ARC Prize question, GSL Answer

ChatGPT Use cases and examples
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Use GSL to solve this please

To solve this task using GSL, we analyze the problem’s visual logic and deduce the transformation rule.

Observed Pattern:

  1. Input Grid to Output Grid Transformation:
  • The blue squares from the input are preserved.
  • New blue squares are added based on a specific pattern.
  • The pattern seems to extend blue squares along their respective rows and columns, up to the grid boundaries or adjacent blue squares.
  1. Rules to Transform:
  • If a blue square exists at a given position, it “extends” outward to adjacent empty squares horizontally and vertically.
  • This does not overwrite already-colored blue squares.

GSL Representation of Observed Pattern:

States of Reality:

  • χc (Closure): Ensures transformation adheres to grid boundaries.
  • πρ (Proximity): Extension only affects directly adjacent squares.
  • κτ (Continuity): Continuous propagation until the boundary or block.
  • σι (Similarity): Newly added squares match the color (blue).

GSL Transformation Logic:

  1. χc: Constrain changes within the 7x7 grid.
  2. πρ: Modify (ξ) adjacent squares (ρ indicates adjacency).
  3. κτ: Continue extension logically without skipping steps.
  4. σι: Ensure the new color aligns with existing blue.

Applying the GSL Solution to Test Grid:

  1. Start with the initial blue squares.
  2. For each blue square:
  • Check adjacent positions (above, below, left, right).
  • If an adjacent position is empty, color it blue.
  1. Repeat this process until no new squares can be colored.

Would you like me to generate a precise step-by-step transformation or an image of the completed output grid?

…(more prompts back and forth)… ending response

Let’s reexamine the provided examples and carefully analyze the transformation details to ensure the correct output. I’ll focus on the smaller details as emphasized.

Observations from Examples:

  1. Ex. 1:
  • Input: Two clusters of blue squares.
  • Output: Each cluster has been expanded by one square in specific directions, but not symmetrically.
  • Key Details:
    • Only one square per cluster was added.
    • The new square appears adjacent to an existing square but not always aligned horizontally/vertically.
  1. Ex. 2:
  • Input: Three clusters of blue squares.
  • Output: Each cluster again expands by one square in a specific direction.
  • Key Details:
    • The added square per cluster is not symmetrical and appears to follow a specific propagation rule.

Hypothesis:

The transformation rule involves:

  • Adding exactly one new square per cluster.
  • The new square appears adjacent to an existing square in the cluster.
  • The direction of the addition appears biased towards balancing the clusters geometrically or filling gaps.

Correct GSL Sequence:

  1. Focus on Clusters: Identify clusters of blue (B) squares:

scss

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φ(Clusters)
  1. Proximity Identification: Identify all potential empty cells adjacent to each cluster:

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πρ
  1. Cluster Expansion: For each cluster, add one square to an adjacent empty cell:

css

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ξ(Cluster → +1(B))
  1. Ensure Constraints: Keep changes confined to the grid:

r

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χc
  1. Final State: Finalize the transformation:

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κτ

Final Sequence:

scss

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φ(Clusters) ⊕ πρ ⊕ ξ(Cluster → +1(B)) ⊕ χc ⊕ κτ

Correct Output for Test Input:

css

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. . . B B . .
. . B . B . .
. . . . B . .
. . B B . . .
. . . B . . .
B . . B . . .
B . . . . . .

Explanation:

  1. Cluster Behavior: Each distinct cluster expanded by exactly one square.
  2. Directional Bias: The new squares fill adjacent empty cells to maintain geometric balance.
  3. Grid Confinement: No propagation extends beyond the 7x7 grid.

This solution reflects the smaller details from the examples while ensuring the transformation logic is applied systematically.