How do I approach Coding & Algorithms interview questions?

Coding & Algorithms questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master coding & algorithms interviews.

What difficulty level is this interview question?

This is a medium difficulty Coding & Algorithms question, commonly asked during Onsite rounds at J.P. Morgan.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at J.P. Morgan during technical interviews.

Solve tiling, probability, and nested-radical puzzles

Quick Overview

This multi-part question evaluates combinatorics and algorithmic skills through 2×N and 3×N domino tiling problems, analysis of an infinite nested square root, and exact discrete probability for selecting couples, testing competencies in recurrence relations, dynamic programming reasoning, limit/closed-form evaluation, and combinatorial counting within the Coding & Algorithms domain. It is commonly asked to assess a candidate's ability to recognize and derive mathematical recurrences, translate infinite expressions into closed forms, and perform precise probability counting, representing a blend of conceptual mathematical understanding and practical algorithmic application.

You are given several independent math/coding puzzles.

1) Domino tiling a 2×N board

You have a 2×N rectangle and an unlimited number of 1×2 and 2×1 dominoes.

Task: Determine the number of distinct ways to tile the 2×N rectangle exactly.
Input: An integer N .
Output: The number of tilings.
Notes: Tiles may be rotated; a tiling is different if the set of covered cell pairs differs.

2) Domino tiling a 3×N board

You have a 3×N rectangle and the same unlimited 1×2 / 2×1 dominoes.

Task: Determine the number of distinct ways to tile the 3×N rectangle exactly.
Input: An integer N .
Output: The number of tilings.

3) Infinite nested square root

Define

$x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}$

Task: Compute the value of x .

4) Combinatorics probability (couples)

There are 5 couples (10 people total). You randomly select 4 people uniformly from the 10.

Task: Compute the probability that among the 4 selected people, there are exactly 2 complete couples (i.e., you selected both partners from exactly two couples).

Solve tiling, probability, and nested-radical puzzles

Quick Overview