Quant Research Interview Question - JP Morgan Chase
If you’re preparing for a quant research interview at JP Morgan Chase or any other top quantitative finance firm, it’s crucial to understand not just technical concepts, but also how to approach and solve unique, brain-teasing questions. These problems often test your mathematical intuition, logical reasoning, and creativity—skills highly valued in quantitative research roles. In this article, we’ll tackle some classic and challenging quant interview questions, breaking down solutions, and using analogies to make the concepts crystal clear. If you’re aiming for a top quant position, read on to boost your interview game!
Quant Research Interview Question - JP Morgan Chase
1. Should You Go First in the Quarter-on-Table Game?
1.1 Problem Statement
You play a game with another person. The rule is: you take turns putting a quarter (a coin of fixed size) on a round table (the table can be any size, but must be perfectly symmetrical). The goal is to keep placing quarters on the table so that no coin overlaps. If you cannot place a quarter because there is no room left, you lose. The question is: Should you go first?
1.2 Restating the Problem
Imagine you and a friend are playing “coin Tetris” on a round table. Each person must place a quarter so that it doesn’t overlap with any existing coin, and the coin must lie flat on the table. Once there’s no legal move, the player who can’t go loses.
1.3 Solution Strategy
Let’s use analogies and step-by-step logic to determine whether going first is an advantage.
- Symmetry: The table is perfectly round and both players use identical coins.
- Initial Move: The first move can be at any location on the table, but the center is special due to symmetry.
- Mirroring: After the first move, can the second player always respond in a way that mirrors the first player's moves?
1.4 The Mirror Strategy Analogy
Imagine standing in front of a mirror: Whatever action you take, your reflection does the exact same thing, but on the opposite side. In this game, if the first player places the quarter exactly in the center of the table, the second player can always mirror any subsequent move with respect to the center. Since the table is round and symmetrical, every legal move the first player makes can be mirrored by the second player.
If the first player does not start at the center, the second player can take the center, and then mirror every subsequent move.
1.5 Step-by-Step Reasoning
- First Move:
- If you go first, place the quarter at the exact center.
- Subsequent Moves:
- For every move your opponent makes, mirror that move across the center (i.e., do the exact opposite move, same distance from center, in the opposite direction).
- Result:
- Each time your opponent finds a spot, you will have a corresponding symmetrical spot to place your quarter.
- Since you went first, you will always have the last move, guaranteeing a win.
1.6 Conclusion and Generalization
Yes, you should go first, and your first move should be at the center of the table. By doing so, you can always mirror your opponent’s moves, ensuring you never run out of legal placements before your opponent.
Analogy: It’s like always having a backup plan that works for every move your opponent makes, thanks to the symmetry of the table.
2. How Many N in [100, 400] Such That \( N^N \) Is a Perfect Square?
2.1 Problem Statement
How many integers \( N \) in the interval \( [100, 400] \) are there such that \( N^N \) is a perfect square?
2.2 Understanding the Problem
We’re looking for numbers \( N \) between 100 and 400 (inclusive) such that when you raise \( N \) to the power of \( N \), the result is a perfect square. A perfect square is a number that can be written as \( k^2 \) for some integer \( k \).
2.3 Mathematical Analysis
Let’s analyze when \( N^N \) is a perfect square.
Recall that: \[ N^N = (N)^{N} \] We want \( N^N = k^2 \) for some integer \( k \).
A number is a perfect square if all exponents in its prime factorization are even. So, the exponent of every prime factor in \( N^N \) must be even.
Prime Factorization Approach
Suppose \( N = p_1^{a_1} p_2^{a_2} p_3^{a_3} \cdots p_k^{a_k} \) is the prime factorization of \( N \).
Then: \[ N^N = (p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k})^N = p_1^{a_1 N} p_2^{a_2 N} \cdots p_k^{a_k N} \] For this to be a perfect square, each exponent \( a_i N \) must be even.
Key Insight
But \( a_i N \) is even if either:
- \( a_i \) is even, or
- \( N \) is even.
But since \( a_i \) comes from the factorization of \( N \), let’s look for a simpler pattern.
