006 • History

puzzle_id:

4-0d3d703e

description:

test input

timestamp:

24.356.195518

call_count:

6

Begin puzzle: 4-0d3d703e

example_1

input

[[3 1 2]
 [3 1 2]
 [3 1 2]]

output

[[4 5 6]
 [4 5 6]
 [4 5 6]]

images

../../../../_images/000-example_1_input17.png ../../../../_images/000-example_1_output17.png

observations

The user wants me to analyze the first example of an ARC task. I need to describe the input and output grids, use code to analyze them, and then formulate a natural language program that explains the transformation.

Plan:

  1. Describe the input grid.

  2. Describe the output grid.

  3. Use code to analyze the input and output grids (size, colors, differences).

  4. Formulate a natural language program for the transformation.

Okay, let’s analyze the first example.

The input grid is a 3x3 grid. The first column is green, the second column is blue, and the third column is red.

The output grid is also a 3x3 grid. The first column is yellow, the second column is gray, and the third column is magenta.

input:
  width: 3
  height: 3
  colors:
    3: 3 (green)
    1: 3 (blue)
    2: 3 (red)
  objects:
    - size: 3, position: column 1, color: green
    - size: 3, position: column 2, color: blue
    - size: 3, position: column 3, color: red

output:
  width: 3
  height: 3
  colors:
    4: 3 (yellow)
    5: 3 (gray)
    6: 3 (magenta)
  objects:
    - size: 3, position: column 1, color: yellow
    - size: 3, position: column 2, color: gray
    - size: 3, position: column 3, color: magenta

Now let’s use code to identify the differences.

def analyze_grid(grid):
    height = len(grid)
    width = len(grid[0])
    colors = {}
    for row in grid:
        for cell in row:
            colors[cell] = colors.get(cell, 0) + 1
    return {"width": width, "height": height, "colors": colors}

input_grid = [[3, 1, 2], [3, 1, 2], [3, 1, 2]]
output_grid = [[4, 5, 6], [4, 5, 6], [4, 5, 6]]

input_analysis = analyze_grid(input_grid)
output_analysis = analyze_grid(output_grid)

print("Input analysis:", input_analysis)
print("Output analysis:", output_analysis)

differences = {}
if input_analysis['width'] != output_analysis['width'] or input_analysis['height'] != output_analysis['height']:
    differences['size_change'] = 'yes'
else:
    differences['size_change'] = 'no'

changed_cells = 0
color_changes = {}
for r in range(input_analysis['height']):
    for c in range(input_analysis['width']):
        if input_grid[r][c] != output_grid[r][c]:
            changed_cells += 1
            old_color = input_grid[r][c]
            new_color = output_grid[r][c]
            if old_color not in color_changes:
                color_changes[old_color] = new_color

differences['cells_changed'] = changed_cells
differences['color_changes'] = color_changes

print("Differences:", differences)

differences:
  cells_changed: 9
  colors_changed: {3: 4, 1: 5, 2: 6}
  transformation:
    - It looks like each color in the input is consistently mapped to a new color in the output.
    - Green (3) maps to Yellow (4).
    - Blue (1) maps to Gray (5).
    - Red (2) maps to Magenta (6).
    - The grid dimensions remain the same.

Natural language program:

The input grid is a 3x3 grid with three distinct colors arranged in columns: green in the first, blue in the second, and red in the third. The output grid has the same dimensions. To generate the output grid, iterate through each pixel in the input grid. If the pixel is green, change its color to yellow. If the pixel is blue, change its color to gray. If the pixel is red, change its color to magenta. The positions of the colors remain the same; only the colors themselves are changed according to this mapping.

example_2

input

[[2 3 8]
 [2 3 8]
 [2 3 8]]

output

[[6 4 9]
 [6 4 9]
 [6 4 9]]

images

../../../../_images/001-example_2_input17.png ../../../../_images/001-example_2_output17.png

observations

The user wants me to analyze the second example of the ARC task. I need to follow the same steps as before: describe the input and output grids, use code to analyze them, and formulate a natural language program for the transformation.

Plan:

  1. Describe the input grid.

  2. Describe the output grid.

  3. Use code to analyze the input and output grids.

  4. Formulate a natural language program for the transformation.

Okay, let’s analyze the second example.

The input grid is a 3x3 grid. The first column is red, the second column is green, and the third column is azure.

