Every concept, endless questions
Each concept is a small financial model. Read how it works, then draw a real interview question from it: forwards, backwards or what-if, with fresh numbers and a worked solution every time.
Forward runs a concept the usual way. Inverse runs it backwards, which is what separates understanding from memorizing.
Brainteasers
2 conceptsFair value of a betting game
An option to try again is worth money, and the way to price it is backwards. Without the option a die is worth its average face. With one reroll you compare each face you might see against that average: keep it if it is higher, throw it back if it is lower, and the value of the game is the average of those decisions. A second reroll is priced the same way against the one-reroll value, and so on. Each extra reroll is worth less than the last, because the bar you are rerolling against keeps rising. This is the same logic that prices an American option: at every point, compare what you hold with what waiting is worth.
Clock angles and rate puzzles
Clock problems are relative-speed problems wearing a costume. The minute hand runs at 6° a minute, the hour hand at 0.5°, so the gap between them changes at exactly 5.5° a minute — and once you hold that number, every question is one division. The angle at any time is the hour hand’s position minus the minute hand’s; the next time they meet is the current gap divided by 5.5; the number of meetings in twelve hours is eleven, because the minute hand laps the hour hand once for every full lap it gains. Interviewers ask these to see whether you reach for the mechanism or try to remember an answer.
Classic puzzles
55 puzzlesMost brainteasers don't come in variations, so these stay as fixed questions.
A bat and ball cost $1.10 in total, and the bat costs $1 more than the ball; what does the ball cost?
Five machines make five parts in five minutes; assuming identical independent machines at a constant rate, how long do 100 machines need for 100 parts?
A patch doubles in area each day and fills a pond on day 30; on what day was the pond half full?
How many cuts are needed to divide a straight rope into eight pieces if each cut affects only one piece?
A drawer contains only black and white socks; without seeing colors, how many socks guarantee a matching-color pair?
6 people each shake hands with every other person exactly once; how many handshakes occur?
8 people each shake hands with every other person exactly once; how many handshakes occur?
10 people each shake hands with every other person exactly once; how many handshakes occur?
12 people each shake hands with every other person exactly once; how many handshakes occur?
14 people each shake hands with every other person exactly once; how many handshakes occur?
A bag contains ample socks in 3 colors; how many blind draws guarantee two socks of the same color?
A bag contains ample socks in 4 colors; how many blind draws guarantee two socks of the same color?
A bag contains ample socks in 5 colors; how many blind draws guarantee two socks of the same color?
A bag contains ample socks in 6 colors; how many blind draws guarantee two socks of the same color?
A bag contains ample socks in 7 colors; how many blind draws guarantee two socks of the same color?
You toss a fair coin until the first head; what is the expected number of tosses?
A family has two children, each independently equally likely to be a boy or girl; given that the older child is a girl, what is the probability both are girls?
In the classic three-door Monty Hall game, the host knows the prize location, always reveals an unchosen goat, and always offers a switch; should you switch?
Two fair coins are tossed; given that at least one is heads, what is the probability both are heads?
You have eight identical-looking balls, exactly one heavier, and a balance scale; how can you identify it in two weighings?
An ant starts at one corner of a unit cube and travels along edges to the opposite corner using only shortest paths; how many such paths exist?
A fair six-sided die is rolled once; before seeing it, you may choose the payoff equal to the roll or a guaranteed 3.6. Which has higher expected value?
On a grid, you must move exactly 3 steps right and two steps up, with no backward moves; how many shortest paths exist?
On a grid, you must move exactly 4 steps right and two steps up, with no backward moves; how many shortest paths exist?
On a grid, you must move exactly 5 steps right and two steps up, with no backward moves; how many shortest paths exist?
On a grid, you must move exactly 6 steps right and two steps up, with no backward moves; how many shortest paths exist?
On a grid, you must move exactly 7 steps right and two steps up, with no backward moves; how many shortest paths exist?
Two runners start together on a circular track; one completes a lap in 4 minutes and the other in 8 minutes, running the same direction at constant speeds. When does the faster runner first lap the slower?
Two runners start together on a circular track; one completes a lap in 5 minutes and the other in 10 minutes, running the same direction at constant speeds. When does the faster runner first lap the slower?
Two runners start together on a circular track; one completes a lap in 6 minutes and the other in 12 minutes, running the same direction at constant speeds. When does the faster runner first lap the slower?
Two runners start together on a circular track; one completes a lap in 7 minutes and the other in 14 minutes, running the same direction at constant speeds. When does the faster runner first lap the slower?
Two runners start together on a circular track; one completes a lap in 8 minutes and the other in 16 minutes, running the same direction at constant speeds. When does the faster runner first lap the slower?
Two ropes each burn in exactly one hour but at uneven rates; how can you measure 45 minutes with a lighter?
You have nine coins, exactly one heavier, and a balance scale; how can you find it in two weighings?
A fair coin is tossed until two consecutive heads appear; what is the expected number of tosses?
You can roll a fair die once and either keep the result or replace it with one final roll; what is the optimal expected payoff?
Three switches outside a room control three incandescent bulbs inside; you may enter the room once and can safely detect warmth. How can you map them?
Two people independently arrive uniformly between noon and 1 pm and each waits 15 minutes; what is the probability they meet?
You have two eggs and a 100-floor building with a monotone break threshold; what minimum worst-case number of drops identifies the threshold?
Each draw independently selects one of 3 equally likely coupon types; what is the expected number of draws to collect all types?
Each draw independently selects one of 4 equally likely coupon types; what is the expected number of draws to collect all types?
Each draw independently selects one of 5 equally likely coupon types; what is the expected number of draws to collect all types?
Each draw independently selects one of 6 equally likely coupon types; what is the expected number of draws to collect all types?
Each draw independently selects one of 7 equally likely coupon types; what is the expected number of draws to collect all types?
A fair coin is tossed until the pattern HT appears; what is the expected waiting time, and how does it compare with HH?
Two independent uniform points are chosen on a unit stick to cut it into three pieces; what is the probability the pieces form a triangle?
A prisoner sees 99 independently fair black-or-white hats ahead of them and must announce one color before the others answer in sequence; everyone hears prior answers. What strategy guarantees at least 99 correct answers among 100 prisoners?
A fair coin is tossed indefinitely; why is the expected payoff of doubling a $1 stake after each loss not a risk-free finite-capital profit strategy?
You observe a random permutation of n distinct candidate ranks sequentially and can select one candidate irrevocably; what is the classic large-n strategy for maximizing the chance of selecting the best?
A fair random walk on the integers starts at zero; it eventually hits +1 with probability one. Does that imply a finite expected hitting time?
A fair die has integer faces 1 through 8; you may keep the first result or replace it with one final roll. What is the optimal rule and expected payoff?
A fair die has integer faces 1 through 10; you may keep the first result or replace it with one final roll. What is the optimal rule and expected payoff?
A fair die has integer faces 1 through 12; you may keep the first result or replace it with one final roll. What is the optimal rule and expected payoff?
A fair die has integer faces 1 through 14; you may keep the first result or replace it with one final roll. What is the optimal rule and expected payoff?
A fair die has integer faces 1 through 16; you may keep the first result or replace it with one final roll. What is the optimal rule and expected payoff?