Mental math and brainteasers for trading interviews

What kinds of math do trading interviews ask?

Four families. Arithmetic under time pressure, sometimes as a timed online test. Probability, mostly dice, cards, and coins. Expected value, often framed as a game you would pay to play. And market making, where you quote a price on an uncertain number. The sample below is from our S&T technicals bank, which mixes these with product questions.

Which arithmetic tricks matter most?

Most fast arithmetic is rounding to an easy number and then correcting for what you added or took away. Learn these six until you use them without thinking, and say the shortcut out loud when you use one.

• Round and correct: 17 times 24 is 17 times 25 (425) minus 17, which is 408.
• Difference of squares: 37 times 43 is (40 minus 3) times (40 plus 3), so 1,600 minus 9, which is 1,591.
• Squares near 100: 98 squared is 10,000 minus 400 plus 4, which is 9,604.
• Squares ending in 5: for 45 squared, multiply 4 by 5 to get 20, then append 25: 2,025.
• Percents by pieces: 15% of 380 is 10% (38) plus 5% (19), which is 57.
• Fractions you should know cold: eighths (1/8 is 0.125, 3/8 is 0.375, 7/8 is 0.875) and sixteenths (5/16 is 0.3125).

Can you work these eleven problems?

Try each one before reading the answer, and time yourself. The arithmetic ones should take under ten seconds once you know the trick; the probability ones under a minute with the reasoning said out loud.

1. A stock falls 20% then rises 20%. Where is it? Down 4%. 100 goes to 80, and 20% of 80 is 16, so 96. Percentage moves on different bases do not cancel.
2. What is the sum of the integers from 1 to 100? 5,050. Pair the ends: 1 and 100, 2 and 99, and so on make 50 pairs of 101.
3. What is the probability of at least one six in three rolls of a die? About 42%. Use the complement: the chance of no six is 5/6 cubed, 125/216, so the answer is 91/216.
4. Heads you win 2 dollars, tails you lose 1. What is it worth? Expected value is 0.5 times 2 minus 0.5 times 1, which is 0.50 dollars a flip. Pay anything under 0.50 and you have an edge.
5. Roll a die and take the face value, or reroll once and take the second roll. What is the game worth? 4.25. The reroll is worth 3.5, so keep 4, 5, or 6 and reroll 1, 2, or 3. Value is (4 plus 5 plus 6) divided by 6, which is 2.5, plus one half times 3.5, which is 1.75.
6. One fair coin and one double headed coin. You pick one at random, flip it, and get heads. What is the chance it is the double headed coin? Two thirds. Heads has probability 1 from the double headed coin and 1/2 from the fair one, so the odds are 1 to 1/2, or 2 to 1.
7. Two cards from a standard deck. What is the chance both are aces? 4/52 times 3/51, which is 1/13 times 1/17, or 1/221, under half a percent.
8. On average, how many rolls of a die until the first six? Six. When each try succeeds with probability p, the expected number of tries is 1/p.
9. What is the angle between the clock hands at 3:15? 7.5 degrees. The minute hand is at 90 degrees. The hour hand moves half a degree per minute, so after 15 minutes past 3 it is at 90 plus 7.5.
10. How many trailing zeros does 100 factorial have? 24. Each zero needs a 5 paired with a 2, and 5s are scarcer. 100 divided by 5 is 20, plus 4 more from 25, 50, 75, and 100, which each carry a second 5.
11. Roll two dice and get paid the higher one. What is the expected payout? 161/36, about 4.47. The chance the maximum is k is (2k minus 1)/36, so sum k times (2k minus 1) for k from 1 to 6: 1, 6, 15, 28, 45, 66, total 161.

How do you answer a make me a market question?

Give a bid and an offer on the number, with a width that reflects how unsure you are, and be ready to trade either side. Say how you got your midpoint, then quote around it. If the interviewer buys from you, they think your offer is low, so you move up; if they sell to you, you move down. The point is not being right about the number. It is quoting two sided prices you would honor and updating when someone trades with you.

Example: make me a market on the number of windows in this building. You estimate floors and windows per floor out loud, say ten floors and about forty windows each, so roughly 400. You quote 350 bid, 450 offered. They buy at 450. You ask yourself what they know, and move to 450 bid, 520 offered.

How should you practice?

Ten minutes a day beats a two hour cram. Speed comes from reps, and calm comes from having seen the shape of the problem before.

1. Daily arithmetic drills with a timer: two digit multiplication, percentages, and fractions to decimals.
2. Three probability or expected value problems a day, said out loud, as if to an interviewer.
3. Once a week, play a market making game with a friend: guess an unknown number and trade around it.
4. Write down every problem you miss and redo it two days later.
5. Practice on paper and in your head, since some interviews allow neither paper nor a calculator.

What are they really scoring?

The process as much as the answer. An interviewer would rather hear "one minus the chance of no sixes, so one minus 125 over 216" and a small arithmetic slip than a correct number with no explanation. Talk while you work, check your answer against common sense, and if you get it wrong, say so and fix it. Staying calm when stuck is part of the test.

Questions students ask

Are brainteasers still used in trading interviews?

Probability and expected value puzzles are, especially at quant firms and on options and market making desks. Pure riddles with a trick answer are less common than they used to be.

Can I use a calculator in a trading interview?

Usually not in a live interview. Timed online math tests often forbid them too. Practice without one.

How fast do I need to be?

Fast enough that the arithmetic never slows the reasoning. Two digit multiplication in a few seconds is a good target. Speed matters most on timed online tests.

Is this the same as a quant interview?

It overlaps. Quant trading interviews go deeper into probability, statistics, and often coding. The problems here are a floor for quant and roughly the ceiling for a markets desk.

What if I freeze on a problem?

Restate it, try a smaller version, and talk through it. Interviewers often give a hint to see how you use it.