The output grid is also a 3x3 grid. The first column is magenta, the second column is yellow, and the third column is maroon.

input:
  width: 3
  height: 3
  colors:
    2: 3 (red)
    3: 3 (green)
    8: 3 (azure)
  objects:
    - size: 3, position: column 1, color: red
    - size: 3, position: column 2, color: green
    - size: 3, position: column 3, color: azure

output:
  width: 3
  height: 3
  colors:
    6: 3 (magenta)
    4: 3 (yellow)
    9: 3 (maroon)
  objects:
    - size: 3, position: column 1, color: magenta
    - size: 3, position: column 2, color: yellow
    - size: 3, position: column 3, color: maroon

Now let’s use code to identify the differences.

def analyze_grid(grid):
    height = len(grid)
    width = len(grid[0])
    colors = {}
    for row in grid:
        for cell in row:
            colors[cell] = colors.get(cell, 0) + 1
    return {"width": width, "height": height, "colors": colors}

input_grid = [[2, 3, 8], [2, 3, 8], [2, 3, 8]]
output_grid = [[6, 4, 9], [6, 4, 9], [6, 4, 9]]

input_analysis = analyze_grid(input_grid)
output_analysis = analyze_grid(output_grid)

print("Input analysis:", input_analysis)
print("Output analysis:", output_analysis)

differences = {}
if input_analysis['width'] != output_analysis['width'] or input_analysis['height'] != output_analysis['height']:
    differences['size_change'] = 'yes'
else:
    differences['size_change'] = 'no'

changed_cells = 0
color_changes = {}
for r in range(input_analysis['height']):
    for c in range(input_analysis['width']):
        if input_grid[r][c] != output_grid[r][c]:
            changed_cells += 1
            old_color = input_grid[r][c]
            new_color = output_grid[r][c]
            if old_color not in color_changes:
                color_changes[old_color] = new_color

differences['cells_changed'] = changed_cells
differences['color_changes'] = color_changes

print("Differences:", differences)

differences:
  cells_changed: 9
  colors_changed: {2: 6, 3: 4, 8: 9}
  transformation:
    - It appears each color in the input maps consistently to a new color in the output.
    - Red (2) maps to Magenta (6).
    - Green (3) maps to Yellow (4).
    - Azure (8) maps to Maroon (9).
    - The grid dimensions remain the same.

Natural language program:

The input grid is a 3x3 grid with three distinct colors arranged in columns: red in the first, green in the second, and azure in the third. The output grid has the same dimensions. To generate the output grid, iterate through each pixel in the input grid. If the pixel is red, change its color to magenta. If the pixel is green, change its color to yellow. If the pixel is azure, change its color to maroon. The positions of the colors remain the same; only the colors themselves are changed according to this mapping.

example_3

input

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

output

[[1 9 2]
 [1 9 2]
 [1 9 2]]

images

../../../../_images/002-example_3_input12.png ../../../../_images/002-example_3_output12.png

observations

The user wants me to analyze the third example of the ARC task and formulate a natural language program for the transformation. I need to describe the input and output grids, use code to analyze them, and then describe the transformation.

Plan:

  1. Describe the input grid.

  2. Describe the output grid.

  3. Use code to analyze the input and output grids.

  4. Formulate a natural language program for the transformation.

example_3

input

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

output

[[1 9 2]
 [1 9 2]
 [1 9 2]]

images

import matplotlib.pyplot as plt
import numpy as np

def show_grid(grid, title):
    cmap = plt.cm.colors.ListedColormap([(0.93, 0.93, 0.93), (0.12, 0.58, 1.00), (0.86, 0.20, 0.16), (0.31, 0.80, 0.19), (0.90, 0.78, 0.00), (0.33, 0.33, 0.33), (0.90, 0.23, 0.64), (0.90, 0.47, 0.08), (0.53, 0.85, 0.94), (0.57, 0.07, 0.19)])
    bounds = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
    norm = plt.cm.colors.BoundaryNorm(bounds, cmap.N)

    data = np.array(grid)
    fig, ax = plt.subplots()
    ax.imshow(data, cmap=cmap, norm=norm)

    ax.grid(which='major', axis='both', linestyle='-', color='k', linewidth=2)
    ax.set_xticks(np.arange(-.5, len(grid[0])-.5, 1))
    ax.set_yticks(np.arange(-.5, len(grid)-.5, 1))
    ax.set_xticklabels([])
    ax.set_yticklabels([])
    ax.set_title(title)
    for i in range(data.shape[0]):
        for j in range(data.shape[1]):
            ax.text(j, i, str(data[i, j]), ha="center", va="center", color="black")
    plt.show()

input_grid = [[5, 8, 6], [5, 8, 6], [5, 8, 6]]
output_grid = [[1, 9, 2], [1, 9, 2], [1, 9, 2]]

show_grid(input_grid, "Input Grid")
show_grid(output_grid, "Output Grid")

observations

The input grid is a 3x3 grid. The first column is gray, the second column is azure, and the third column is magenta.

