Debt Capital Markets puzzles, solved step by step

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All topicsLeverage, coverage and cash flow9Mental maths and numeracy8Estimation and market sizing7Logic and brainteasers8Cost of capital and valuation riddles7Bond pricing and yield7Compounding, PIK and fees6Issuance and refinancing arithmetic8Credit spreads and default probability8Duration and convexity8Capital structure and recovery8Probability and expected value10Yield curve and forward rates6
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  1. 037You have a 3-litre and a 4-litre bottle and unlimited water. How do you measure exactly 2 litres and exactly 5 litres, and which whole-litre amounts up to 7 can you make?Logic and brainteasersWarm upNomuraNew York · 2026

    Try it first

    How many of the amounts 1 to 7 litres can you measure, counting water held across both bottles?

    Show the worked solution

    For 2 litres: fill the 3, pour it into the 4, fill the 3 again and top up the 4; exactly 2 litres stay in the 3-litre bottle. For 5: fill the 4, pour into the 3 to leave 1, empty the 3, move the 1 into it, then fill the 4, for 1 plus 4. Every whole amount from 1 to 7 is possible, because 3 and 4 differ by 1.

    What moves are you actually allowed?

    Three: fill a bottle to the top, empty it, or pour from one into the other until the first is empty or the second is full. The only amounts you can know for certain are full bottles and what is left after a pour stops at a full bottle, so every measurement is built from 3s and 4s. Think of it as making change with only Rs 3 and Rs 4 coins, where you are also allowed to hand coins back.

    Every fill, pour and empty moves water in steps of 3 and 42 litres0/30/4Start3/30/4Fill the 30/33/4Pour 3 into 43/33/4Fill the 3 again2/34/4Pour into 4 until full5 litres0/34/4Fill the 43/31/4Pour 4 into 30/31/4Empty the 31/30/4Pour the 1 into 31/34/4Fill the 4Read each pair as litres in the 3-litre bottle / 4-litre bottle. Lime marks the target amount.
    Four moves leave 2 litres in the 3-litre bottle after the 4-litre bottle is topped up, and five moves put 1 litre in the 3-litre bottle beside a full 4-litre bottle, which together hold exactly 5 litres.

    Why can you reach every amount from 1 to 7?

    Because 4 minus 3 is 1, and once you can make 1 you can make anything by adding bottles. The amounts you can measure are exactly the combinations of 3 and 4 that fit in the bottles, and since the two sizes share no common factor, every whole litre up to their total of 7 is reachable. One litre: fill the 4 and pour into the 3. Three and four: fill one bottle. Five: 1 plus a full 4. Six: 3 in each. Seven: both full.

    The relationship
    gcd⁡(3,4)=1  ⇒  3a+4b can equal any whole number\gcd(3, 4) = 1 \;\Rightarrow\; 3a + 4b \text{ can equal any whole number}
    gcdthe greatest common divisor, the largest number dividing both sizes
    a, bhow many times you add or remove each bottle's volume, positive or negative
    What it says in wordsWhen the bottle sizes share no factor, their combinations reach every whole number.

    The same rule tells you when a puzzle has no answer. With a 4-litre and a 6-litre bottle, every amount you can make is even, so 5 litres is impossible, and you can say so without trying a single pour. Interviewers like that sentence more than the pouring itself, because it shows you found the structure rather than a lucky sequence.

    Where candidates lose it

    Candidates start pouring at random and lose track of the state, which in a phone interview is fatal because the interviewer cannot see your paper. Say each state as a pair, litres in the 3 then litres in the 4, after every move.

    The second miss is solving 2 litres and freezing on 5, which cannot fit in either bottle. The question is asking for water held across both bottles, and saying that out loud is half the answer.

    What the interviewer asks next

    • With a 5-litre and a 7-litre bottle, what is the fewest number of moves to measure 1 litre?
    • Can you measure 5 litres with a 4-litre and a 6-litre bottle? Prove it either way.
    • How does this relate to what bond sizes you can build from fixed lot sizes?

    Asked at Nomura, Equity Capital Markets, New York, 2026 (Wall Street Oasis): How much water can you fill using 1 3liter and 1 4liter bottle using each other?