006 • History

puzzle_id:

6-0520fde7

description:

init working

timestamp:

24.329.064813

call_count:

6

Begin puzzle: 6-0520fde7

example_1

input

[[1 0 0 5 0 1 0]
 [0 1 0 5 1 1 1]
 [1 0 0 5 0 0 0]]

output

[[0 0 0]
 [0 2 0]
 [0 0 0]]

images

../../../../_images/000-example_1_input5.png ../../../../_images/000-example_1_output5.png

observations

The input grid is a 3x7 grid containing blue (1), white (0), and gray (5) pixels. The gray pixels form a vertical line in the center. The blue pixels are clustered around the gray line.

The output grid is a 3x3 grid. It contains only white (0) and red (2) pixels. A single red pixel is located in the center.

code_execution:

import numpy as np

input_grid = np.array([[1, 0, 0, 5, 0, 1, 0],
                      [0, 1, 0, 5, 1, 1, 1],
                      [1, 0, 0, 5, 0, 0, 0]])

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

print("Input Grid Shape:", input_grid.shape)
print("Output Grid Shape:", output_grid.shape)

print("\nInput Grid Color Counts:")
for color, count in np.nditer(np.unique(input_grid, return_counts=True)):
    print(f"{color}: {count}")

print("\nOutput Grid Color Counts:")
for color, count in np.nditer(np.unique(output_grid, return_counts=True)):
    print(f"{color}: {count}")

code_execution_result: 1

Input Grid Shape: (3, 7)
Output Grid Shape: (3, 3)

Input Grid Color Counts:
0: 11
1: 7
5: 3

Output Grid Color Counts:
0: 8
2: 1

input:
  width: 7
  height: 3
  colors:
    - 0: 11
    - 1: 7
    - 5: 3
  objects:
    - size: 3x1, position: center, color: gray
    - size: various, position: around central gray line, color: blue

output:
  width: 3
  height: 3
  colors:
    - 0: 8
    - 2: 1
  objects:
    - size: 1x1, position: center, color: red

differences:
  cells_changed: 9
  colors_changed: Gray and blue pixels are replaced by white and one red pixel in the center.
transformation:
  - The input grid is reduced in size.
  - The central gray column is replaced with a single red pixel in the center of the output grid.
  - All other pixels are replaced with white.

The transformation appears to involve a reduction in size, focusing on the central column of the input grid. The central gray column is transformed into a single red pixel in the center of the smaller output grid. All other pixels become white. A possible natural language program could be: “Reduce the grid size, replace the central column with a red pixel, and fill the rest with white”.

example_2

input

[[1 1 0 5 0 1 0]
 [0 0 1 5 1 1 1]
 [1 1 0 5 0 1 0]]

output

[[0 2 0]
 [0 0 2]
 [0 2 0]]

images

../../../../_images/001-example_2_input5.png ../../../../_images/001-example_2_output5.png

observations

Example 2 Analysis

The input grid is a 3x7 grid consisting of blue (1), white (0), and gray (5) pixels. The gray pixels again form a vertical central line, with blue pixels clustered around it, similar to Example 1.

The output grid is a 3x3 grid. It contains white (0) and red (2) pixels. Three red pixels are arranged in an L-shape pattern.

code_execution:

import numpy as np

input_grid = np.array([[1, 1, 0, 5, 0, 1, 0],
                      [0, 0, 1, 5, 1, 1, 1],
                      [1, 1, 0, 5, 0, 1, 0]])

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

print("Input Grid Shape:", input_grid.shape)
print("Output Grid Shape:", output_grid.shape)

print("\nInput Grid Color Counts:")
for color, count in np.nditer(np.unique(input_grid, return_counts=True)):
    print(f"{color}: {count}")

print("\nOutput Grid Color Counts:")
for color, count in np.nditer(np.unique(output_grid, return_counts=True)):
    print(f"{color}: {count}")

code_execution_result: 1

Input Grid Shape: (3, 7)
Output Grid Shape: (3, 3)

Input Grid Color Counts:
0: 8
1: 10
5: 3

Output Grid Color Counts:
0: 6
2: 3

input:
  width: 7
  height: 3
  colors:
    - 0: 8
    - 1: 10
    - 5: 3
  objects:
    - size: 3x1, position: center, color: gray
    - size: various, position: around central gray line, color: blue

output:
  width: 3
  height: 3
  colors:
    - 0: 6
    - 2: 3
  objects:
    - size: 3x1, position: L-shape around center, color: red

differences:
  cells_changed: 9
  colors_changed: Gray and blue pixels are replaced by white and red pixels forming an L shape.
transformation:
  - The input grid is reduced in size.
  - The gray column is replaced, and some nearby pixels are changed to red pixels forming an L shape.
  - The remaining pixels are set to white.

