Derivatives Foundation puzzles, solved step by step

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100
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Hard
29
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All topicsMental maths and estimation9Random walks and Markov chains7Conditional probability and Bayes7Volatility and correlation7Option pricing intuition7Expected value and optimal stopping10Market making11Option payoffs and no-arbitrage10Probability and counting11Distributions and statistics8Games and logic8Betting and sizing5
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Showing 1–3 of 3 · filtered from 100Clear filters
  1. 043In your head, no paper: convert 3/32 to a decimal, work out 38 x 42, and give 1/7 to four decimal places.Mental maths and estimationWarm upBelvedere TradingChicago · 2021

    Try it first

    What is 38 x 42?

    Show the worked solution

    0.09375, 1,596 and 0.1429. For 3/32, halve 1 five times to get 1/32 = 0.03125 and multiply by 3. For 38 x 42, both sit 2 away from 40, so the product is 40 squared minus 2 squared, 1,600 - 4. For 1/7, the repeating block is 142857, because 7 x 142857 = 999,999, so 1/7 = 0.142857... which rounds to 0.1429. Each one is a known anchor plus one step.

    Why does the interviewer ask three small sums in a row?

    A shopkeeper who totals a bill in his head is not doing long addition; he rounds to the nearest hundred and fixes the difference. Trading desks ask quick sums to see whether you reach for an anchor you already know, because that is how prices get checked in the two seconds before someone else trades. The three questions here each have a short route: fractions with a power of two below them are halvings, products of numbers either side of a round number are a difference of squares, and sevenths are one repeating block of six digits. The speed comes from knowing which route fits, and the interviewer listens to the route as much as to the number.

    Three anchors replace three long divisions: halve, square, and know the sevenths3/32: halve five times1/2 = 0.51/4 = 0.251/8 = 0.1251/16 = 0.06251/32 = 0.03125x 3: 0.03125 x 3= 0.0937538 x 42: around 4040 x 402 x 2 out(40 - 2)(40 + 2) = 40^2 - 2^2= 1,600 - 4= 1,5961/7: one cycle of six0.142857 142857 ...7 x 142857 = 999999so 1/7 = 142857 / 999999the other sevenths reuse it:2/7 = 0.2857143/7 = 0.428571rounded to 4 places:fifth digit is 5, so round up= 0.1429Each answer rests on a fact you already know, so the interviewer hears a method, not a guess
    Halving 1 five times gives 1/32 = 0.03125 and three of those make 0.09375, a 40 by 40 square with a 2 by 2 corner removed shows 38 x 42 = 1,596, and the six-digit cycle 142857 gives 1/7 = 0.1429 to four places.

    What is the route for each one?

    For 3/32, keep halving from a half: 0.5, 0.25, 0.125, 0.0625, 0.03125. Every halving adds a digit, and 32 is five halvings, so 1/32 is 0.03125 and three of them are 0.09375. Bond traders do this all day, because US Treasury prices are quoted in 32nds. For 38 x 42, notice that both numbers sit 2 from 40, so the product is 40 squared minus 2 squared, 1,596, a trick that works for any pair placed evenly around a round number. The same trick gives 47 x 53 = 2,500 - 9 = 2,491. For 1/7, remember that 7 x 142857 = 999,999. So 1/7 is 142857 divided by 999,999, which is 0.142857 repeating, and the fifth decimal is 5, so it rounds up to 0.1429.

    The relationship
    332=3×125=3×0.03125,(a−b)(a+b)=a2−b2,17=142857999999=0.142857‾\frac{3}{32} = 3 \times \frac{1}{2^5} = 3 \times 0.03125, \qquad (a - b)(a + b) = a^2 - b^2, \qquad \frac17 = \frac{142857}{999999} = 0.\overline{142857}
    2 to the 532, five halvings of 1
    a, bthe round midpoint, 40, and the distance to each number, 2
    the barthe block of six digits that repeats forever
    What it says in wordsTurn each sum into a fact you already know plus one small step.

    What does a strong candidate add after the answers?

    Add the sanity check and the neighbours. A product of two numbers around a centre is always a little below the centre squared, so 1,596 must be under 1,600. Three 32nds must be just under a tenth, since 3.2 of them would make exactly 0.1. The sevenths share the same six digits in rotation, 2/7 = 0.285714 and 3/7 = 0.428571, so knowing one seventh gives you all six. The limitation is honest too: anchors cover the common cases, and for an awkward product like 37 x 46 you fall back on splitting, 37 x 46 = 37 x 50 - 37 x 4 = 1,850 - 148 = 1,702. Saying when you switch method sounds better than pretending one trick does everything.

    Where candidates lose it

    The common loss is starting long multiplication for 38 x 42 aloud, carrying digits and losing track. The interviewer gives you a pair either side of 40 on purpose; a candidate who does not spot it looks slow even if the answer is right.

