Derivatives Foundation puzzles, solved step by step

About 12 billion, with an honest range of 10 to 14 billion. A floor plate of 4,000 square metres at 3.8 metres a floor gives 1.52 million cubic metres over 100 floors; take half as usable, 0.76 million. A golf ball is 4.27 centimetres across, about 41 cubic centimetres, so 24,500 would fit in a solid cubic metre. Spheres leave gaps: at a packing factor of 0.64 that is 15,700 per cubic metre, and 0.76 million times 15,700 is 11.9 billion.

Why build a chain instead of guessing a number?

If someone asks how much rice a storeroom holds, you do not guess a tonnage. You measure the room, decide how much of it is shelving, and know how much a sack holds. An estimation question is marked on whether each link in the chain is stated, sized and defensible, not on the final figure. Here the chain is floors, floor plate, floor height, usable fraction, ball volume, packing factor. Say the six links before you fill any of them in, so the interviewer can follow and correct a single link rather than the whole answer.

Where does the packing factor come from, and why does it matter so much?

Pour marbles into a jar and shake it: they settle at roughly 64% of the jar's volume filled, with the rest air. Stack them by hand in the tightest possible pattern and you reach about 74%. Tip them in gently without shaking and you can be as low as 56%. The floor plate and height are things you can look up or pace out, but the packing factor is a physical assumption you must own, and it moves the answer from 10.4 to 13.8 billion on its own. Say which one you are using and why: balls poured into a building are a shaken random pile, so 0.64 is the defensible middle.

What do you say when the interviewer pushes on the usable fraction?

Lifts, stairs, structural columns, ducts and ceiling voids take a large slice of a tower, and nobody knows it to the percent. Give a range and show which way it pushes the answer: 40% usable takes you to about 9.5 billion, 60% to about 14 billion, so the final figure is somewhere between 10 and 15 billion whatever you assume. A candidate who reports a single number to three figures has missed the point of the question. A candidate who says 12 billion, give or take a third, with the reasons, has answered it.

About 8,400 people, with an honest range of roughly 5,558 to 12,994. Of 45 floors, take 40 as ordinary office floors after plant rooms, lobby and trading floors. A floor plate of 3,000 square metres with 70% usable gives 2,100 square metres of desk space per floor. At 10 square metres per person that is 210 people a floor, and 40 floors give 8,400. The widest assumption is the space per person, 8 to 12 square metres, which alone moves the answer by half.

Why build from the floor rather than from the bank?

If you wanted to know how many people a wedding hall holds, you would not guess from the size of the family; you would pace the hall and think about chairs per row. An estimate is only as good as the link the listener can check, and anyone who has worked on an office floor has a feel for how many desks it holds. So the chain is office floors, floor plate, usable share, square metres per person. Say the four links before any number, so the interviewer can argue with one of them rather than with the whole answer.

Which assumption should you spend your time on?

The one with the widest range. Floors run 38 to 42, a ratio of 1.11; the plate 2,700 to 3,300, a ratio of 1.22; the usable share 65% to 75%, 1.15; but space per person runs 8 to 12 square metres, a ratio of 1.5, so it moves the answer most. Check it against a floor you know: a trading floor packs people at 6 to 8 square metres, a floor of meeting rooms and offices spreads them at 15 or more. If the interviewer gives you one fact, ask for that one.

What would you add about occupancy?

That desks and people are different counts. With hot-desking, a floor of 210 desks might be home to 250 or 300 people who are not all in on the same day, so a question about who works in the building can give a larger answer than a question about who is in it. Say which one you are answering. The limitation to state: the usable share and the space per person are guesses from general experience, not measurements, and a real number would come from the building's floor plans and the badge-in data.

249 trailing zeros. A zero at the end is a factor of 10, and a 10 is a 2 times a 5. Among 1 to 1,000 there are 994 factors of 2 but only 249 factors of 5, so the 5s decide. Multiples of 5 contribute 200, multiples of 25 add a second 5 each for 40 more, multiples of 125 add 8, and 625 adds 1: 200 + 40 + 8 + 1 = 249.

