$$Question: A linguist observes that the number of unique words in a text grows with the square of the page count. If a 10-page document contains 220 unique words, how many unique words would a 15-page document have, assuming the growth pattern holds? - Deep Underground Poetry

Understanding the Context

The Science Behind the Pattern

If the number of unique words grows proportionally to the square of page count, this implies a mathematical relationship:
Unique Words = k × (Page Count)²
Using real data — a 10-page document with 220 unique words — we calculate the constant:
220 = k × (10)² → k = 220 / 100 = 2.2
Thus, for a 15-page document:
Unique Words = 2.2 × (15)² = 2.2 × 225 = 495

This suggests a 15-page document would contain approximately 495 unique words, reflecting the nonlinear increase in lexical variety as text length increases.

Key Insights

Why This Matters in U.S. Content Creation

At a time when writers and platforms seek data-backed guidance, the square-of-page growth pattern highlights a natural effect: longer texts enable richer vocabulary without constant repetition. This pattern helps communicate value in reusable writing models, content strategy, and language education. For U.S.-focused creators, understanding this trend can refine editorial approaches and improve content longevity. It also supports conversations about quality, depth, and audience engagement in a landscape where word choice shapes readership and retention.

Common Questions About the Pattern

How does page count truly affect word diversity?
As pages increase, more space allows for varied topics, nuances, and vocabulary expansion—especially in professional or academic writing—resulting in higher lexical density.

Final Thoughts

Can this formula apply to all text types?
While ideal for structured, page-length content like reports or essays, the relationship may soften in constrained formats