    The second is 1/7 as 0.1428, truncating instead of rounding. The next digit is 5, so the fourth decimal rounds up. Say the repeating block first, then round, and the slip disappears.

    What the interviewer asks next

    • What is 5/16 as a decimal, and 7/64?
    • Work out 67 x 73 and 96 x 104 the same way.
    • What is 5/7 to four places?
    • Estimate 1/13 to three decimal places.

    Asked at Belvedere Trading, Trading, Chicago, 2021 (Wall Street Oasis): 3/32 mental math, 38*42, crossing the bridge in the shortest amount of time

  2. 056In your head, no paper: 998 x 1,003. Then 2.5% of 4,860. Then 99 squared.Mental maths and estimationWarm upOld Mission CapitalChicago · 2015Old Mission CapitalChicago · 2015

    Try it first

    998 x 1,003. Say it inside five seconds.

    Show the worked solution

    1,000,994, then 121.5, then 9,801. 998 x 1,003 is (1,000 - 2)(1,000 + 3) = 1,000,000 + 1,000 - 6. For 2.5% of 4,860, take 10%, which is 486, and then a quarter of it, 121.5. For 99 squared, (100 - 1) squared is 10,000 - 200 + 1 = 9,801. In every case the move is the same: go to the nearest round number and fix the small error afterwards.

    Why is the round number always the first move?

    Buying seven items at Rs 99 each, nobody multiplies 99 by 7; you take Rs 700 and hand back Rs 7. Multiplying by a round number is free, and the correction is a small product you can hold in your head, so every mental multiplication is a round-number product plus or minus a correction. For 998 x 1,003 the round number is 1,000 for both, so the product is 1,000,000, the two cross terms are + 3 x 1,000 and - 2 x 1,000, which net to + 1,000, and the only real work is the sign of 2 x 3 at the end: a minus times a plus is a minus, so subtract 6.

    Move to the nearest round number, then fix the small error998 x 1,003(1,000 - 2)(1,000 + 3)1,000,000+ 1,000 x (3 - 2) = + 1,000- 2 x 3 = - 61,000,9942.5% of 4,86010% of 4,860= 486a quarter of 486= 121.5121.599 squared(100 - 1) squared10,000- 2 x 100 = - 200+ 1 x 1 = + 19,801
    Each calculation becomes a round-number product with a correction: 998 x 1,003 is a million plus 1,000 minus 6, which is 1,000,994; 2.5% of 4,860 is a tenth, 486, then a quarter, 121.5; and 99 squared is 10,000 minus 200 plus 1, which is 9,801.

    How do you handle percentages that are not round?

    Build the percentage out of ones you can do instantly. 2.5% is a quarter of 10%, so take a tenth of 4,860, which is 486, and quarter it: half is 243, half again is 121.5. The same habit covers 7.5% as 10% minus a quarter of 10%, 15% as 10% plus half of that, and 12.5% as an eighth. On a desk this is how you convert a basis-point move to rupees while someone is still talking: 25 basis points on Rs 4,860 crore of notional is the same 121.5, in crore.

    The relationship
    (a−b)(a+c)=a2+a(c−b)−bc(a−b)2=a2−2ab+b2(a-b)(a+c) = a^2 + a(c-b) - bc \qquad (a-b)^2 = a^2 - 2ab + b^2
    athe nearest round number, here 1,000 or 100
    b, cthe small distances from the round number, here 2 and 3, or 1
    bcthe small product that carries the sign most people get wrong
    What it says in wordsA product near a round number is the round square plus the round number times the net offset, minus the product of the two offsets.

    What does the interviewer listen for beyond the answer?

    Speed, but also the sanity check. 998 x 1,003 must be a little above a million because 998 x 1,003 is roughly 1,000 x 1,001, and 99 squared must end in 1 because 9 x 9 does, so a candidate who says 1,000,994 and 9,801 without pausing shows both the trick and the check. Where this breaks down is numbers that are not near anything round, such as 47 x 83; there the move is to split one factor, 47 x 80 + 47 x 3, and accept that it takes two beats rather than one.

    Where candidates lose it

    The slip in 998 x 1,003 is the sign of the last term: candidates add 6 and say 1,001,006, or drop the cross terms and say 999,994. Say the expansion out loud, plus 1,000 then minus 6, and the sign looks after itself.

    The slip in 99 squared is forgetting the plus 1 and answering 9,800. The check is the last digit: 9 times 9 ends in 1, so 9,800 cannot be right.

    What the interviewer asks next

    • 1,002 x 997, same method.
    • What is 12.5% of 4,860?
    • 101 squared minus 99 squared, without squaring either.
    • A bond moves 35 basis points on Rs 2,400 crore. How many crore is that?

