliamsRichmond · 2025

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    Answer inside five seconds.

    Show the worked solution

    Day 29. The coverage doubles every day, so it halves when you step back one day. Full on day 30 means half on day 29, a quarter on day 28 and one thirty-second on day 25. Most of the growth in a doubling process arrives right at the end.

    Why is day 15 wrong?

    Day 15 would be right if the pond gained the same area every day, like a tank filling from a tap. Doubling is different: it is money compounding at 100% a day. Under doubling, each step back in time halves the amount, so half full is always exactly one step before full. On day 15 the pond is only one part in about 32,768 covered.

    Pond covered, last six days: each bar is double the one beforehalf the pond1/32Day 251/16Day 261/8Day 271/4Day 281/2Day 29FullDay 30the answer
    The pond is one thirty-second covered on day 25 and still only one quarter covered on day 28. It reaches half on day 29 and is full on day 30, because each day's bar is double the one before.

    Why would a banking interviewer ask this?

    It is a test of whether you think in growth rates or in straight lines. The same instinct that answers day 15 also overestimates how much of a compounding investment's value arrives early. Say that link in one sentence after you answer: it shows you know why the question is on the table.

    Where candidates lose it

    Answering day 15 is the whole trap, and many strong candidates say it because they are answering fast. The interviewer is watching whether you pause for one second and ask what kind of growth this is.

    Say day 29, then give the one-line reason: doubling forward means halving backward.

    What the interviewer asks next

    • On which day is the pond a quarter covered?
    • If you start with two lily pads, when is the pond full?
    • Where do you see this pattern in a DCF or an LBO?

    Asked at Harris Williams, Generalist, Richmond, 2025 (Wall Street Oasis): The lilypad doubles in size everyday, until the 30th day, when the pond is full

  2. 059A fund invests Rs 100 crore. Option one returns Rs 300 crore at the end of year 5. Option two returns Rs 100 crore through a dividend recap at the end of year 2 and Rs 200 crore at exit in year 5. Both are 3.0x. Which has the higher IRR, and by how much?Growth and compoundingHardMoelis & CompanyNew York · 2025

    Try it first

    Before you solve: how big is the IRR gap?

    Show the worked solution

    Option two, by about 10 points: roughly 34.8% against 24.6%. Option one triples the money in five years, so its IRR is 3 to the power of one fifth, minus 1, about 24.6%. Option two returns a third of the proceeds in year 2. Test 35%: the year-2 cheque is worth 54.9 today and the year-5 cheque 44.6, together 99.5, just under 100, so the IRR is a shade under 35%.

    Why does the same multiple give two different IRRs?

    Imagine lending a friend Rs 100 and getting Rs 300 back. If the Rs 300 arrives in one go after five years, fine. If Rs 100 comes back after two years, you have your stake back early and can lend it again. The money multiple counts rupees; the IRR counts rupees and how long each one spends away from you. Option two returns a third of the proceeds three years sooner, so the money is tied up for less time on average and the yearly rate is higher.

    Same 3.0x, but option two hands back a third of the money three years earlyOption oneYr 0Yr 1Yr 2Yr 3Yr 4Yr 5-100+3003.0x the moneyIRR 24.6%Option twoYr 0Yr 1Yr 2Yr 3Yr 4Yr 5-100+100+2003.0x the moneyIRR 34.8%recap: stake back early
    Both options turn Rs 100 crore into Rs 300 crore, but option two returns Rs 100 crore at year 2 through a recap, which lifts its IRR to 34.8% against 24.6% for a single exit at year 5.

    How do you get the second IRR without a calculator?

    Option one is clean. You know 2x in five years is about 15% a year and 3x is about 25%, and the exact figure is 24.6%. Option two has no neat formula, so pick a round guess and test it. At 35%, the year-2 cheque is worth 100 over 1.35 squared, about 54.9, and the year-5 cheque is worth 200 over 1.35 to the fifth, about 44.6: 99.5 in total, just short of the 100 invested, so the IRR sits just under 35%. One trial, said out loud, is all the interviewer needs.

    The relationship
    100=300(1+r1)5⇒r1=24.6%100=100(1+r2)2+200(1+r2)5⇒r2≈34.8%100 = \frac{300}{(1+r_1)^5} \Rightarrow r_1 = 24.6\% \qquad 100 = \frac{100}{(1+r_2)^2} + \frac{200}{(1+r_2)^5} \Rightarrow r_2 \approx 34.8\%
    r_1IRR of option one, a single exit in year 5
    r_2IRR of option two, the recap plus the exit
    100the Rs 100 crore invested at the start
    What it says in wordsThe IRR is the discount rate at which the cash coming back is worth exactly what went in.

    Why do sponsors like recaps, and what is the catch?

    Funds are often judged on IRR, so pulling cash out early through a dividend recapThe company borrows new debt and pays the money out to its owners as a dividend. lifts the number without changing the multiple. The catch is that the recap is paid for with new borrowing, so the company carries more debt for its last three years and the exit equity is smaller and riskier. A strong answer names all three effects: a higher IRR, the same multiple, more leverage. IRR rewards speed and the multiple rewards size, and investors look at both.

    Where candidates lose it

    The trap is saying the IRRs are equal because both deals are 3.0x. The multiple has no clock in it; the IRR does, and the question is built to see whether you notice.

    The second loss is freezing because option two has no closed-form answer. Guess a rate, discount both cheques and adjust: one trial at 35% lands within a fraction of a point.

    What the interviewer asks next

    • What if the recap paid Rs 100 crore at the end of year 1 instead of year 2?
    • The recap's extra debt cuts the exit proceeds from Rs 200 crore to Rs 170 crore. Is option two still ahead on IRR?
    • Why might a fund's investors care more about the money multiple than the IRR?

    Asked at Moelis & Company, Generalist, New York, 2025 (Wall Street Oasis): 5 techs in 30 mins covering accounting, m&a math, LBO math, DCF math, and debt