003 • History

puzzle_id:

2-0ca9ddb6

description:

example_3

timestamp:

24.357.081407

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_input21.png ../../../../_images/000-example_1_output21.png

observations

The user wants me to analyze an input-output pair of grids from an ARC task and derive a natural language program that describes the transformation from input to output.

I need to:

  1. Describe the input and output grids.

  2. Use code to analyze the grids, focusing on color counts, changed pixels, and object properties.

  3. Formulate a hypothesis about the transformation rule.

  4. Express the transformation rule as a natural language program.

Example 1 Analysis:

Input grid: 9x9, mostly white (0) with a single red (2) pixel and a single blue (1) pixel. Output grid: 9x9, the red pixel remains, the blue pixel remains, and new yellow (4) and orange (7) pixels have appeared around them.

Let’s analyze the changes:

  • The red pixel at (3, 2) remains at (3, 2).

  • The blue pixel at (6, 6) remains at (6, 6).

  • Yellow pixels appear at (2, 1), (2, 3), (4, 1), (4, 3). These are adjacent (horizontally and vertically) to the red pixel.

  • Orange pixels appear at (5, 6), (6, 5), (6, 7), (7, 6). These are adjacent (horizontally and vertically) to the blue pixel.

It seems like the transformation involves identifying non-white pixels and adding adjacent pixels of specific colors based on the original color.

Okay, I will analyze the provided input and output grids for example_1.

The input grid is a 9x9 grid that is mostly white. There is a single red pixel and a single blue pixel. The red pixel is located in the fourth row, third column (using 0-based indexing, so at coordinates (3, 2)). The blue pixel is located in the seventh row, seventh column (at coordinates (6, 6)).

The output grid is also a 9x9 grid. The red and blue pixels remain in their original positions. New pixels of yellow and orange have appeared around the red and blue pixels respectively. Around the red pixel, yellow pixels have appeared in the cells directly adjacent (horizontally and vertically). Around the blue pixel, orange pixels have appeared in the cells directly adjacent (horizontally and vertically).

import numpy as np

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, input_counts = np.unique(input_grid, return_counts=True)
output_colors, output_counts = np.unique(output_grid, return_counts=True)

print("Input grid dimensions:", input_grid.shape)
print("Output grid dimensions:", output_grid.shape)
print("Input color counts:", dict(zip(input_colors, input_counts)))
print("Output color counts:", dict(zip(output_colors, output_counts)))

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

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

differences:
  cells_changed: 8
  colors_changed:
    - white to yellow around red
    - white to orange around blue
  transformation:
    - for each non-white pixel in the input, add adjacent pixels of a specific color in the output.
    - the color to add depends on the color of the original pixel.

Natural Language Program:

The input grid is a 9x9 grid. Identify all non-white pixels in the input grid.

For each red pixel found in the input grid:

  1. Locate the position of the red pixel.

  2. In the output grid, fill the cells directly adjacent (up, down, left, right) to the original red pixel with yellow. The original red pixel remains red in the output.

For each blue pixel found in the input grid:

  1. Locate the position of the blue pixel.

  2. In the output grid, fill the cells directly adjacent (up, down, left, right) to the original blue pixel with orange. The original blue pixel remains blue in the output.

Initialize the output grid as a copy of the input grid. Iterate through each cell in the input grid. If a cell is red, change its adjacent white cells in the output grid to yellow. If a cell is blue, change its adjacent white cells in the output grid to orange. The color palette consists of white, blue, red, yellow, and orange.

Okay, proceed to the next example.

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_input21.png ../../../../_images/001-example_2_output21.png

observations

The user wants me to analyze the second example of the ARC task and formulate a natural language program for the transformation.

Plan:

  1. Describe the input and output grids of example 2.

  2. Analyze the differences between the input and output grids using code. Focus on color changes and their locations relative to the original non-white pixels.

