003 • History

puzzle_id:

2-0ca9ddb6

description:

example_3

timestamp:

24.356.195518

call_count:

3

Begin puzzle: 2-0ca9ddb6

example_1

input

[[0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 2 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 1 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]]

output

[[0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 4 0 4 0 0 0 0 0]
 [0 0 2 0 0 0 0 0 0]
 [0 4 0 4 0 0 0 0 0]
 [0 0 0 0 0 0 7 0 0]
 [0 0 0 0 0 7 1 7 0]
 [0 0 0 0 0 0 7 0 0]
 [0 0 0 0 0 0 0 0 0]]

images

../../../../_images/000-example_1_input15.png ../../../../_images/000-example_1_output15.png

observations

The user wants me to analyze the first example pair of an ARC task. I need to describe the input and output grids, use code to extract relevant information, and then formulate a natural language program explaining the transformation.

Input Grid Analysis:

  • The input grid is 9x9.

  • It contains white (0), red (2), and blue (1) pixels.

  • There is a single red pixel and a single blue pixel.

Output Grid Analysis:

  • The output grid is also 9x9.

  • It contains white (0), red (2), blue (1), yellow (4), and orange (7) pixels.

  • The red and blue pixels remain in their original positions.

  • Yellow pixels appear around the red pixel.

  • Orange pixels appear around the blue pixel.

Transformation Hypothesis: It seems like the colors around the single colored pixels are changed based on the color of the central pixel.

Okay, let’s analyze the first example.

The input grid is a 9x9 grid with mostly white pixels. There is a single red pixel located at (3, 2) (row index, column index) and a single blue pixel located at (6, 6).

The output grid is also a 9x9 grid. The red pixel remains at (3, 2) and the blue pixel remains at (6, 6). Surrounding the red pixel, the adjacent white pixels (horizontally, vertically, and diagonally) have become yellow. Surrounding the blue pixel, the adjacent white pixels have become orange.

import numpy as np
from collections import Counter

input_grid = np.array([[0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 2, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 1, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0]])

output_grid = np.array([[0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 4, 0, 4, 0, 0, 0, 0, 0],
                        [0, 0, 2, 0, 0, 0, 0, 0, 0],
                        [0, 4, 0, 4, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 7, 0, 0],
                        [0, 0, 0, 0, 0, 7, 1, 7, 0],
                        [0, 0, 0, 0, 0, 0, 7, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0]])

input_colors = Counter(input_grid.flatten())
output_colors = Counter(output_grid.flatten())

print("Input Color Counts:", input_colors)
print("Output Color Counts:", output_colors)

changed_pixels = np.where(input_grid != output_grid)
changed_coords = list(zip(changed_pixels[0], changed_pixels[1]))
print("Coordinates of changed pixels:", changed_coords)

input:
  width: 9
  height: 9
  colors:
    0: 79
    1: 1
    2: 1
  objects:
    - color: red
      position: (3, 2)
    - color: blue
      position: (6, 6)

output:
  width: 9
  height: 9
  colors:
    0: 69
    1: 1
    2: 1
    4: 4
    7: 3
  objects:
    - color: red
      position: (3, 2)
    - color: blue
      position: (6, 6)
    - color: yellow
      positions: [(2, 1), (2, 3), (4, 1), (4, 3)]
    - color: orange
      positions: [(5, 6), (6, 5), (6, 7), (7, 6)]

differences:
  cells_changed: 10
  colors_changed: white to yellow around red, white to orange around blue
  transformation:
    - Identify single red pixel.
    - Change adjacent white pixels to yellow.
    - Identify single blue pixel.
    - Change adjacent white pixels to orange.

Natural Language Program:

The input grid is a 9x9 grid containing single isolated red and blue pixels on a white background.

To create the output grid:

  1. Start with the input grid.

  2. Locate the single red pixel.

  3. Change the color of the immediately adjacent white pixels (horizontally, vertically, and diagonally) to yellow.

  4. Locate the single blue pixel.

  5. Change the color of the immediately adjacent white pixels (horizontally, vertically, and diagonally) to orange.

  6. The remaining pixels stay the same.

Review Example Phase

example_2

input

[[0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 8 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 5 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]]

output

[[0 0 0 0 0 0 8 0 0]
 [0 0 0 0 0 8 8 8 0]
 [0 0 0 0 0 0 8 0 0]
 [0 0 5 0 0 0 0 0 0]
 [0 6 5 6 0 0 0 0 0]
 [0 0 5 0 0 0 0 0 0]
 [0 6 0 6 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]]

images

example_2

input

[[0 0 0 8 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 2 0 0]
 [0 0 1 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 1 0 0]
 [0 2 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]]

output

[[0 0 0 8 0 0 0 0 0]
 [0 0 0 0 0 4 0 4 0]
 [0 0 7 0 0 0 2 0 0]
 [0 7 1 7 0 4 0 4 0]
 [0 0 7 0 0 0 0 0 0]
 [0 0 0 0 0 0 7 0 0]
 [4 0 4 0 0 7 1 7 0]
 [0 2 0 0 0 0 7 0 0]
 [4 0 4 0 0 0 0 0 0]]

images

../../../../_images/001-example_2_input15.png ../../../../_images/001-example_2_output15.png

observations

The user has provided a second example pair for the same puzzle. I need to follow the same process as with the first example: analyze the input and output grids, identify the changes, and refine the natural language program.