The transformation in this example is similar to the first, with grid size reduction. However, instead of a single central red pixel, we see three red pixels forming an “L” shape. The positions of the red pixels seem related to the positions of the blue and gray pixels in the input. A more refined natural language program might be: “Reduce the grid size; replace the central gray column and some adjacent pixels with red pixels forming an ‘L’ shape; fill the rest with white.”

example_3

input

[[0 0 1 5 0 0 0]
 [1 1 0 5 1 0 1]
 [0 1 1 5 1 0 1]]

output

[[0 0 0]
 [2 0 0]
 [0 0 2]]

images

../../../../_images/002-example_3_input4.png ../../../../_images/002-example_3_output4.png

observations

Example 3 Analysis

The input grid is a 3x7 grid. It contains white (0), blue (1), and gray (5) pixels. The gray pixels form a vertical line in the center. The blue pixels are somewhat clustered around the gray line but less densely than in the previous examples.

The output grid is a 3x3 grid. It contains white (0) and red (2) pixels. Two red pixels are placed in the corners, at positions (1,0) and (2,2).

code_execution:

import numpy as np

input_grid = np.array([[0, 0, 1, 5, 0, 0, 0],
                      [1, 1, 0, 5, 1, 0, 1],
                      [0, 1, 1, 5, 1, 0, 1]])

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

print("Input Grid Shape:", input_grid.shape)
print("Output Grid Shape:", output_grid.shape)

print("\nInput Grid Color Counts:")
for color, count in np.nditer(np.unique(input_grid, return_counts=True)):
    print(f"{color}: {count}")

print("\nOutput Grid Color Counts:")
for color, count in np.nditer(np.unique(output_grid, return_counts=True)):
    print(f"{color}: {count}")

code_execution_result: 1

Input Grid Shape: (3, 7)
Output Grid Shape: (3, 3)

Input Grid Color Counts:
0: 9
1: 7
5: 3

Output Grid Color Counts:
0: 7
2: 2

input:
  width: 7
  height: 3
  colors:
    - 0: 9
    - 1: 7
    - 5: 3
  objects:
    - size: 3x1, position: center, color: gray
    - size: various, position: around central gray line, color: blue

output:
  width: 3
  height: 3
  colors:
    - 0: 7
    - 2: 2
  objects:
    - size: 1x1, position: top-left and bottom-right corners, color: red

differences:
  cells_changed: 9
  colors_changed: Gray and blue pixels are replaced with white, and two red pixels appear in opposite corners.
transformation:
  - The input grid is reduced in size.
  - Two red pixels are placed in opposite corners of the output grid.
  - The remaining pixels are set to white.

The transformation in Example 3 is again a size reduction. The placement of the red pixels is different; they are now in opposite corners. The natural language program is becoming more complex. A possible description could be: “Reduce the grid size; place red pixels in the top-left and bottom-right corners; fill remaining cells with white.” The relationship between the input and the placement of the red pixels requires further investigation.

examples summary

Based on the three examples, the transformation involves the following steps:

  1. Size Reduction: The input grid (always 3x7) is reduced to a 3x3 output grid. This reduction discards the outer columns and retains the central 3x3 area.

  2. Central Column Focus: The transformation primarily focuses on the central column of the input grid (the column containing gray pixels). The color and distribution of pixels in this central column strongly influences the output.

  3. Color Replacement: The central column (gray pixels) and surrounding pixels are replaced with red (2) and white (0) pixels in the 3x3 output grid. The specific arrangement of red and white pixels varies, depending on the pattern of blue (1) and gray (5) pixels in the input’s central region.

  4. Red Pixel Placement Rules (Hypothesized): The pattern suggests that the placement of red pixels is not arbitrary. The number and positions of the red pixels seem to be a function of the density and position of the blue pixels around the central gray column in the original input image.

    • Example 1: One central red pixel; sparsely distributed blue pixels.

    • Example 2: Three red pixels in an L-shape; denser distribution of blue pixels.

    • Example 3: Two red pixels in opposite corners; moderately distributed blue pixels.

Code to explore patterns and validate hypotheses:

code_execution:

import numpy as np

examples = [
    (np.array([[1, 0, 0, 5, 0, 1, 0], [0, 1, 0, 5, 1, 1, 1], [1, 0, 0, 5, 0, 0, 0]]), np.array([[0, 0, 0], [0, 2, 0], [0, 0, 0]])),
    (np.array([[1, 1, 0, 5, 0, 1, 0], [0, 0, 1, 5, 1, 1, 1], [1, 1, 0, 5, 0, 1, 0]]), np.array([[0, 2, 0], [0, 0, 2], [0, 2, 0]])),
    (np.array([[0, 0, 1, 5, 0, 0, 0], [1, 1, 0, 5, 1, 0, 1], [0, 1, 1, 5, 1, 0, 1]]), np.array([[0, 0, 0], [2, 0, 0], [0, 0, 2]]))
]

for input_grid, output_grid in examples:
    print("Input:")
    print(input_grid)
    print("\nOutput:")
    print(output_grid)
    center_column = input_grid[:, 3]
    print("\nCenter column:", center_column)
    blue_count = np.sum(input_grid == 1)
    print("Total blue pixels:", blue_count)

