003 • History

puzzle_id:: 3-1e0a9b12
description:: example_3
timestamp:: 24.357.081407
call_count:: 3

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

INSTRUCTIONS:

above is a pair of example input and output grids

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.
- How to identify 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

You will examine and analyze the example grids

For each example pair, your goal is to derive a natural language description of the transformation rule that explains how the input is changed to produce the output. This “natural language program” should describe the steps or logic involved in the transformation.

the natural language program should be sufficient for an intelligent agent to perform the operation of generating an output grid from the input, without the benefit of seeing the examples. So be sure that the provide

context for understanding the input grid (objects, organization and important colors) particularly context for how to identify the ‘objects’
process for initializing the output grid (copy from input or set size and fill)
describe the color palette to be used in the output
describe how to determine which pixels should change in the output

For example, it might state:

copy input to working output
identify sets of pixels in blue (1) rectangles in working grid
identify to largest rectangle
set the largest rectangle’s pixels to red (2)

But remember - any information that describe the story of the transformations is desired. Be flexible and creative.