005 • History

puzzle_id:: 3-1e0a9b12
description:: test input
timestamp:: 24.357.081407
call_count:: 5

Begin puzzle: 3-1e0a9b12

example_1

input

[[0 4 0 9]
 [0 0 0 0]
 [0 4 6 0]
 [1 0 0 0]]

output

[[0 0 0 0]
 [0 0 0 0]
 [0 4 0 0]
 [1 4 6 9]]

images

observations

The user wants me to analyze an input-output pair from an ARC task and describe the transformation rule in natural language.

First, I will describe the input and output grids visually. Then, I will use code to analyze the grids and identify the differences and potential transformation rules. Finally, I will formulate a natural language program based on my findings.

Okay, let’s analyze the provided example.

The input grid is a 4x4 grid with white (0), yellow (4), maroon (9), blue (1), and magenta (6) pixels. The yellow pixels form two separate objects. The output grid is also a 4x4 grid with the same color palette. It appears some pixels have moved or changed colors.

import numpy as np
from collections import Counter

input_grid = np.array([[0, 4, 0, 9],
                       [0, 0, 0, 0],
                       [0, 4, 6, 0],
                       [1, 0, 0, 0]])

output_grid = np.array([[0, 0, 0, 0],
                        [0, 0, 0, 0],
                        [0, 4, 0, 0],
                        [1, 4, 6, 9]])

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

print("Input Grid:")
print(input_grid)
print("\nOutput Grid:")
print(output_grid)

print("\nInput Color Counts:")
print(input_colors)
print("\nOutput Color Counts:")
print(output_colors)

input_width = input_grid.shape[1]
input_height = input_grid.shape[0]
output_width = output_grid.shape[1]
output_height = output_grid.shape[0]

print(f"\nInput Grid Dimensions: {input_width}x{input_height}")
print(f"Output Grid Dimensions: {output_width}x{output_height}")

changed_pixels = []
for y in range(input_height):
    for x in range(input_width):
        if input_grid[y, x] != output_grid[y, x]:
            changed_pixels.append(((y, x), input_grid[y, x], output_grid[y, x]))

print("\nChanged Pixels ( (row, col), old_color, new_color ):")
for pixel in changed_pixels:
    print(f"  {pixel}")

input:
  width: 4
  height: 4
  colors:
    0: 10
    4: 2
    9: 1
    1: 1
    6: 1
  objects:
    - size: 1, position: (0, 1), color: yellow
    - size: 1, position: (2, 1), color: yellow
    - size: 1, position: (0, 3), color: maroon
    - size: 1, position: (3, 0), color: blue
    - size: 1, position: (2, 2), color: magenta

output:
  width: 4
  height: 4
  colors:
    0: 10
    4: 2
    9: 1
    1: 1
    6: 1
  objects:
    - size: 1, position: (2, 1), color: yellow
    - size: 1, position: (3, 1), color: yellow
    - size: 1, position: (3, 3), color: maroon
    - size: 1, position: (3, 0), color: blue
    - size: 1, position: (3, 2), color: magenta

differences:
  cells_changed: 4
  colors_changed:
    - yellow: moved from (0, 1) to (3, 1)
    - maroon: moved from (0, 3) to (3, 3)
    - magenta: moved from (2, 2) to (3, 2)
    - white: changed to white at (0, 1), (0, 3), (2, 2)
    - white: changed from white at (3, 1), (3, 3), (3, 2)
  transformation:
    - Non-white pixels in the input appear to have moved to the bottom row of the output.
    - The order of the non-white pixels in the bottom row of the output seems to correspond to their column position in the input.

Natural Language Program:

Context: The input is a 4x4 grid containing pixels of white, yellow, maroon, blue, and magenta. The non-white pixels appear as single, disconnected objects.

