This is a collection of resolved Langton's Ant rules up to 12 colors (resolved means producing a highway, wedge, or spacefiller).

Basic rules of Langton's Ant (for new users):
A square grid is populated by an ant that moves depending on the square it's on:

Rule 1: If the square is white, the ant changes it to red and turns left.
Rule 2: If the square is red, the ant changes it to white and turns right.

The general rule that simple rules create complex behavior applies here.

Phase 1: Order
In the first few hundred steps, the ant produces simple (and often symmetric) patterns.

Phase 2: Chaos
After several hundred steps, the ant loses its orderly behavior and becomes chaotic. During this period, the ant makes very irregular patterns that have barely any order in them.

Phase 3: Highway
After 9,977 steps, the ant makes a regular and orderly pattern, often called the Highway. In the highway, the ant moves 2 squares diagonally every 104 steps, meaning the speed is c/52 diagonal.

These rules can be extended to multiple colors. However, not all rules produce highways; LRL has been run for over 272 billion steps without a highway.



Only rules starting with L are in this collection because rules starting with R are just mirror images, along the Y-axis if the ant starts out facing up.

Contents:
- 1 2-color rule (The classic Langton's Ant)
- 1 3-color rule
- 2 4-color rules
- 2 5-color rules
- 7 6-color rules
- 15 7-color rules
- 23 8-color rules
- 41 9-color rules
- 98 10-color rules
- 163 11-color rules
- 334 12-color rules

Record holders for each color count:
2 colors: LR, classic highway at 9,977 steps.
3 colors: LLR, highway at 39 steps.
4 colors: LLRL, highway at ~256,000 steps.
5 colors: LLRLL, spacefiller at 1,865 steps.
6 colors: LLLLRL, highway at ~178,800,000 steps.
7 colors: LLRLRLR, highway at ~1,888,700,000 steps.
8 colors: LLRLRRLR, highway at ~607,000,000 steps.
9 colors: LLLLLLLRL, highway at ~756,000,000 steps.
10 colors: LLLLLLRRRL, highway at ~2,130,000,000 steps.
11 colors: LRLRRRRLLRL, wedge at ~250,900,000 steps.

Notable rules in this collection:
LR: The classic Langton's Ant.
LLR: Immediate highway.
LLRL: Simplest ant to last relatively long. (~256,000 steps)
LLRLL: Simplest ant to produce a spacefiller.
LLLLRL: Lasts unusually long for an ant this simple. Highway at ~178,800,000 steps.
LLRLRLL: Simplest ant to produce a wedge.
LLRLRLR: Similar to LLLLRL. Highway at ~1,888,700,000 steps.
LLRLRRLLL: Makes a wedge that fills 1/4 of the plane.
LLLLLLLLRL: The longest lasting rule so far. Highway at ~2,130,000,000 steps.
LLLLRRRRRL: Makes a highway that goes on for a while, then fails.
LLLRLLRLLL: Makes an unusually-shaped wedge that fills 1/4 of the plane.
LLLRRRLLRR: Makes many long, highway-like extrusions out of the main anthill.
LLRLRRLLLL: Makes an extremely wide highway. Interestingly textured chaos.
LLRLRRLRLL: Another extremely wide highway. The rulestring is palindromic.
LLRRLRLRLL: Makes many failed highways before being successful at ~59,995,000 steps.
LLRRRLLLLR: Unusual chaos at the boundary, makes many highway and wedge-like extrusions.
LLRRRLRLRL: Makes very interesting patterns at the boundary.
LLRRRLRRRL: Makes a triangular wedge. Another triangle is contained within the wedge.
LRLRLRRLRL: Simplest ant to produce an oblique highway.
LLLLLRRLRLL: Makes an interestingly-shaped highway.
LLLLLRRRRRL: Not only unusually-textured chaos in the interior, but also many highway-like extrusions out of the boundary.
LLRLLLRRRRR: Very interesting exterior chaos. This chaotic rule might have snuck into the collection.
LLRLRRLLLLL: Makes an interesting wedge that fills half of the plane.
LLRLRRLRLLL: Makes a failed wedge to the left, assuming the ant starts out facing up.
LRLRLRRRLLR: Makes an oblique highway with displacement (3,5) after 500,000 steps.
LRLRRRRLLRL: Makes a wedge at ~250,900,000 steps.
LRRRRLLLRRR: Makes an internal highway that goes in a spiral.


Notable rules that are chaotic:
LRL: Chaos, not proven to be eternal.
LLRR: Symmetric chaos, all symmetric chaos eternal since the ant must return to the origin infinitely many times.
LRRL: More symmetric chaos.
LLLRR: Simplest ant to grow at a logarithmic (?) rate.
LRRRRRLLR: Grows a square containing a chaotically bouncing highway.

If you're hungry for a proof, then read on:

Claim: If a highway has a displacement of (a, b), then a and b must have the same parity.

Proof: Imagine that the grid is a checkerboard pattern as follows:

xoxoxoxoxoxoxoxox
oxoxoxoxoxoxoxoxo
xoxoxoxoxoxoxoxox
oxoxoxoxoxoxoxoxo
xoxoxoxoxoxoxoxox
oxoxoxoxoxoxoxoxo
xoxoxoxoxoxoxoxox
oxoxoxoxoxoxoxoxo

In a given rule, the ant can only turn left or right, moving to an adjacent square in the process. If the ant is vertical in a square labeled 'x', then the ant must be horizontal in a square labeled 'o', and vice versa. As a consequence, if the displacement of a given highway is (a, b), then a and b must sum to an even number in order for the ant to remain in the same orientation. The only way this can happen is if a and b are either both even or both odd. In either case, the parity is the same, thus proving the statement above. QED