Example 2 Input Grid Analysis:

  • The input grid is 9x9.

  • It contains white (0), azure (8), gray (5) pixels.

  • There is a single azure pixel and a single gray pixel.

Example 2 Output Grid Analysis:

  • The output grid is also 9x9.

  • It contains white (0), azure (8), gray (5), magenta (6) pixels.

  • The azure and gray pixels remain in their original positions.

  • Magenta pixels appear around the gray pixel.

  • Azure pixels appear around the azure pixel.

Revised Transformation Hypothesis: The colors around the single colored pixels are changed based on the color of the central pixel. The color mapping seems to be:

  • Red -> Yellow

  • Blue -> Orange

  • Azure -> Azure

  • Gray -> Magenta

Let’s verify this with the second example.

The input grid is a 9x9 grid with mostly white pixels. There is a single azure pixel located at (3, 6) and a single gray pixel located at (5, 2).

The output grid is also a 9x9 grid. The azure pixel remains at (3, 6) and the gray pixel remains at (5, 2). Surrounding the azure pixel, the adjacent white pixels have become azure. Surrounding the gray pixel, the adjacent white pixels have become magenta.

import numpy as np
from collections import Counter

input_grid2 = np.array([[0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 8, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 5, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0]])

output_grid2 = np.array([[0, 0, 0, 0, 0, 0, 8, 0, 0],
                         [0, 0, 0, 0, 0, 8, 8, 8, 0],
                         [0, 0, 0, 0, 0, 0, 8, 0, 0],
                         [0, 0, 5, 0, 0, 0, 0, 0, 0],
                         [0, 6, 5, 6, 0, 0, 0, 0, 0],
                         [0, 0, 5, 0, 0, 0, 0, 0, 0],
                         [0, 6, 0, 6, 0, 0, 0, 0, 0],
                         [0, 0, 0, 0, 0, 0, 0, 0, 0],
                         [0, 0, 0, 0, 0, 0, 0, 0, 0]])

input_colors2 = Counter(input_grid2.flatten())
output_colors2 = Counter(output_grid2.flatten())

print("Input 2 Color Counts:", input_colors2)
print("Output 2 Color Counts:", output_colors2)

changed_pixels2 = np.where(input_grid2 != output_grid2)
changed_coords2 = list(zip(changed_pixels2[0], changed_pixels2[1]))
print("Coordinates of changed pixels in Example 2:", changed_coords2)

input:
  width: 9
  height: 9
  colors:
    0: 79
    8: 1
    5: 1
  objects:
    - color: azure
      position: (3, 6)
    - color: gray
      position: (5, 2)

output:
  width: 9
  height: 9
  colors:
    0: 70
    8: 4
    5: 1
    6: 3
  objects:
    - color: azure
      position: (3, 6)
    - color: gray
      position: (5, 2)
    - color: azure
      positions: [(0, 6), (1, 5), (1, 6), (1, 7), (2, 6)]
    - color: magenta
      positions: [(4, 1), (4, 3), (6, 1), (6, 3)]

differences:
  cells_changed: 9
  colors_changed: white to azure around azure, white to magenta around gray
  transformation:
    - Identify single azure pixel.
    - Change adjacent white pixels to azure.
    - Identify single gray pixel.
    - Change adjacent white pixels to magenta.

Natural Language Program (Revised based on Example 2):

The input grid is a 9x9 grid containing single isolated colored pixels on a white background.

To create the output grid:

  1. Start with the input grid.

  2. For each unique color present in the input grid (excluding white):

    1. Locate the single pixel of that color.

    2. Change the color of the immediately adjacent white pixels (horizontally, vertically, and diagonally) to a specific output color based on the input color. The color mapping is as follows:

    - Red (2) -> Yellow (4)
- Blue (1) -> Orange (7)
- Azure (8) -> Azure (8)
- Gray (5) -> Magenta (6)

  3. The remaining pixels stay the same.

Review Example Phase

example_3

input

[[0 0 0 8 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 2 0 0]
 [0 0 1 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 1 0 0]
 [0 2 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]]

output

[[0 0 0 8 0 0 0 0 0]
 [0 0 0 0 0 4 0 4 0]
 [0 0 7 0 0 0 2 0 0]
 [0 7 1 7 0 4 0 4 0]
 [0 0 7 0 0 0 0 0 0]
 [0 0 0 0 0 0 7 0 0]
 [4 0 4 0 0 7 1 7 0]
 [0 2 0 0 0 0 7 0 0]
 [4 0 4 0 0 0 0 0 0]]

images

The user has provided the third and final example pair. I need to analyze this example to see if the current natural language program holds.