Process:

Initialize an empty output grid of the same dimensions as the input grid, filled with white pixels.
Identify all the non-white pixels in the input grid along with their colors and column positions.
Place these non-white pixels in the last row of the output grid, maintaining their relative order based on their original column position in the input. The leftmost non-white pixel in the input (smallest column index) should be placed in the leftmost available cell in the last row of the output, and so on.
Any remaining cells in the output grid will remain white.

example_2

input

[[0 0 0 0 0 9]
 [0 0 0 8 0 0]
 [0 0 0 0 0 0]
 [4 0 0 0 0 0]
 [4 0 7 8 0 0]
 [4 0 7 0 0 0]]

output

[[0 0 0 0 0 0]
 [0 0 0 0 0 0]
 [0 0 0 0 0 0]
 [4 0 0 0 0 0]
 [4 0 7 8 0 0]
 [4 0 7 8 0 9]]

images

observations

The user wants me to analyze the second example pair of the ARC task 3-1e0a9b12 and formulate a natural language program describing the transformation.

Plan:

Describe the input and output grids visually.
Use code to analyze the grids, focusing on color counts, changed pixels, and object properties.
Formulate a natural language program based on the analysis.
Capture the analysis and program in YAML blocks.

Example 2 Input Grid: The input grid is 6x6. It contains white, maroon, azure, yellow, and orange pixels. There’s a vertical stack of three yellow pixels on the left side. A maroon pixel is in the top right corner. Two azure pixels are present, one above the other. Two orange pixels are stacked vertically in the lower left quadrant.

Example 2 Output Grid: The output grid is also 6x6. The three yellow pixels remain in the same position. The two orange pixels remain in the same position. The azure pixels appear to have moved to the bottom row. The maroon pixel has also moved to the bottom row.

Hypothesis: It appears that all the non-yellow and non-orange single-pixel objects are moved to the last row, maintaining their relative column order.

Let’s use code to verify this.

observations

The input grid is a 6x6 grid. It features white (0), maroon (9), azure (8), yellow (4), and orange (7) pixels. There appears to be a vertical line of three yellow pixels on the left edge. An isolated maroon pixel is located in the top right corner. Two azure pixels form a vertical line in the upper middle section. Two orange pixels are stacked vertically in the lower left quadrant.

The output grid is also a 6x6 grid with the same color palette. The three yellow pixels on the left remain in their original positions. The two orange pixels also appear to be in the same positions as the input. The azure and maroon pixels seem to have moved to the bottom row.

import numpy as np
from collections import Counter

input_grid = np.array([[0, 0, 0, 0, 0, 9],
                       [0, 0, 0, 8, 0, 0],
                       [0, 0, 0, 0, 0, 0],
                       [4, 0, 0, 0, 0, 0],
                       [4, 0, 7, 8, 0, 0],
                       [4, 0, 7, 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],
                        [4, 0, 0, 0, 0, 0],
                        [4, 0, 7, 8, 0, 0],
                        [4, 0, 7, 8, 0, 9]])

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

print("Input Grid:")
print(input_grid)
print("\nOutput Grid:")
print(output_grid)

print("\nInput Color Counts:")
print(input_colors)
print("\nOutput Color Counts:")
print(output_colors)

input_width = input_grid.shape[1]
input_height = input_grid.shape[0]
output_width = output_grid.shape[1]
output_height = output_grid.shape[0]

print(f"\nInput Grid Dimensions: {input_width}x{input_height}")
print(f"Output Grid Dimensions: {output_width}x{output_height}")

changed_pixels = []
for y in range(input_height):
    for x in range(input_width):
        if input_grid[y, x] != output_grid[y, x]:
            changed_pixels.append(((y, x), input_grid[y, x], output_grid[y, x]))

print("\nChanged Pixels ( (row, col), old_color, new_color ):")
for pixel in changed_pixels:
    print(f"  {pixel}")

input:
  width: 6
  height: 6
  colors:
    0: 27
    9: 1
    8: 2
    4: 3
    7: 2
  objects:
    - size: 3, positions: [(3, 0), (4, 0), (5, 0)], color: yellow # Vertical line
    - size: 1, position: (0, 5), color: maroon
    - size: 1, position: (1, 3), color: azure
    - size: 1, position: (4, 3), color: azure
    - size: 1, position: (4, 2), color: orange
    - size: 1, position: (5, 2), color: orange

