Number Walks

An interactive cellular walk simulator tracking number paths and elastic collisions on a bounded grid.

Settings & Mode

Interaction Mode

Sit back and watch the mathematical spiral emerge. Use Auto Play or Step controls below.

Grid Size (Odd: 5–101)

Animation Delay (ms) 30ms

Placed Numbers: 0

Last Added: -

Walk Steps: -

Collisions: -

The Governing Laws of the Walk

This simulation visualizes a structured path-building logic that maps integer values to spatial grid coordinates. The walk begins with a zero in the center cell and is generated iteratively using the following mathematical axioms:

Spawn and Launch: Each new number N (starting at 1) spawns at the center 0 cell. Its initial departure vector rotates 90° clockwise with each iteration:
- N = 1 starts heading Right
- N = 2 starts heading Down
- N = 3 starts heading Left
- N = 4 starts heading Up
- N = 5 wraps back and starts heading Right, and so on.
Counting and Walking: The number N attempts to travel exactly N steps. Steps are only counted when landing on a vacant cell.
Elastic Collisions: If the path collides with a cell that is already occupied:
- The step is not counted.
- The walker lands on the occupied cell and immediately rotates 90° clockwise.
- The walker resumes counting and moving in this new direction from the next cell.
Boundary Reflections (U-Turns): If the walker hits the grid boundary wall:
- It performs a double-right-turn bounce sequence.
- First, it turns 90° clockwise and steps. Then, it turns 90° clockwise again and steps.
- It resumes counting and walking in the opposite direction, offset by one row or column.
- These boundary recovery maneuvers do not count towards the N steps.
Settling: When the walker completes its N-th step on a vacant cell, the walk halts, and the number N is permanently placed in that cell, acting as a new obstacle for all subsequent walks.