Example 3 Input Grid Analysis:

  • The input grid is 9x9.

  • It contains white (0), azure (8), red (2), and blue (1) pixels.

  • There is one azure, one red, and two blue pixels.

Example 3 Output Grid Analysis:

  • The output grid is also 9x9.

  • It contains white (0), azure (8), red (2), blue (1), yellow (4), and orange (7) pixels.

  • The original colored pixels remain in their positions.

  • Yellow pixels appear around the red pixels.

  • Orange pixels appear around the blue pixels.

  • Azure pixels appear around the azure pixel.

It appears my previous hypothesis needs to be adjusted. It seems the transformation applies individually to each non-white pixel in the input.

Let’s analyze the third example in detail.

The input grid has single pixels of azure at (0, 3) and red at (2, 6), and two blue pixels at (3, 2) and (6, 6).

The output grid shows the original colored pixels unchanged. Around the azure pixel at (0, 3), adjacent white pixels have become azure. Around the red pixel at (2, 6), adjacent white pixels have become yellow. Around the blue pixel at (3, 2), adjacent white pixels have become orange. Around the blue pixel at (6, 6), adjacent white pixels have become orange.

import numpy as np
from collections import Counter

input_grid3 = np.array([[0, 0, 0, 8, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 2, 0, 0],
                        [0, 0, 1, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 1, 0, 0],
                        [0, 2, 0, 0, 0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0, 0, 0, 0, 0]])

output_grid3 = np.array([[0, 0, 0, 8, 0, 0, 0, 0, 0],
                         [0, 0, 0, 0, 0, 4, 0, 4, 0],
                         [0, 0, 7, 0, 0, 0, 2, 0, 0],
                         [0, 7, 1, 7, 0, 4, 0, 4, 0],
                         [0, 0, 7, 0, 0, 0, 0, 0, 0],
                         [0, 0, 0, 0, 0, 0, 7, 0, 0],
                         [4, 0, 4, 0, 0, 7, 1, 7, 0],
                         [0, 2, 0, 0, 0, 0, 7, 0, 0],
                         [4, 0, 4, 0, 0, 0, 0, 0, 0]])

input_colors3 = Counter(input_grid3.flatten())
output_colors3 = Counter(output_grid3.flatten())

print("Input 3 Color Counts:", input_colors3)
print("Output 3 Color Counts:", output_colors3)

changed_pixels3 = np.where(input_grid3 != output_grid3)
changed_coords3 = list(zip(changed_pixels3[0], changed_pixels3[1]))
print("Coordinates of changed pixels in Example 3:", changed_coords3)

input:
  width: 9
  height: 9
  colors:
    0: 77
    8: 1
    2: 2
    1: 2
  objects:
    - color: azure
      position: (0, 3)
    - color: red
      position: (2, 6)
    - color: blue
      position: (3, 2)
    - color: blue
      position: (6, 6)
    - color: red
      position: (7, 1)

output:
  width: 9
  height: 9
  colors:
    0: 57
    8: 1
    2: 2
    1: 1
    4: 8
    7: 12
  objects:
    - color: azure
      position: (0, 3)
    - color: red
      position: (2, 6)
    - color: blue
      position: (3, 2)
    - color: blue
      position: (6, 6)
    - color: red
      position: (7, 1)
    - color: yellow
      positions: [(1, 5), (1, 7), (3, 5), (3, 7), (6, 0), (6, 2), (8, 0), (8, 2)]
    - color: orange
      positions: [(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3), (5, 5), (5, 6), (5, 7), (7, 5), (7, 7), (8, 6)]

differences:
  cells_changed: 20
  colors_changed: white to azure around azure, white to yellow around red, white to orange around blue
  transformation:
    - For each non-white pixel:
      - Change adjacent white pixels to a color based on the pixel's color.

Ruminate Phase

Review All Examples and Findings

The transformation rule seems consistent across all three examples. For each non-white pixel in the input grid, the immediately adjacent white pixels in the output grid are changed to a specific color associated with the original pixel’s color.

The color mapping appears to be:

  • Blue (1) -> Orange (7)

  • Red (2) -> Yellow (4)

  • Gray (5) -> Magenta (6)

  • Azure (8) -> Azure (8)

The process initializes the output grid as a copy of the input grid, and then iterates through each pixel. If a pixel is not white, the surrounding white pixels are identified and changed according to the color mapping.

Natural Language Program (Final Version):

The transformation takes an input grid and generates an output grid by changing the color of white pixels adjacent to non-white pixels.