The output grid is also a 3x3 grid. The first column is blue, the second column is maroon, and the third column is red.

input:
  width: 3
  height: 3
  colors:
    5: 3 (gray)
    8: 3 (azure)
    6: 3 (magenta)
  objects:
    - size: 3, position: column 1, color: gray
    - size: 3, position: column 2, color: azure
    - size: 3, position: column 3, color: magenta

output:
  width: 3
  height: 3
  colors:
    1: 3 (blue)
    9: 3 (maroon)
    2: 3 (red)
  objects:
    - size: 3, position: column 1, color: blue
    - size: 3, position: column 2, color: maroon
    - size: 3, position: column 3, color: red

def analyze_grid(grid):
    height = len(grid)
    width = len(grid[0])
    colors = {}
    for row in grid:
        for cell in row:
            colors[cell] = colors.get(cell, 0) + 1
    return {"width": width, "height": height, "colors": colors}

input_grid = [[5, 8, 6], [5, 8, 6], [5, 8, 6]]
output_grid = [[1, 9, 2], [1, 9, 2], [1, 9, 2]]

input_analysis = analyze_grid(input_grid)
output_analysis = analyze_grid(output_grid)

print("Input analysis:", input_analysis)
print("Output analysis:", output_analysis)

differences = {}
if input_analysis['width'] != output_analysis['width'] or input_analysis['height'] != output_analysis['height']:
    differences['size_change'] = 'yes'
else:
    differences['size_change'] = 'no'

changed_cells = 0
color_changes = {}
for r in range(input_analysis['height']):
    for c in range(input_analysis['width']):
        if input_grid[r][c] != output_grid[r][c]:
            changed_cells += 1
            old_color = input_grid[r][c]
            new_color = output_grid[r][c]
            if old_color not in color_changes:
                color_changes[old_color] = new_color

differences['cells_changed'] = changed_cells
differences['color_changes'] = color_changes

print("Differences:", differences)

differences:
  cells_changed: 9
  colors_changed: {5: 1, 8: 9, 6: 2}
  transformation:
    - Each color in the input maps consistently to a new color in the output.
    - Gray (5) maps to Blue (1).
    - Azure (8) maps to Maroon (9).
    - Magenta (6) maps to Red (2).
    - The grid dimensions remain the same.

Natural language program:

The input grid is a 3x3 grid with three distinct colors arranged in columns: gray in the first, azure in the second, and magenta in the third. The output grid has the same dimensions. To generate the output grid, iterate through each pixel in the input grid. If the pixel is gray, change its color to blue. If the pixel is azure, change its color to maroon. If the pixel is magenta, change its color to red. The positions of the colors remain the same; only the colors themselves are changed according to this mapping.

example_4

input

[[9 4 2]
 [9 4 2]
 [9 4 2]]

output

[[8 3 6]
 [8 3 6]
 [8 3 6]]

images

../../../../_images/003-example_4_input2.png ../../../../_images/003-example_4_output2.png

observations

The user wants me to analyze the fourth example of the ARC task and formulate a natural language program for the transformation. I need to describe the input and output grids, use code to analyze them, and then describe the transformation.

Plan:

  1. Describe the input grid.

  2. Describe the output grid.

  3. Use code to analyze the input and output grids.

  4. Formulate a natural language program for the transformation.

example_4

input

[[9 4 2]
 [9 4 2]
 [9 4 2]]

output

[[8 3 6]
 [8 3 6]
 [8 3 6]]

images

import matplotlib.pyplot as plt
import numpy as np

def show_grid(grid, title):
    cmap = plt.cm.colors.ListedColormap([(0.93, 0.93, 0.93), (0.12, 0.58, 1.00), (0.86, 0.20, 0.16), (0.31, 0.80, 0.19), (0.90, 0.78, 0.00), (0.33, 0.33, 0.33), (0.90, 0.23, 0.64), (0.90, 0.47, 0.08), (0.53, 0.85, 0.94), (0.57, 0.07, 0.19)])
    bounds = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
    norm = plt.cm.colors.BoundaryNorm(bounds, cmap.N)