Why count 5s and not 10s?

Imagine packing gift boxes that each need one lid and one base, from a pile of 994 lids and 249 bases. You can make 249 boxes, and the extra lids are useless. A trailing zero is a box: it needs one factor of 2 and one factor of 5, and in the product 1 x 2 x 3 x ... x 1,000 the 5s run out long before the 2s. Counting multiples of 10 misses every zero that comes from pairing a 5 in one number with a 2 in another; 4 x 5 is 20, which ends in a zero though neither factor is a multiple of 10. So the question is simply: how many times does 5 divide into 1,000 factorial?

How do you count the 5s without missing the doubled ones?

Count in layers. Every multiple of 5 carries at least one 5: 1,000 divided by 5 is 200. Every multiple of 25 carries a second 5 that the first layer did not see: 1,000 over 25 is 40. Multiples of 125 carry a third, 8 of them, and 625 carries a fourth, once. Each layer counts only the extra 5 that the layer before it missed, which is why you add the plain quotients and never multiply. The sum is 249. The same method gives 24 zeros for 100 factorial, 20 plus 4, and the layering stops as soon as the power of 5 exceeds the number.

What does the interviewer learn from how you say it?

The answer is a test of whether you decompose a question into its prime factors rather than reach for the obvious count. Say the pairing argument first, then the layered count, then check the shape: the answer is a little under n over 4, because 200 + 40 + 8 + 1 is a geometric series that sums towards 1,000 over 4. That check is useful on the follow-ups. For the number of factors of 2 the same layering gives 994; for 3s it gives 498, so in base 6, where a trailing zero needs a 2 and a 3, 1,000 factorial ends in 498 zeros. The limitation is worth a sentence: this counts zeros at the end, not zeros anywhere in the 2,568-digit number, which is a different and much harder question.

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.

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.

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.

42, 120 and 65. In the first, the differences 4, 6, 8, 10 rise by 2, so the next difference is 12 and the term is 42; the terms are n(n + 1). In the second, each term is the previous one times 1, 2, 3, 4, so the next multiplier is 5 and the term is 120; these are the factorials. In the third, the differences 2, 4, 8, 16 double, so add 32 to get 65; the terms are 2 to the n plus 1.

What order should you test patterns in?

A mechanic with an unknown rattle checks the cheap, common causes first and the exotic ones last. Sequence questions reward the same discipline: write the first differences, then the second differences, and only if neither settles try ratios and then rules that mix the two, such as double and subtract one. Each test takes a few seconds and most interview sequences give way to one of the first three. The order matters because the interviewer is timing you; the candidate who stares at the numbers hoping to recognise them is slower than the one who writes a row of differences under them without thinking.

How does each of the three give way?

For 2, 6, 12, 20, 30, the first differences are 4, 6, 8, 10 and the second differences are all 2. Constant second differences mean the sequence is a quadratic in n, so the next difference is 12 and the next term 42, and the closed form is n(n + 1): 1 x 2, 2 x 3, up to 6 x 7. For 1, 1, 2, 6, 24, the differences 0, 1, 4, 18 tell you nothing, so try ratios: 1, 2, 3, 4. The next ratio is 5 and the term is 120; the terms are 0!, 1!, 2!, 3!, 4!, and 5! is 120. For 3, 5, 9, 17, 33, the differences are 2, 4, 8, 16, a doubling, so the next difference is 32 and the term is 65. Spot the rule in one line too: each term is twice the previous minus 1, and every term is a power of two plus 1, so the next is 2 to the 6 plus 1.

Is the answer really unique, and what should you say if pushed?

Strictly, no. Any five numbers can be continued by any sixth, because a polynomial of degree five can be passed through all six points. For example n(n + 1) + (n - 1)(n - 2)(n - 3)(n - 4)(n - 5) matches 2, 6, 12, 20, 30 exactly and then gives 162. The expected answer is the simplest rule that fits, and the way to show you know that is to name the rule, not just the number. That habit carries over to the desk: a pattern in five data points is a hypothesis, and the trader who states the rule can test it on the sixth point, while the one who only extrapolates cannot tell when the pattern has broken. If the interviewer offers a sequence that resists all three tests, try alternating terms, or interleaved sequences, before guessing.