  3. Refine the hypothesis about the transformation rule based on the second example.

  4. Update the natural language program to be consistent with both examples.

Example 2 Analysis:

Input grid: 9x9, mostly white with single pixels of azure (8), red (2) (two instances), and blue (1) (two instances). Output grid: 9x9, the azure, red, and blue pixels remain. New yellow (4) and orange (7) pixels appear around the red and blue pixels.

Let’s analyze the changes in Example 2:

  • Azure pixel at (0, 3) remains at (0, 3). No new pixels around it.

  • Red pixel at (2, 6) remains at (2, 6). Yellow pixels appear at (1, 6), (3, 6), (2, 5).

  • Blue pixel at (3, 2) remains at (3, 2). Orange pixels appear at (2, 2), (4, 2), (3, 1).

  • Blue pixel at (6, 6) remains at (6, 6). Orange pixels appear at (5, 6), (7, 6), (6, 5), (6, 7).

  • Red pixel at (7, 1) remains at (7, 1). Yellow pixels appear at (6, 1), (8, 1), (7, 0).

It still appears that non-white pixels trigger the addition of adjacent pixels. The color of the added pixels seems to be dependent on the color of the original pixel. Azure seems to be an exception, not triggering any additions.

Okay, I will analyze the provided input and output grids for example_2.

The input grid is a 9x9 grid, predominantly white, with single pixels of azure (8) at (0, 3), red (2) at (2, 6), blue (1) at (3, 2), blue (1) at (6, 6), and red (2) at (7, 1).

The output grid is also a 9x9 grid. The azure, red, and blue pixels from the input remain in their original positions. New pixels of yellow (4) and orange (7) have appeared around the red and blue pixels respectively, similar to the first example. The azure pixel does not have any new pixels around it.

Around the red pixel at (2, 6), yellow pixels have appeared at (1, 6), (3, 6), and (2, 5). Around the blue pixel at (3, 2), orange pixels have appeared at (2, 2), (4, 2), and (3, 1). Around the blue pixel at (6, 6), orange pixels have appeared at (5, 6), (7, 6), (6, 5), and (6, 7). Around the red pixel at (7, 1), yellow pixels have appeared at (6, 1), (8, 1), and (7, 0).

It reinforces the idea that red pixels trigger the appearance of adjacent yellow pixels, and blue pixels trigger the appearance of adjacent orange pixels. The azure pixel does not seem to trigger any change.

import numpy as np

input_grid = 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_grid = 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_colors, input_counts = np.unique(input_grid, return_counts=True)
output_colors, output_counts = np.unique(output_grid, return_counts=True)

print("Input grid dimensions:", input_grid.shape)
print("Output grid dimensions:", output_grid.shape)
print("Input color counts:", dict(zip(input_colors, input_counts)))
print("Output color counts:", dict(zip(output_colors, output_counts)))

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

input:
  width: 9
  height: 9
  colors:
    0: 75
    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)

differences:
  cells_changed: 15
  colors_changed:
    - white to yellow around red
    - white to orange around blue
  transformation:
    - for each red pixel in the input, add adjacent yellow pixels in the output.
    - for each blue pixel in the input, add adjacent orange pixels in the output.
    - azure pixels do not trigger any changes.

Natural Language Program:

The input grid is a 9x9 grid. The color palette includes white, blue, red, yellow, orange, and azure.

Initialize the output grid as a direct copy of the input grid.

Iterate through each cell in the input grid.

If a cell contains a red pixel:

  1. Identify the coordinates of the red pixel.

  2. In the output grid, change the color of the directly adjacent white pixels (up, down, left, right) to yellow. The original red pixel remains red.

If a cell contains a blue pixel:

  1. Identify the coordinates of the blue pixel.

  2. In the output grid, change the color of the directly adjacent white pixels (up, down, left, right) to orange. The original blue pixel remains blue.

Azure pixels in the input grid do not cause any changes to adjacent pixels in the output grid. They remain azure in the output grid.

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_input15.png ../../../../_images/002-example_3_output15.png

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.

See also