2.4 Let’s Try Small Examples
- If \( N \) is even, then \( N^N \) is always a perfect square.
Let’s check: \[ 2^2 = 4 = 2^2 \quad \text{(Perfect square)} \] \[ 4^4 = 256 = 16^2 \quad \text{(Perfect square)} \] \[ 6^6 = 46,656 = 216^2 \quad \text{(Perfect square)} \] So, for even \( N \), \( N^N \) is a perfect square.
- If \( N \) is odd, what happens?
Let’s check: \[ 3^3 = 27 \quad \text{(not a perfect square)} \] \[ 5^5 = 3125 \quad \text{(not a perfect square)} \] Let’s factorize: For odd \( N \), \( N \) is odd, so the exponent is odd, and the base is odd.
2.5 General Mathematical Proof
Let’s suppose \( N = m^2 \) is a perfect square. Is \( N^N \) always a perfect square?
Let’s try: \[ N = 9, N^N = 9^9 = (3^2)^9 = 3^{18} = (3^9)^2 \quad \text{(Perfect square)} \] So, for perfect square \( N \), \( N^N \) is a perfect square.
But the key is: \( N^N \) is a perfect square if and only if \( N \) is even or \( N \) itself is a perfect square.
2.6 General Condition
Let’s formalize:
- If \( N \) is even, \( N^N \) is a perfect square.
- If \( N \) is odd and not a perfect square, \( N^N \) is not a perfect square.
- If \( N \) is odd and a perfect square, \( N^N \) is a perfect square.
Thus, all even \( N \) in [100,400], and all odd perfect squares in [100,400].
2.7 Counting the Values
Let’s count the number of even \( N \) in [100,400]:
- The first even number is 100, the last is 400.
- Number of even numbers = \( \frac{400 - 100}{2} + 1 = 151 \).
Now count the odd perfect squares in [100,400]:
- Perfect squares between 100 and 400: 121, 169, 225, 289, 361.
- Corresponding to \( 11^2, 13^2, 15^2, 17^2, 19^2 \).
Number of odd perfect squares = 5.
However, some of those odd perfect squares may already be included if they are even (but perfect squares of odd numbers are odd).
2.8 Final Answer
The answer is:
- Number of even \( N \) in [100,400]: 151
- Number of odd perfect squares in [100,400]: 5
Total = 151 + 5 = 156
2.9 Summary Table
| Type | Number | Explanation |
|---|---|---|
| Even N | 151 | All even integers between 100 and 400 inclusive |
| Odd perfect square N | 5 | Odd perfect squares between 100 and 400 (121, 169, 225, 289, 361) |
| Total | 156 | Sum of previous two rows |
3. Bubble Sort: Probability the Array Is Sorted After One Iteration
3.1 Problem Statement
Run one iteration of bubble sort on an array. What is the probability that the array is now sorted?
3.2 Understanding Bubble Sort
Bubble sort repeatedly steps through the list, compares adjacent pairs, and swaps them if they are in the wrong order. After one full pass (iteration), the largest element is moved to the end of the array. But the problem asks: After one iteration, is the array sorted, and what’s the probability of this happening?
3.3 Analogous Example
Imagine a group of children standing in a row by height. Each time you go down the line, you compare each pair and swap if the taller is before the shorter. After one full pass, the tallest child is guaranteed to be at the end, but are all the children now in order?
3.4 Mathematical Analysis
Let’s suppose the array has \( n \) distinct elements. The question: What is the probability that an arbitrary random array is sorted after one iteration of bubble sort?
- After one pass, the largest element is at the end.
- For the array to be sorted, the remaining \( n-1 \) elements must have been in sorted order before the pass.
3.5 Step-by-Step Reasoning
- The Only Way the Array Gets Sorted:
The array will be sorted after one pass if and only if, before the pass, the array was sorted except for the largest element, which could have been anywhere. - Counting Permutations:
The number of arrays where the largest element is at any position and the rest are sorted is \( n \) (since the largest can be at any of \( n \) positions in a sorted array of \( n-1 \) elements). - Total Number of Arrays:
The total number of possible arrays is \( n! \). - Probability:
\[ P = \frac{n}{n!} \] . But wait, this fails to account for cases where the elements before the maximum are out of order but get fixed by the bubble sort pass before the maximum element bubbles through.