    data = np.array(grid)
    fig, ax = plt.subplots()
    ax.imshow(data, cmap=cmap, norm=norm)

    ax.grid(which='major', axis='both', linestyle='-', color='k', linewidth=2)
    ax.set_xticks(np.arange(-.5, len(grid[0])-.5, 1))
    ax.set_yticks(np.arange(-.5, len(grid)-.5, 1))
    ax.set_xticklabels([])
    ax.set_yticklabels([])
    ax.set_title(title)
    for i in range(data.shape[0]):
        for j in range(data.shape[1]):
            ax.text(j, i, str(data[i, j]), ha="center", va="center", color="black")
    plt.show()

input_grid = [[9, 4, 2], [9, 4, 2], [9, 4, 2]]
output_grid = [[8, 3, 6], [8, 3, 6], [8, 3, 6]]

show_grid(input_grid, "Input Grid")
show_grid(output_grid, "Output Grid")

observations

The input grid is a 3x3 grid. The first column is maroon, the second column is yellow, and the third column is red.

The output grid is also a 3x3 grid. The first column is azure, the second column is green, and the third column is magenta.

input:
  width: 3
  height: 3
  colors:
    9: 3 (maroon)
    4: 3 (yellow)
    2: 3 (red)
  objects:
    - size: 3, position: column 1, color: maroon
    - size: 3, position: column 2, color: yellow
    - size: 3, position: column 3, color: red

output:
  width: 3
  height: 3
  colors:
    8: 3 (azure)
    3: 3 (green)
    6: 3 (magenta)
  objects:
    - size: 3, position: column 1, color: azure
    - size: 3, position: column 2, color: green
    - size: 3, position: column 3, color: magenta

def analyze_grid(grid):
    height = len(grid)
    width = len(grid[0])
    colors = {}
    for row in grid:
        for cell in row:
            colors[cell] = colors.get(cell, 0) + 1
    return {"width": width, "height": height, "colors": colors}

input_grid = [[9, 4, 2], [9, 4, 2], [9, 4, 2]]
output_grid = [[8, 3, 6], [8, 3, 6], [8, 3, 6]]

input_analysis = analyze_grid(input_grid)
output_analysis = analyze_grid(output_grid)

print("Input analysis:", input_analysis)
print("Output analysis:", output_analysis)

differences = {}
if input_analysis['width'] != output_analysis['width'] or input_analysis['height'] != output_analysis['height']:
    differences['size_change'] = 'yes'
else:
    differences['size_change'] = 'no'

changed_cells = 0
color_changes = {}
for r in range(input_analysis['height']):
    for c in range(input_analysis['width']):
        if input_grid[r][c] != output_grid[r][c]:
            changed_cells += 1
            old_color = input_grid[r][c]
            new_color = output_grid[r][c]
            if old_color not in color_changes:
                color_changes[old_color] = new_color

differences['cells_changed'] = changed_cells
differences['color_changes'] = color_changes

print("Differences:", differences)

differences:
  cells_changed: 9
  colors_changed: {9: 8, 4: 3, 2: 6}
  transformation:
    - Each color in the input maps consistently to a new color in the output.
    - Maroon (9) maps to Azure (8).
    - Yellow (4) maps to Green (3).
    - Red (2) maps to Magenta (6).
    - The grid dimensions remain the same.

Natural language program:

The input grid is a 3x3 grid with three distinct colors arranged in columns: maroon in the first, yellow in the second, and red in the third. The output grid has the same dimensions. To generate the output grid, iterate through each pixel in the input grid. If the pixel is maroon, change its color to azure. If the pixel is yellow, change its color to green. If the pixel is red, change its color to magenta. The positions of the colors remain the same; only the colors themselves are changed according to this mapping.

examples summary

The user wants me to analyze all the examples and come up with a general rule for the transformation. Looking at the individual examples, it seems like a direct color mapping is happening. I need to collect all the mappings and see if there is a consistent rule.

Plan:

  1. Collect the color mappings from each example.

  2. Check for consistency in the mappings.

  3. Formulate a general natural language program.

Ruminate Phase

Review All Examples and Findings

Across all four examples, the input and output grids have the same dimensions (3x3). The core transformation appears to be a direct mapping of colors from the input grid to the output grid. The positions of the colors within the grid remain consistent; only the color values change.

