Editing Openai/69558c0c-d6e4-800b-8304-eab31684eaee (section)

=== 2. “Unique walking paths” without revisiting coordinates ===

This is the dominant term — it dwarfs everything else combined.

You are counting simple paths (no repeated coordinate) on a 3D grid.

===== Known result from graph theory: =====

The number of simple paths in a finite grid grows super-exponentially.

A very conservative lower bound:

Number of paths≥2N\text{Number of paths} \ge 2^{N}Number of paths≥2N
Where NNN is the number of nodes.

Here:

21.386×10152^{1.386 \times 10^{15}}21.386×1015
That number is so large it cannot be written in scientific notation in any meaningful way.
For comparison:
* Estimated atoms in the observable universe: ~10⁸⁰
* This number has ~4.17 × 10¹⁴ decimal digits

📌 This single requirement already dominates the total task count.

✔ Finite
❌ Physically, computationally, or conceptually enumerable? No.