Known result: The correct number of permutations of {1..n} sorted after one bubble pass = 2^(n−1)
Probability =\(P = \frac{2^{n-1}}{n!} \)
4. Dice Game: Should You Play?
4.1 Problem Statement
You roll two dice. If you get (6,6), you win 100. If you get (6, not 6) or (not 6, 6), you lose x. In all other cases, you reroll. When is it favorable to play this game (what is the threshold for x)?
4.2 Understanding the Problem
You roll two dice repeatedly until you get either:
- Both 6s (win 100)
- Exactly one 6 (lose x)
Other outcomes are ignored; you reroll.
4.3 Calculating Probabilities
Let’s enumerate possible outcomes:
- Total outcomes per roll: \(6 \times 6 = 36\)
- (6,6): 1 outcome
- (6, not 6): 5 outcomes
- (not 6, 6): 5 outcomes
- Other (not 6, not 6): \(5 \times 5 = 25\) outcomes
So:
- P(win) = \(1/36\)
- P(lose) = \(5/36 + 5/36 = 10/36\)
- P(reroll) = \(25/36\)
4.4 Expected Value Calculation
Let \(E\) be the expected value of the game.
\[ E = P(\text{win}) \times 100 + P(\text{lose}) \times (-x) + P(\text{reroll}) \times E \]
Plug in the numbers: \[ E = \frac{1}{36} \times 100 + \frac{10}{36} \times (-x) + \frac{25}{36} \times E \]
Solve for \(E\): \[ E - \frac{25}{36}E = \frac{100}{36} - \frac{10x}{36} \] \[ \frac{11}{36}E = \frac{100 - 10x}{36} \] \[ E = \frac{100 - 10x}{11} \]
4.5 When Should You Play?
You should play if \(E > 0\): \[ \frac{100 - 10x}{11} > 0 \implies 100 - 10x > 0 \implies x < 10 \]
So, you should play if and only if \(x < 10\).
4.6 Analogy
Think of this as a slot machine: If the cost for a “near miss” (rolling a single 6) is less than 10, the expected payout favors you in the long run.
4.7 Code Example for Simulation
import random
def play_game(x, trials=100000):
total = 0
for _ in range(trials):
while True:
d1 = random.randint(1, 6)
d2 = random.randint(1, 6)
if d1 == 6 and d2 == 6:
total += 100
break
elif d1 == 6 or d2 == 6:
total -= x
break
# else: reroll
return total / trials
print(play_game(9)) # Should be > 0
print(play_game(10)) # Should be ≈ 0
print(play_game(11)) # Should be < 0
Conclusion: Mastering Quant Interview Questions
Quant interviews at JP Morgan Chase and similar firms are designed to test your ability to break down complex problems, spot patterns, and reason mathematically. The four problems above illustrate the types of logical and mathematical thinking you’ll need:
- Using symmetry and mirroring strategies in combinatorial games
- Analyzing properties of numbers and their powers
- Understanding the mechanics and edge cases of algorithms like bubble sort
- Modeling expected value and risk in repeated random processes
To prepare effectively, practice solving these types of problems step by step, use analogies to internalize the logic, and always double-check your reasoning. With this approach, you’ll greatly increase your chances of success in quant research interviews at any top firm, including JP Morgan Chase.
Frequently Asked Questions (FAQ)
Q: Are these questions actually asked in JP Morgan Chase quant interviews?
Yes, these and similar questions have been reported in interviews for quant roles at top financial institutions, including JP Morgan Chase, Citadel, Jane Street, and others.
Q: How should I approach such brain-teaser questions in interviews?
Stay calm, break down the problem, use examples and analogies, and clearly explain your reasoning out loud. Interviewers look for your thought process as much as the final answer.
Q: What other topics should I study for quant interviews?
Brush up on probability, combinatorics, statistics, algorithms, linear algebra, and brainteasers. Practice coding and mental math as well.
Ready for your quant interview at JP Morgan Chase? Practice these questions and similar ones regularly to sharpen your mind and stand out from the competition!
