003 • Response

puzzle_id:: 3-1e0a9b12
description:: example_3
timestamp:: 24.357.081407
call_count:: 3
model:: gemini-2.0-flash-thinking-exp-1219

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.

time
response time	total elapsed
14.082	38.66

usage
token type	call count	total used
Prompt	6573	12878
Response	1800	4822
Total	8373	17700
Cached	0	0