Challenge Your Algorithm Skills with Four Progressive Puzzle Problems

Introduction

The computing world evolves rapidly, but algorithms remain a constant foundation; mastering them is essential for efficient programming.

This article offers four mathematical‑puzzle‑style algorithm problems for engineers who already know sorting and searching and want to deepen their skills.

Puzzle Q1 – Entry Level

Difficulty: ★

Problem: Determine how many distinct paths a robot can take in 12 moves without revisiting a position, moving only forward, backward, left or right.

Solution idea: Model the robot’s position with coordinates, avoid previously visited cells, and use depth‑first search to enumerate valid paths.

Answer: 324,932 possible paths.

Puzzle Q2 – Beginner

Difficulty: ★★

Problem: From numbers 1 to N, select 7 composite numbers such that the maximum number of “friend” layers needed to connect any two numbers is 6; find the smallest N.

Solution idea: Treat numbers sharing a non‑trivial common divisor as friends, analyze divisor relationships, and construct chains of products (e.g., a×b, b×c, …) ensuring the required connectivity.

Answer: The set [4, 26, 39, 33, 55, 35, 49] satisfies the condition.

Puzzle Q3 – Intermediate

Difficulty: ★★

Problem: Count IP addresses that use each decimal digit 0‑9 exactly once and whose 32‑bit binary representation is symmetric.

Solution idea: Enumerate all 9! permutations for the decimal part, generate 16‑bit palindromic binary halves (65,536 possibilities), combine them, and convert to decimal IPs.

Answer: 8 such IP addresses.

Puzzle Q4 – Advanced

Difficulty: ★★★

Problem: Arrange a 6×5 classroom seating so that every seat’s immediate neighbors (front, back, left, right) are of the opposite gender; count distinct arrangements.

Solution idea: Perform exhaustive search over the 30 seats, prune invalid configurations, and use memoization to speed up the search.

Answer: 13,374,192 possible seatings.

Implementation

All solutions are implemented in Ruby; code listings (e.g., 08.01, 19.01, 40.01, 40.02, 68.01, 68.02) illustrate depth‑first search, combinatorial generation, and pruning techniques.