Zigzag Number Spiral

9 months ago

The article discusses a zigzag number spiral grid where numbers are arranged in a spiral-like manner, reversing direction at the edges.
The goal is to find a closed-form expression for the number at the mth row and nth column, denoted as f(m, n).
The spiral touches the left edge on even-numbered rows and the top edge on odd-numbered columns, leading to specific patterns.
Formulas for edge numbers are derived: f(m, 1) = m² if m is even, and (m-1)² + 1 if m is odd; similarly for f(1, n).
A piecewise formula for f(m, n) is developed based on whether m or n is greater and their parity (odd/even).
The piecewise formula is simplified into a single closed-form expression: f(m, n) = (max(m, n))² - max(m, n) + 1 + (-1)^max(m, n) * (m - n).
An alternative approach focuses on diagonal numbers, leading to the same closed-form expression.
The diagonal analysis shows f(n, n) = n² - n + 1 for all positive integers n, simplifying the general formula.

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