Jane Street Interview Questions

You are given two identical eggs and a 100-story building. Your task is to determine the highest floor from which you can drop an egg without it breaking.

How can you invert a pyramid of coins by moving only three coins?

You flip a fair coin repeatedly and stop flipping after three heads in a row occur. What is the expected number of flips required?

How many handshakes occur if every person in a room shakes hands with every other person exactly once?

Given that the probability it rains on Sunday is 40% and the probability it rains on the weekend (Saturday or Sunday) is 60%, what is the probability it rains on Saturday?

There are 10 castles, numbered 1 through 10, with respective values of 1 to 10 points. You have 100 soldiers to distribute among the castles in any way you choose, and your opponent does the same independently. For each castle, the player with more soldiers wins that castle's points; in the event of a tie, no one receives points for that castle. Additionally, for each castle you win, you lose 0.2 points for every soldier you have more than your opponent at that location. All 100 soldiers must be deployed. Formulate a strategy to maximize your expected score.

1. How many shortest paths exist from one corner of a chessboard to the opposite corner? 2. What is the smallest positive integer that has exactly 28 divisors?

What is the probability that the sum of the numbers is even when tossing two dice?

Two players each have a die. I have a 20-sided die numbered 1-20, and the other player has a 30-sided die numbered 1-30. Both players roll their respective die. If my number is greater, the other player pays me the value of my die roll in dollars. If the other player's number is greater, I pay them the value of their die roll in dollars. If we roll the same number, I pay the other player that number in dollars. What is the expected value of my winnings or losses for a single round of this game?

Suppose two players play a game where Player A and then Player B each pick an integer between 1 and 30. Then, a 30-sided die is rolled. Whoever guessed closer to the value of the roll wins an amount of money equal to the value of the roll from the other player. Given the choice, should you go first or second? What number should you choose? What is the expected value of your position?

Given a 4x4 chessboard, can a knight start from any square and visit every other square exactly once without revisiting any square?

Design a heap data structure that supports adding elements and removing the top element, with the following constraints: (1) The topmost element must always be less than the topmost element in the left and right child subtrees; (2) The left child subtree must have as many or one more element than the right child subtree. The heap should also support a 'min' function (returns the value at the top of the tree) and an 'empty' function (returns an empty tree). Provide a description and implementation of such a data structure.

You have two decks of cards. Each deck contains both red and black cards in equal proportion. One deck has 52 cards, and the other has 104 cards, both consisting of half red and half black cards. You may choose which deck to play with. Then, you draw two cards at random from your chosen deck. If both cards are red, you win a prize. Which deck should you choose to maximize your chance of winning, and why?

There are 30 blue balls and 30 red balls, and two urns. Your opponent may arrange the balls in the two urns in any way he chooses, without telling you the arrangement. You then select one urn and draw a ball at random from it. You win $10 if you draw a blue ball, and $0 otherwise. How much should you be willing to pay to play this game?

What is the smallest positive integer whose digits multiply to 108?

What is the probability of drawing two red cards in succession from a standard deck of 52 cards and from a double deck of 104 cards?

An unfair die with 12 faces has the number 11 with a probability of 40%, while the other faces are equally likely. You and another player are playing a game where whoever is closer to the correct answer wins. What is your optimal strategy, and should you choose to go first or second?

How much would you pay to play a game where you roll a die and receive the number of dots rolled in dollars?