output:
  width: 6
  height: 6
  colors:
    0: 27
    9: 1
    8: 2
    4: 3
    7: 2
  objects:
    - size: 3, positions: [(3, 0), (4, 0), (5, 0)], color: yellow # Vertical line
    - size: 1, position: (5, 5), color: maroon
    - size: 1, position: (5, 3), color: azure
    - size: 1, position: (4, 3), color: azure
    - size: 1, position: (4, 2), color: orange
    - size: 1, position: (5, 2), color: orange

differences:
  cells_changed: 4
  colors_changed:
    - maroon: moved from (0, 5) to (5, 5)
    - azure: moved from (1, 3) to (5, 3)
    - white: changed to white at (0, 5), (1, 3)
    - white: changed from white at (5, 5), (5, 3)
  transformation:
    - Non-white pixels that are not part of the contiguous yellow or orange blocks appear to have moved to the bottom row.
    - The order of these moved pixels in the bottom row corresponds to their column position in the input.

Natural Language Program:

Context: The input is a 6x6 grid containing pixels of white, maroon, azure, yellow, and orange. There is a vertical line of three yellow pixels on the left edge and a vertical line of two orange pixels in the lower left. Other non-white pixels appear as single, disconnected objects.

Process:

Initialize an empty output grid of the same dimensions as the input grid, filled with white pixels.
Identify the contiguous blocks of yellow and orange pixels in the input grid. Copy these blocks directly to the same positions in the output grid.
Identify all remaining non-white, single-pixel objects in the input grid, along with their colors and column positions.
Place these remaining non-white pixels in the last row of the output grid, maintaining their relative order based on their original column position in the input. The leftmost of these pixels in the input (smallest column index) should be placed in the leftmost available empty (white) cell in the last row of the output, and so on.
Any remaining cells in the output grid will remain white.

example_3

input

[[0 0 0 1 0]
 [0 3 0 0 0]
 [0 3 0 1 2]
 [6 0 0 0 0]
 [0 3 0 0 0]]

output

[[0 0 0 0 0]
 [0 0 0 0 0]
 [0 3 0 0 0]
 [0 3 0 1 0]
 [6 3 0 1 2]]

images

observations

The user wants me to analyze the third example pair of the ARC task 3-1e0a9b12 and formulate a natural language program describing the transformation.

Plan:

Describe the input and output grids visually.
Use code to analyze the grids, focusing on color counts, changed pixels, and object properties.
Formulate a natural language program based on the analysis of all three examples.
Capture the analysis and program in YAML blocks.

Example 3 Input Grid: The input grid is 5x5. It contains white, blue, green, red, and magenta pixels. There are three green pixels forming a vertical line. There are two blue pixels, one in the top right and one in the middle right. A single red pixel is in the bottom right and a single magenta pixel is on the left side.

Example 3 Output Grid: The output grid is also 5x5. The green pixels seem to have shifted down. The blue pixels also seem to have shifted down. The magenta pixel has also shifted down. The red pixel has also shifted down. It appears all the non-white pixels have moved down, with potential interactions in the bottom row.

Let’s use code to verify this and refine the transformation rule by considering all three examples.

observations

The input grid is a 5x5 grid. It contains white (0), blue (1), green (3), red (2), and magenta (6) pixels. There is a vertical line of three green pixels in the second column. There are two isolated blue pixels, one in the first row and one in the third row. A single red pixel is present in the third row, and a single magenta pixel is in the fourth row.

The output grid is also a 5x5 grid with the same color palette. The green pixels appear to have shifted downwards. The blue pixels also seem to have moved downwards. The magenta pixel has moved downwards. The red pixel has also moved downwards. It seems all non-white pixels have moved down, occupying the bottom rows.

import numpy as np
from collections import Counter

input_grid = np.array([[0, 0, 0, 1, 0],
                       [0, 3, 0, 0, 0],
                       [0, 3, 0, 1, 2],
                       [6, 0, 0, 0, 0],
                       [0, 3, 0, 0, 0]])

output_grid = np.array([[0, 0, 0, 0, 0],
                        [0, 0, 0, 0, 0],
                        [0, 3, 0, 0, 0],
                        [0, 3, 0, 1, 0],
                        [6, 3, 0, 1, 2]])

