# Double-ended colourful Nim

> Source: https://ollybritton.com/notes/random/games/double-ended-colourful-nim/ · Updated: 2024-12-05

Some partial results about the Grundy values for double-ended colourful Nim, as discussed at the end of [Winning colourful Nim with the eeny-meeny rule](https://ollybritton.com/blog/colourful-nim/). At the moment, apart from these basic cases I've got no idea in order to how efficiently compute the values for an arbitrary position. The hope would be for a convenient rule like the eeny-meeny rule.

### No colour changes
$$
\mathcal G(x) \mapsto x
$$

### One colour change
$$
\mathcal G(x, y) \mapsto x \oplus y
$$
This is intuitive, you're effectively playing on two separate Nim piles.

### Two colour changes
$$
\mathcal G(1, y, 1) = \begin{cases}
1 &\text{if }y =1 \\
0 &\text{otherwise}
\end{cases}
$$
$$
\mathcal G(1, y, 2) = \begin{cases}
2 &\text{if }y=1 \\
1 &\text{if }y=2 \text{ or }y \ge 4 \\
3 &\text{if } y = 3
\end{cases}
$$
$$
\mathcal G(1, y, 3) = \begin{cases}
3 &\text{if } y = 1 \\
2 &\text{if } y = 2 \text{ or } y\ge 4 \\
1 &\text{if } y = 3
\end{cases}
$$
$$
\mathcal G(1, y, 4) = \begin{cases}
4 &\text{if }y =1,2,3,\text{or }7 \\
3 &\text{otherwise}
\end{cases}
$$
$$
\mathcal G(1, y, 5) = \begin{cases}
5 &\text{if }y=1,2,3, \text{or }5 \\
4 &\text{if }y=4,6,\text{or }y \ge 8 \\
3 &\text{if }y = 7
\end{cases}
$$
$$
\mathcal G(1, y, 6) \stackrel?= \begin{cases}
8 &\text{if }y=1,2,3,4,5 \\
7 &\text{otherwise }
\end{cases}
$$
(haven't proven the last one)

### Other rules
- $\mathcal G(x_1, \ldots, x_n) = \mathcal G(x_n, \ldots, x_1)$ (because the game is symmetric)
- Guess:
	- $\mathcal G(x, y, z) = 0$ iff $x = z$ and $y \ne x, z$
	- If $x = y = z$, then $\mathcal G(x, y, z) = x$.
	- (probably wouldn't be hard by induction)

### Table of values

| (n1, n2, n3) | G(n1, n2, n3) |
| ------------ | ------------- |
| $(1,1,1)$    | $1$           |
| $(2,1,1)$    | $2$           |
| $(3,1,1)$    | $3$           |
| $(1,2,1)$    | $0$           |
| $(2,2,1)$    | $1$           |
| $(3,2,1)$    | $2$           |
| $(1,3,1)$    | $0$           |
| $(2,3,1)$    | $3$           |
| $(3,3,1)$    | $1$           |
| $(1,1,2)$    | $2$           |
| $(2,1,2)$    | $0$           |
| $(3,1,2)$    | $1$           |
| $(1,2,2)$    | $1$           |
| $(2,2,2)$    | $2$           |
| $(3,2,2)$    | $3$           |
| $(1,3,2)$    | $3$           |
| $(2,3,2)$    | $0$           |
| $(3,3,2)$    | $2$           |
| $(1,1,3)$    | $3$           |
| $(2,1,3)$    | $1$           |
| $(3,1,3)$    | $0$           |
| $(1,2,3)$    | $2$           |
| $(2,2,3)$    | $3$           |
| $(3,2,3)$    | $0$           |
| $(1,3,3)$    | $1$           |
| $(2,3,3)$    | $2$           |
| $(3,3,3)$    | $3$           |
[Here's a longer list of calculations for bigger positions](double-ended-grundy.txt).

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt