Weekly Problem: The Grid Independence Challenge
A puzzle at the intersection of graph theory and visual thinking
The Problem
Consider a 4×4 grid where each cell can be either selected or not selected. Two cells cannot both be selected if they share an edge (top, right, bottom, or left).
Your Challenge:
- What is the maximum number of cells you can select following this rule?
- Can you prove your answer is optimal?
- What pattern emerges in your solution?
Think about chess pieces that don’t attack each other…
Why This Matters
This puzzle illustrates a fundamental concept in independent set problems: the relationship between graph structure and maximum independent set size. The grid structure creates a special case that appears in many real applications, from processor scheduling to wireless network design.
Think About:
- How would your solution strategy change for a 5×5 grid?
- What if diagonal neighbors couldn't be selected either?
- Can you find a general formula for an n×n grid?
Weekly Problem: The Grid
