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  1. 006You have a 3-litre bottle, a 4-litre bottle and an unlimited tap, and no way of marking the bottles. How do you measure exactly 2 litres, and which whole-litre volumes can you make at all?Logic and brainteasersCoreNomuraNew York · 2026

    Try it first

    What is the fewest number of fills and pours that leaves exactly 2 litres in one bottle?

    Show the worked solution

    Fill the 3-litre bottle, pour it into the 4, fill the 3 again and pour until the 4 is full: 2 litres are left in the 3-litre bottle. That is four moves. Because 3 and 4 share no common factor, fills and pours can build every whole number of litres, so any total from 1 to 7 is possible across the two bottles.

    What does each move actually measure?

    Think of paying for something with only Rs 3 and Rs 4 coins. You can pay Rs 2 by handing over two Rs 3 coins and taking a Rs 4 coin back as change. The bottles work the same way. Every fill adds a full bottle and every pour into a full bottle sets one bottle's worth aside, so the volumes you can make are sums and differences of 3s and 4s. Two litres is 3 plus 3 minus 4: you draw two small bottles from the tap, the big bottle swallows 4 of those 6 litres, and 2 are left.

    Four moves to 2 litres: the 4-litre bottle has room for exactly 1 more1. Fill the 33 L bottle4 L bottle(3, 0)2. Pour it into the 43 L bottle4 L bottle(0, 3)3. Fill the 3 again3 L bottle4 L bottle(3, 3)4. Top up the 43 L bottle4 L bottle(2, 4)Totals the two bottles can hold: every whole number from 1 to 71 L4 less 32 L3 + 3 less 43 Lfill the 34 Lfill the 45 L1 in the 3, plus 46 L2 in the 3, plus 47 Lboth full
    Filling the 3-litre bottle twice and pouring into the 4-litre bottle leaves 2 litres behind in four moves, and because 3 and 4 share no common factor, every total from 1 to 7 litres can be built from the two bottles.

    Is there a second route, and which volumes are out of reach?

    Start from the big bottle and it still works, only slower. Fill the 4, pour into the 3 to leave 1, empty the 3, move the 1 litre across, fill the 4 again and top up the 3, which takes 2 litres and leaves 2 in the big bottle. That is six moves, 4 plus 4 minus 3 minus 3. Saying both routes and choosing the shorter one shows the interviewer you searched the problem rather than remembered it. Write each state as a pair, small bottle first, so neither of you loses track.

    For the second half of the question, use BezoutA result in number theory: whole-number combinations of two numbers produce exactly the multiples of their highest common factor, and nothing else.'s rule: two bottle sizes can measure exactly the multiples of their highest common factor. The highest common factor of 3 and 4 is 1, so every whole number of litres can be built, limited only by capacity: up to 4 in one bottle and 7 across both. Contrast a 4 and a 6: their common factor is 2, so 3 litres is impossible however long you pour.

    The relationship
    2=2×3−1×4gcd⁡(3,4)=12 = 2\times 3 - 1\times 4 \qquad \gcd(3,4) = 1
    2 x 3two fills of the 3-litre bottle
    1 x 4one full 4-litre bottle set aside
    gcd(3,4)the highest common factor of the two sizes, here 1
    What it says in wordsAny volume that is a whole-number mix of 3s and 4s can be poured, and because 3 and 4 share no factor, that covers every whole number.

    Where candidates lose it

    Candidates start pouring at random and narrate a dozen moves, losing track of which bottle holds what. The interviewer cannot follow it and neither can they. Write the state as a pair, small bottle first, and say each pair out loud.

    The second loss is the second half of the question. Showing you can make 2 litres is not the same as saying which volumes are possible. Give the rule, that 3 and 4 share no common factor so every total from 1 to 7 works, and one pair of sizes where it fails.

    What the interviewer asks next

    • With a 4-litre and a 6-litre bottle, can you measure 3 litres?
    • What is the fewest number of moves to leave exactly 1 litre in a bottle?
    • With a 5-litre and a 3-litre bottle, how do you measure 4 litres?

    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?