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6/4/2024

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$x=\underbrace {\{X_{L}} _{x_{L}\lnot \geq y}|X_{R}\}\leq \{Y_{L}|Y_{R}\}=y$

$x=\{\underbrace {X_{L}} _{x_{L}\lnot \geq y}|X_{R}\}\leq \{Y_{L}|Y_{R}\}=y$

Suppose green edges are excluded from hackenbush. Does that permit surreal numbers to model the scoring?

Yes, if you restrict hackenbush to only red and blue edges (excluding green), then surreal numbers can effectively model the scoring. Here's why:

• Surreal numbers excel at two-player, zero-sum games: This is exactly what red-and-blue hackenbush represents. Each player (red and blue) aims to gain an advantage by forcing the other into a losing position.
• Winning positions define surreal numbers: Surreal numbers are constructed by analyzing the set of winning positions available to each player in a game.
• Red and blue control: With only red and blue edges, each player has clear control over their colored edges. Red cuts red, and blue cuts blue. This allows for a clear definition of winning positions based on who can make the last legal move.

Here's how surreal numbers capture the scoring:

• Red edge value: Each blue edge can be considered worth 1 unit.
• Red edge sequence: As you move down a chain of red edges, their value gets halved compared to the previous one. This reflects the diminishing advantage of having more red edges as the opponent can cut them away.
• Summing values: The final surreal number score for a position is calculated by summing the values of all blue edges minus the sum of the values of all red edges.

Therefore, with only red and blue edges, surreal numbers can effectively model the scoring in hackenbush. The system leverages the core strengths of surreal numbers – representing winnable positions and their "size" – to assign a score to a specific arrangement of red and blue edges.

In this equivalency between red-green hackenbush (without green), is the addition of surreal arithmetic equivalent to placing two hackenbush positions side-by-side?

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No, adding surreal numbers in red-blue Hackenbush (without green) isn't always equivalent to placing two hackenbush positions side-by-side. While there's a connection, it's not a perfect match.

Here's why:

• Surreal numbers capture winning potential: In red-blue Hackenbush, surreal numbers represent the combined winnable positions for a player, considering both the individual positions.
• Side-by-side placement creates options: Placing two positions side-by-side essentially creates a new game where the first player can choose to move on either side. This introduces additional winning possibilities compared to the individual positions.

Let's look at an example:

• Position A: Two red edges and one blue edge (surreal number value: 1 - (1/2) - (1/4) = 3/4)
• Position B: One red edge and two blue edges (surreal number value: 2 - (1/2) = 3/2)
• Adding surreal numbers: 3/4 + 3/2 = 9/4.
• Side-by-side combination: The new position offers the first player a choice to cut a red edge from either side. This might create new winning opportunities not reflected in the sum (3/4 + 3/2).

Similarities do exist:

• Despite the differences, there can sometimes be a match between adding surreal numbers and combining positions. This is because some combined positions might not offer the first-mover any significant advantage, and the available winning opportunities become essentially the sum of the original positions.

Overall, surreal number addition provides a theoretical framework to understand the winning potential in combined red-blue Hackenbush positions. However, it doesn't perfectly capture the strategic complexities that arise due to the first-mover choice in the combined game.

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