Let’s collect the color mappings observed in each example:

  • Example 1: 3 (green) -> 4 (yellow), 1 (blue) -> 5 (gray), 2 (red) -> 6 (magenta)

  • Example 2: 2 (red) -> 6 (magenta), 3 (green) -> 4 (yellow), 8 (azure) -> 9 (maroon)

  • Example 3: 5 (gray) -> 1 (blue), 8 (azure) -> 9 (maroon), 6 (magenta) -> 2 (red)

  • Example 4: 9 (maroon) -> 8 (azure), 4 (yellow) -> 3 (green), 2 (red) -> 6 (magenta)

By combining these mappings, we can deduce the complete color transformation rule for this task.

color_mappings = {}

# Example 1
color_mappings[3] = 4
color_mappings[1] = 5
color_mappings[2] = 6

# Example 2
color_mappings[2] = 6
color_mappings[3] = 4
color_mappings[8] = 9

# Example 3
color_mappings[5] = 1
color_mappings[8] = 9
color_mappings[6] = 2

# Example 4
color_mappings[9] = 8
color_mappings[4] = 3
color_mappings[2] = 6

print("Combined Color Mappings:", color_mappings)

# Deduce the full mapping, handling potential inconsistencies
full_mapping = {}
for mapping in [
    {3: 4, 1: 5, 2: 6},
    {2: 6, 3: 4, 8: 9},
    {5: 1, 8: 9, 6: 2},
    {9: 8, 4: 3, 2: 6}
]:
    for k, v in mapping.items():
        if k in full_mapping and full_mapping[k] != v:
            print(f"Inconsistency found for color {k}: {full_mapping[k]} vs {v}")
        full_mapping[k] = v

print("Full Color Mapping:", full_mapping)

The code confirms a consistent one-to-one mapping between the colors in the input and output grids across all examples.

Final Natural Language Program:

The transformation rule involves a direct color replacement based on a fixed mapping.

  1. Initialize the output grid: Create an output grid with the same dimensions (height and width) as the input grid.

  2. Color mapping: Apply the following color transformation to each pixel from the input grid to the output grid:

    • If the input pixel is green (3), the output pixel should be yellow (4).

    • If the input pixel is blue (1), the output pixel should be gray (5).

    • If the input pixel is red (2), the output pixel should be magenta (6).

    • If the input pixel is azure (8), the output pixel should be maroon (9).

    • If the input pixel is gray (5), the output pixel should be blue (1).

    • If the input pixel is magenta (6), the output pixel should be red (2).

    • If the input pixel is maroon (9), the output pixel should be azure (8).

    • If the input pixel is yellow (4), the output pixel should be green (3).

  3. Iterate and apply: Iterate through each cell (pixel) of the input grid. For each pixel, determine its color and apply the corresponding color transformation to the same position in the output grid.

  4. The color palette for the output grid consists of yellow, gray, magenta, maroon, blue, and green.

color_map = {
    3: 4,  # green to yellow
    1: 5,  # blue to gray
    2: 6,  # red to magenta
    8: 9,  # azure to maroon
    5: 1,  # gray to blue
    6: 2,  # magenta to red
    9: 8,  # maroon to azure
    4: 3   # yellow to green
}

Final Natural Language Program:

The transformation between the input and output grids is a direct color replacement based on a fixed mapping.

  1. Initialize the output grid: Create an output grid with the exact same dimensions (height and width) as the input grid.

  2. Color mapping: Apply the following color transformation rule to each pixel:

    • Green (3) becomes Yellow (4)

    • Blue (1) becomes Gray (5)

    • Red (2) becomes Magenta (6)

    • Azure (8) becomes Maroon (9)

    • Gray (5) becomes Blue (1)

    • Magenta (6) becomes Red (2)

    • Maroon (9) becomes Azure (8)

    • Yellow (4) becomes Green (3)

  3. Iterate and transform: Go through each pixel in the input grid, one by one. For each pixel, find its color value. Using the color mapping defined above, determine the corresponding output color for that pixel. Place this new color value in the same position (row and column) in the output grid.

  4. The output grid will only contain the colors present in the mapping’s output values: Yellow, Gray, Magenta, Maroon, Blue, and Green.

test

input

[[8 1 3]
 [8 1 3]
 [8 1 3]]

image

../../../../_images/005-test_input2.png

observations

INSTRUCTIONS:

using the knowledge you have gathered from the previous examples and the step by step natural language program

  • predict what the test output should be

  • use code_execution to validate the output

  • make final adjustments

  • submit final output

See also