    Asked at Old Mission Capital, Prop Trading, Chicago, 2015 (Wall Street Oasis): multiple questions based on arithmetic operations involving multiple digits. These questions are supposed to answered without consulting a calculator
    Asked at Old Mission Capital, Prop Trading, Chicago, 2015 (Wall Street Oasis): Phone interview included arithmetic operations that was supposed to be done without pen or paper

  3. 082Without paper: 56 x 56, then 73 x 74.Mental maths and estimationWarm upAkuna CapitalChicago · 2025DRWLondon · 2026

    Try it first

    Which first move gets 56 x 56 out fastest in your head?

    Show the worked solution

    3,136 and 5,402. Anchor 56 on 50: (50 + 6)^2 = 2,500 + 2 x 50 x 6 + 36 = 3,136. For 73 x 74, square the smaller number and add it once: 73^2 = (70 + 3)^2 = 4,900 + 420 + 9 = 5,329, then 5,329 + 73 = 5,402. Check each a second way: 56^2 = 60 x 52 + 4^2 = 3,120 + 16, and 73 x 74 = 70 x 74 + 3 x 74 = 5,180 + 222.

    Why anchor on a round number instead of multiplying digit by digit?

    Measuring a room, you do not count tiles one by one; you count whole rows, then the part row, then the corner. A square of side 56 is a 50 by 50 block, two 50 by 6 strips and a 6 by 6 corner. Anchoring on 50 replaces one hard multiplication with four numbers you already hold and one addition, which is the whole trick of mental arithmetic under a clock. 2,500, 300, 300 and 36 add to 3,136. The second strip is the one people drop; say both strips aloud.

    Anchor on a round number and the multiplication becomes additions50 x 50 = 2,5006x5050 x 6 = 30036= 30056 = 50 + 62,500 + 300 + 300 + 36 = 3,13673 x 73= 4,900 + 420 + 9 = 5,329one morecolumn: 7374 = 73 + 15,329 + 73 = 5,402a second route for each: 60 x 52 + 16 = 3,136 and 70 x 74 + 3 x 74 = 5,402
    A square of side 56 splits into a 50 by 50 block worth 2,500, two 50 by 6 strips worth 300 each and a 6 by 6 corner worth 36, adding to 3,136, and a 73 by 74 rectangle splits into a 73 by 73 square worth 5,329 plus one extra column of 73, adding to 5,402.

    How do you handle 73 x 74 when neither number is round?

    Consecutive numbers are a square plus the smaller number: 73 x 74 = 73 x 73 + 73. That turns the problem into a square you can anchor, (70 + 3)^2 = 4,900 + 420 + 9 = 5,329, and one more addition, 5,402. Rewriting a product into a square and a correction works because a square has a shape you can reconstruct, and the correction is a number you already have. It also stops a common slip: people who try 70 x 74 then add 3 x 74 get it right too, but 3 x 74 = 222 is where under time pressure the 2 and the 22 get muddled.

    The relationship
    (a+b)2=a2+2ab+b2,n(n+1)=n2+n(a+b)^2 = a^2 + 2ab + b^2, \qquad n(n+1) = n^2 + n
    athe round anchor, 50 or 70
    bthe remainder, 6 or 3
    nthe smaller of two consecutive numbers, 73
    What it says in wordsA square anchors on a round number plus a remainder, and a product of neighbours is the square of the smaller one plus itself.

    Then check, out loud, in a different way. 56^2 is also 60 x 52 + 4^2, because (n + d)(n - d) = n^2 - d^2 with n = 56 and d = 4: 3,120 + 16 = 3,136. And 73 x 74 = 70 x 74 + 3 x 74 = 5,180 + 222 = 5,402. The interviewer times the first answer but listens for the second route; a trader who checks is worth more than one who is merely fast. The limitation: these tricks suit two-digit numbers near a round anchor; for 87 x 93 use the difference of squares around 90 instead.

    Where candidates lose it

    The common loss on 56^2 is dropping one of the two cross terms and saying 2,836. Say two times 50 times 6, not fifty times six, so the doubling is audible to you as well as the interviewer.

    On 73 x 74 the loss is choosing a route with a hard middle step, like 73 x 70 plus 73 x 4, and stalling on 292. Square and add one copy; it is the shortest path and the easiest to check.

    What the interviewer asks next

    • Now 87 x 93.
    • What is 56^2 minus 44^2, without computing either square?
    • Give me 1/56 to two decimal places.

    Asked at Akuna Capital, Prop Trading, Chicago, 2025 (Wall Street Oasis): The mental math problems which were timed, one example was the 56*56
    Asked at DRW, Trader Intern Interview, London, 2026 (Wall Street Oasis): Mental math multiplication (e.g. 73*74)