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

print("Input Grid:")
print(input_grid)
print("\nOutput Grid:")
print(output_grid)

print("\nInput Color Counts:")
print(input_colors)
print("\nOutput Color Counts:")
print(output_colors)

input_width = input_grid.shape[1]
input_height = input_grid.shape[0]
output_width = output_grid.shape[1]
output_height = output_grid.shape[0]

print(f"\nInput Grid Dimensions: {input_width}x{input_height}")
print(f"Output Grid Dimensions: {output_width}x{output_height}")

changed_pixels = []
for y in range(input_height):
    for x in range(input_width):
        if input_grid[y, x] != output_grid[y, x]:
            changed_pixels.append(((y, x), input_grid[y, x], output_grid[y, x]))

print("\nChanged Pixels ( (row, col), old_color, new_color ):")
for pixel in changed_pixels:
    print(f"  {pixel}")

input:
  width: 5
  height: 5
  colors:
    0: 17
    1: 2
    3: 3
    2: 1
    6: 1
  objects:
    - size: 3, positions: [(1, 1), (2, 1), (4, 1)], color: green # Vertical line
    - size: 1, position: (0, 3), color: blue
    - size: 1, position: (2, 3), color: blue
    - size: 1, position: (3, 0), color: magenta
    - size: 1, position: (2, 4), color: red

output:
  width: 5
  height: 5
  colors:
    0: 17
    1: 2
    3: 3
    2: 1
    6: 1
  objects:
    - size: 3, positions: [(2, 1), (3, 1), (4, 1)], color: green # Vertical line
    - size: 1, position: (3, 3), color: blue
    - size: 1, position: (4, 3), color: blue
    - size: 1, position: (4, 0), color: magenta
    - size: 1, position: (4, 4), color: red

differences:
  cells_changed: 10
  colors_changed:
    - blue: moved from (0, 3) to (3, 3)
    - green: moved from (1, 1) to (2, 1)
    - green: moved from (2, 1) to (3, 1)
    - green: moved from (4, 1) to (4, 1) # appears to stay, but conceptually moves
    - blue: moved from (2, 3) to (4, 3)
    - magenta: moved from (3, 0) to (4, 0)
    - red: moved from (2, 4) to (4, 4)
    - white: changed to white at (0, 3), (1, 1), (2, 1), (2, 3), (3, 0), (2, 4)
    - white: changed from white at (3, 3), (2, 1), (3, 1), (4, 3), (4, 0), (4, 4)
  transformation:
    - All non-white pixels appear to have moved to the bottom rows of the output grid.
    - The columns of the non-white pixels are preserved during this downward movement.

Based on the analysis of all three examples, a more comprehensive natural language program can be formulated.

Natural Language Program:

Context: The input is a grid containing pixels of various colors, including white. The non-white pixels can form single objects or contiguous blocks.

Process:

Initialize an empty output grid of the same dimensions as the input grid, filled with white pixels.
Iterate through each column of the input grid from left to right.
Within each column, identify all the non-white pixels and their respective colors.
Place these non-white pixels in the corresponding column of the output grid, starting from the bottom row and moving upwards. The order of the non-white pixels within the column is maintained, such that the topmost non-white pixel in the input column will be placed in the topmost available non-white cell in the output column (from bottom to top).
Any remaining cells in the output grid will remain white.

This program describes a transformation where the non-white pixels “fall” to the bottom of the grid within their respective columns.

examples summary

The user wants me to synthesize the observations from the three examples and refine the natural language program. I need to make sure the program works for all examples.

Example 1: Non-white pixels move to the bottom row, ordered by their original column. Example 2: Non-white pixels not part of the yellow or orange blocks move to the bottom row, ordered by their original column. The yellow and orange blocks stay in place. Example 3: All non-white pixels move to the bottom rows of their original columns.

The rule from Example 3 seems to be the most general and covers the observations in Example 1. Let’s test if it fits Example 2.

In Example 2:

Yellow pixels in column 0 go to the bottom of column 0.
Maroon pixel in column 5 goes to the bottom of column 5.
Azure pixels in column 3 go to the bottom of column 3.
Orange pixels in column 2 go to the bottom of column 2.

This fits the “fall down” rule.

Let’s refine the natural language program based on this.

Summary of Observations:

Across all three examples, the transformation appears to involve moving the non-white pixels towards the bottom of the grid, within their respective columns.

