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  1. 005Without paper, what are 997 x 1,003 and 67 squared?Estimation and mental mathsWarm upAkuna Capitalchicago · 2024

    Try it first

    What is 997 x 1,003?

    Show the worked solution

    997 x 1,003 = 999,991 and 67 squared = 4,489. The first is a difference of squares: the numbers sit 3 either side of 1,000, so the product is 1,000,000 minus 9. The second uses the same identity the other way round: 67 squared is 64 x 70 plus 3 squared, which is 4,480 plus 9. Both take one line once you spot the round number nearby.

    Why does multiplying around a round number work?

    Take a square garden 10 metres a side and reshape it to 13 by 7. The fence is the same length, but the plot shrinks from 100 square metres to 91. It always shrinks by the square of how far you moved each side, here 3 squared, 9. Two numbers spaced equally around a midpoint multiply to the midpoint squared minus the gap squared. For 997 x 1,003 the midpoint is 1,000 and the gap is 3, so the answer is 1,000,000 minus 9.

    Move one strip and a square minus a corner becomes a rectangleb²a x (a - b)stripaasquare minus corner: a² - b²stand the stripon its enda x (a - b)stripa + ba - brectangle: (a - b)(a + b)997 x 1,003 = (1,000 - 3)(1,000 + 3) = 1,000,000 - 9 =999,99167 x 67 = (67 - 3)(67 + 3) + 3² = 64 x 70 + 9 = 4,480 + 9 =4,489
    Removing a b by b corner from an a by a square and standing the leftover strip on its end makes a rectangle a + b wide and a - b tall, which is why 997 x 1,003 is 1,000,000 - 9 = 999,991 and 67 squared is 64 x 70 + 9 = 4,489.
    The relationship
    (a−b)(a+b)=a2−b2a2=(a−b)(a+b)+b2(a-b)(a+b) = a^2 - b^2 \qquad a^2 = (a-b)(a+b) + b^2
    athe round midpoint, or the number being squared
    bthe gap you choose to make a factor round
    What it says in wordsA product around a midpoint is the midpoint squared less the gap squared; run it backwards to square any number.

    How do you square a number like 67 in your head?

    Push it to a round neighbour and repair the difference. Move 3 down to 64 and 3 up to 70, multiply those, then add back the 3 squared you took away: 64 x 70 = 4,480, plus 9 is 4,489. You choose the gap so one factor is round. The expansion route agrees: 67 is 70 minus 3, so 67 squared is 4,900 - 420 + 9, the same 4,489. Two methods landing on one number is your check.

    Why would a fund ask arithmetic at all?

    A trader checks prices, spreads and position sizes in their head all day, and a slip costs money before any spreadsheet catches it. Interviewers use speed on arithmetic like this as a proxy for how quickly you would catch a quote that does not add up. Say the identity as you use it, so that if you slip, the interviewer can see where and you can recover out loud.

    Where candidates lose it

    The trap is grinding 997 x 1,003 column by column under time pressure, dropping a carry and landing on 1,000,009 or 999,909. Two numbers either side of 1,000 is the whole question, and it is there to see whether you notice.

    For 67 squared, candidates who remember 65 squared is 4,225 try to count up from there and lose track of the cross term. Pick the route that gives you one round multiplication, and say it out loud as you go.

    What the interviewer asks next

    • What is 48 x 52?
    • What is 95 squared, and what is the fastest route?
    • Estimate 1.03 to the tenth power in your head.

    Asked at Akuna Capital, Hedge Fund, chicago, 2024 (Wall Street Oasis): focusing on quick probability puzzles, mental math, and some data structure questions