54 people. Each stop halves the load and then adds a third of what is left, so the load is multiplied by 1/2 x 4/3 = 2/3 at every stop. Undoing three stops means dividing 16 by 2/3 three times: 16 to 24 to 36 to 54. The forward check works: 54 to 27 to 36, then 18 to 24, then 12 to 16.

Why work backwards rather than guess a start?

If someone tells you a shirt was marked down by half, then marked up by a third, and now costs Rs 16, you do not guess the original tag; you undo the two moves in reverse order. Each stop is a pair of multiplications, and multiplications are undone by dividing in the opposite order, so the end number walks back to the start without any trial and error. The growth of a third comes last at each stop, so it is undone first: divide by 4/3, which is multiply by 3/4. Then undo the halving by doubling. From 16: times 3/4 is 12, times 2 is 24. Twice more gives 36 and then 54.

What is the one-line version an interviewer wants to hear?

Collapse each stop to a single factor. Losing half is x 1/2 and gaining a third is x 4/3, so a stop is x 2/3, and three stops are x 8/27. The start is 16 x 27/8 = 54. Saying that in one breath shows you saw the structure rather than the arithmetic, which is what the question is for. The limitation is that it only works because every stop has the same two moves; a route where the fractions change needs the step-by-step walk back.

Where do the whole-number checks help you?

The bus carries people, so every intermediate count must be a whole number, and that is a free check. If the backwards walk ever produces a fraction, you have undone the moves in the wrong order. Undoing the halving first from 16 gives 32, then times 3/4 gives 24, which happens to be whole here, but on the next stop it would give 48 then 36, and the order mistake would have cost you nothing visible until the end, where 72 x 3/4 = 54 again by luck. Order matters in general even when the numbers hide it, so say the order out loud.

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.

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.

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.

76. Multiplying by 4 and keeping only the last two digits gives 04, 16, 64, 56, 24, 96, 84, 36, 44, 76, and then 76 x 4 = 304 returns to 04. The cycle has length 10 starting at 4^1, and 3000 is a multiple of 10, so 4^3000 ends like 4^10, in 76. Check: 76 x 76 = 5,776, so 76 reproduces itself under squaring, which is what a power of 4^10 must do.

Why do only the last two digits matter at each step?

When you work out what time it will be 3,000 hours from now, you do not count the hours; you note that the clock face has 24 positions and ask where 3,000 lands on it. Last two digits are a clock with 100 positions. The last two digits of a product are fixed by the last two digits of the factors alone, so the sequence of 4^n mod 100 can only visit 100 states and has to fall into a cycle. Walk it: 04, 16, 64, 56, 24, 96, 84, 36, 44, 76, and 76 x 4 = 304 brings you back to 04. Ten steps, then it repeats.

How do you reduce 3000 without miscounting the start of the cycle?

The cycle begins at 4^1 = 04, not at 4^0 = 01, because 01 is never revisited: once a power of 4 is a multiple of 4 it stays one, and 01 is not. So the positions n = 1, 11, 21 and so on share last digits 04, and the positions n = 10, 20, 30 and so on share 76. 3000 is a multiple of 10, so it sits in the same slot as 10, and 4^10 ends in 76. Candidates who start counting from n = 0 land one step off and say 44.

A second route, worth one sentence: split 100 into 4 and 25. Any 4^n with n at least 1 is 0 mod 4, and 4^10 = 1,048,576 is 1 mod 25, so 4^3000 is 1 mod 25. The number under 100 that is 0 mod 4 and 1 mod 25 is 76. Two methods agreeing is the finish an interviewer wants. The limitation: the cycle trick is only this short because 4 is small; for a base like 7 the cycle mod 100 is 4 steps, for 3 it is 20, and you should check rather than assume.

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.

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.

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.