Example 1: All non-white pixels shift to the last row, maintaining the order based on their original column position.
Example 2: All non-white pixels shift downwards within their columns. The contiguous blocks of yellow and orange pixels also follow this rule.
Example 3: All non-white pixels shift downwards within their respective columns.

The consistent pattern observed is that non-white pixels move towards the bottom of the grid, staying within their original columns.

Code Re-investigation:

Let’s write a function to simulate the “falling down” of pixels and test it against the examples.

import numpy as np

def apply_transformation(input_grid):
    height, width = input_grid.shape
    output_grid = np.zeros_like(input_grid)
    for col in range(width):
        non_white_pixels = [input_grid[row, col] for row in range(height) if input_grid[row, col] != 0]
        num_non_white = len(non_white_pixels)
        for i in range(num_non_white):
            output_grid[height - num_non_white + i, col] = non_white_pixels[i]
    return output_grid

# Example 1
input_grid_1 = np.array([[0, 4, 0, 9],
                         [0, 0, 0, 0],
                         [0, 4, 6, 0],
                         [1, 0, 0, 0]])
output_grid_1 = np.array([[0, 0, 0, 0],
                          [0, 0, 0, 0],
                          [0, 4, 0, 0],
                          [1, 4, 6, 9]])
predicted_output_1 = apply_transformation(input_grid_1)
print("Example 1 Prediction Correct:", np.array_equal(predicted_output_1, output_grid_1))

# Example 2
input_grid_2 = np.array([[0, 0, 0, 0, 0, 9],
                         [0, 0, 0, 8, 0, 0],
                         [0, 0, 0, 0, 0, 0],
                         [4, 0, 0, 0, 0, 0],
                         [4, 0, 7, 8, 0, 0],
                         [4, 0, 7, 0, 0, 0]])
output_grid_2 = np.array([[0, 0, 0, 0, 0, 0],
                          [0, 0, 0, 0, 0, 0],
                          [0, 0, 0, 0, 0, 0],
                          [4, 0, 0, 0, 0, 0],
                          [4, 0, 7, 8, 0, 0],
                          [4, 0, 7, 8, 0, 9]])
predicted_output_2 = apply_transformation(input_grid_2)
print("Example 2 Prediction Correct:", np.array_equal(predicted_output_2, output_grid_2))

# Example 3
input_grid_3 = np.array([[0, 0, 0, 1, 0],
                         [0, 3, 0, 0, 0],
                         [0, 3, 0, 1, 2],
                         [6, 0, 0, 0, 0],
                         [0, 3, 0, 0, 0]])
output_grid_3 = np.array([[0, 0, 0, 0, 0],
                          [0, 0, 0, 0, 0],
                          [0, 3, 0, 0, 0],
                          [0, 3, 0, 1, 0],
                          [6, 3, 0, 1, 2]])
predicted_output_3 = apply_transformation(input_grid_3)
print("Example 3 Prediction Correct:", np.array_equal(predicted_output_3, output_grid_3))

The code confirms that the “falling down” transformation works correctly for all three examples.

Final Natural Language Program:

Context: The input is a grid containing pixels of various colors, including white.

Process:

Initialize an empty output grid of the same dimensions as the input grid, filled with white pixels.
Iterate through each column of the input grid from left to right.
Within the current column, identify all non-white pixels and their colors, preserving their original order from top to bottom.
Place these identified non-white pixels into the corresponding column of the output grid, starting from the bottom row and filling upwards. The topmost non-white pixel from the input column should be placed in the highest available empty cell (from the bottom) in the output column.
After processing all columns, the output grid will represent the transformed input grid.

test

input

[[0 2 0 4 3]
 [5 0 0 0 0]
 [0 0 6 0 0]
 [5 2 0 4 0]
 [5 0 0 0 0]]

image

observations

INSTRUCTIONS:

using the knowledge you have gathered from the previous examples and the step by step natural language program we want to build the test output grid

initial the output grid copy input grid or set size
set pixels in the output grid as described in the program
use code_execution to validate the output - correct configuration of objects and attributes
make final adjustments review properties of this grid and compare with examples, paying attention to similarities and differences in size, colors, and object arrangements.
submit final output grid