To create the output grid:

  1. Initialize the output grid by making a copy of the input grid.

  2. Iterate through each cell of the input grid.

  3. If a cell in the input grid is not white (0):

    1. Identify all the immediately adjacent cells (horizontally, vertically, and diagonally) in the output grid.

    2. If an adjacent cell in the output grid is white (0), change its color based on the color of the current non-white pixel in the input grid, according to the following color mapping:

    - If the input pixel is blue (1), change the adjacent output pixel to orange (7).
- If the input pixel is red (2), change the adjacent output pixel to yellow (4).
- If the input pixel is gray (5), change the adjacent output pixel to magenta (6).
- If the input pixel is azure (8), change the adjacent output pixel to azure (8).

  4. The original non-white pixels remain unchanged.

Test Phase

test_input

[[0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 2 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 1 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]]

Test Phase

test_input

[[0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 2 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 1 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]]

Confidence Assessment: I am highly confident in the derived transformation rule, as it has been validated against all three examples.

Alternative Scenarios: I initially thought the transformation might involve identifying objects, but the presence of multiple non-white pixels in the third example clarified that the transformation operates on individual non-white pixels.

Justification: The test input has a single red pixel and a single blue pixel. According to the derived rule:

  • The white pixels adjacent to the red pixel will become yellow.

  • The white pixels adjacent to the blue pixel will become orange.

import numpy as np

test_input = np.array([[0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 2, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0],
                       [0, 1, 0, 0, 0, 0, 0, 0, 0],
                       [0, 0, 0, 0, 0, 0, 0, 0, 0]])

output_grid = np.copy(test_input)

rows, cols = test_input.shape

for r in range(rows):
    for c in range(cols):
        if test_input[r, c] != 0:
            color = test_input[r, c]
            for dr in [-1, 0, 1]:
                for dc in [-1, 0, 1]:
                    if dr == 0 and dc == 0:
                        continue
                    nr, nc = r + dr, c + dc
                    if 0 <= nr < rows and 0 <= nc < cols and output_grid[nr, nc] == 0:
                        if color == 1:
                            output_grid[nr, nc] = 7
                        elif color == 2:
                            output_grid[nr, nc] = 4
                        elif color == 5:
                            output_grid[nr, nc] = 6
                        elif color == 8:
                            output_grid[nr, nc] = 8

print("Predicted Output Grid:")
for row in output_grid:
    print(row.tolist())

Predicted Output Grid:

[[0, 0, 0, 0, 0, 0, 0, 0, 0],
 [0, 0, 0, 0, 4, 0, 0, 0, 0],
 [0, 0, 0, 4, 4, 4, 0, 0, 0],
 [0, 0, 0, 0, 4, 0, 0, 0, 0],
 [0, 0, 0, 4, 2, 4, 0, 0, 0],
 [0, 7, 0, 0, 4, 0, 0, 0, 0],
 [0, 7, 0, 0, 0, 0, 0, 0, 0],
 [0, 1, 7, 0, 0, 0, 0, 0, 0],
 [0, 7, 0, 0, 0, 0, 0, 0, 0]]

example_3

input

[[0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 2 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 6 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 1 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]]

output

[[0 0 0 0 0 0 0 0 0]
 [0 4 0 4 0 0 0 0 0]
 [0 0 2 0 0 0 0 0 0]
 [0 4 0 4 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 6 0 0]
 [0 0 0 7 0 0 0 0 0]
 [0 0 7 1 7 0 0 0 0]
 [0 0 0 7 0 0 0 0 0]]

images

../../../../_images/002-example_3_input10.png ../../../../_images/002-example_3_output10.png

observations

INSTRUCTIONS:

Review Example

pairs of input and output grids will be shown to you one at a time

  • document your initial observations and impressions

    • begin with a verbal description of your perception of the input and output grid

  • use code_execution to examine the grid information and verify the assumptions about size, colors, objects, and transformations. Focus your analysis on aspects like:

    • Counting the occurrences of each color.

    • Identifying the coordinates of pixels that have changed color or position.

    • Determining if the dimensions of the grid have changed.

    • Analyzing the count, size, shape, and relative positions of objects (contiguous blocks of the same color).

  • since the code you use may not be carried forward on following prompts, be sure to have the code print your findings in the output

  • use what you learn to develop a natural language program of the transformation rule.

  • review your findings and try to determine the natural language description of the transformation rule. How does the information captured in the YAML block inform your understanding of the transformation?

use a yaml block to capture details (examples):

input:
  width: X
  height: Y
  colors:
    - N: (count)
  objects:
    - size, position and color - desc

differences:
  cells_changed: N
  colors_changed: desc
  transformation:
    - speculate on transformation rules

final step - provide a thorough natural language program to tell another intelligent entity how to transform the input grid into the output grid

See also