    #Analysis of blue pixel distribution relative to center

    central_3x3 = input_grid[0:3,1:4]
    print("\nCentral 3x3:")
    print(central_3x3)
    blue_in_central_3x3 = np.sum(central_3x3 == 1)
    print("Blue pixels in central 3x3:", blue_in_central_3x3)

    print("-"*20)

code_execution_result: 1

Input:
[[1 0 0 5 0 1 0]
 [0 1 0 5 1 1 1]
 [1 0 0 5 0 0 0]]

Output:
[[0 0 0]
 [0 2 0]
 [0 0 0]]

Center column: [5 5 5]
Total blue pixels: 7

Central 3x3:
[[0 0 5]
 [1 0 5]
 [0 0 5]]
Blue pixels in central 3x3: 1
--------------------
Input:
[[1 1 0 5 0 1 0]
 [0 0 1 5 1 1 1]
 [1 1 0 5 0 1 0]]

Output:
[[0 2 0]
 [0 0 2]
 [0 2 0]]

Center column: [5 5 5]
Total blue pixels: 10

Central 3x3:
[[1 0 5]
 [0 1 5]
 [1 0 5]]
Blue pixels in central 3x3: 3
--------------------
Input:
[[0 0 1 5 0 0 0]
 [1 1 0 5 1 0 1]
 [0 1 1 5 1 0 1]]

Output:
[[0 0 0]
 [2 0 0]
 [0 0 2]]

Center column: [5 5 5]
Total blue pixels: 9

Central 3x3:
[[0 1 5]
 [1 0 5]
 [1 1 5]]
Blue pixels in central 3x3: 4
--------------------

The code analysis shows a correlation between the number of blue pixels in the central 3x3 area of the input and the pattern of red pixels in the output. Further investigation might reveal a precise rule, but it’s clear that the blue pixel distribution in that central area is a key factor.

Revised Natural Language Program:

“Reduce the input grid to a 3x3 grid. Analyze the distribution of blue pixels within the central 3x3 region of the input grid. Based on this distribution, replace some cells in the 3x3 output grid with red pixels. Fill the remaining cells with white pixels.” This program is still imprecise concerning the exact mapping of blue pixel distribution to red pixel placement, but it more accurately reflects the observed transformation.

test

input

[[1 0 1 5 1 0 1]
 [0 1 0 5 1 0 1]
 [1 0 1 5 0 1 0]]

image

../../../../_images/004-test_input3.png

observations

Here’s a report analyzing the test input and applying the refined natural language program developed from the previous examples:

Test Input Analysis:

The test input grid is a 3x7 grid, consistent with the previous examples. It contains white (0), blue (1), and gray (5) pixels. The gray pixels form a central vertical column. The blue pixels are distributed around the central column.

code_execution:

import numpy as np

test_input = np.array([[1, 0, 1, 5, 1, 0, 1],
                      [0, 1, 0, 5, 1, 0, 1],
                      [1, 0, 1, 5, 0, 1, 0]])

print("Test Input Grid Shape:", test_input.shape)
print("\nTest Input Color Counts:")
for color, count in np.nditer(np.unique(test_input, return_counts=True)):
    print(f"{color}: {count}")

central_3x3 = test_input[0:3, 1:4]
print("\nCentral 3x3:")
print(central_3x3)
blue_in_central_3x3 = np.sum(central_3x3 == 1)
print("\nBlue pixels in central 3x3:", blue_in_central_3x3)

code_execution_result: 1

Test Input Grid Shape: (3, 7)

Test Input Color Counts:
0: 8
1: 10
5: 3

Central 3x3:
[[0 1 5]
 [1 0 5]
 [0 1 5]]

Blue pixels in central 3x3: 3

Applying the Refined Natural Language Program:

  1. Size Reduction: The 3x7 input is reduced to a 3x3 grid, focusing on the central 3x3 region.

  2. Blue Pixel Distribution Analysis: The central 3x3 region of the input has three blue pixels. Comparing this to the previous examples:

    • Example 1 (1 blue pixel in central 3x3): One central red pixel.

    • Example 2 (3 blue pixels in central 3x3): Three red pixels in an L-shape.

    • Example 3 (4 blue pixels in central 3x3): Two red pixels in opposite corners.

The test input has 3 blue pixels in the central 3x3 region, suggesting a pattern similar to Example 2 (L-shape of red pixels).

  1. Predicted Output: Based on this analysis, a likely output is a 3x3 grid with three red pixels arranged in an L-shape, surrounded by white pixels. Therefore the predicted output is:

[[0 2 0]
 [0 0 2]
 [0 2 0]]

Final Prediction:

The predicted output grid is:

[[0 2 0]
 [0 0 2]
 [0 2 0]]

This prediction is based on the observed pattern in the previous examples, where the number and arrangement of blue pixels within the central 3x3 area of the input strongly influence the output. The count of 3 blue pixels in the central 3x3 area of the test input leads to the predicted L-shaped arrangement of red pixels in the output.

initialize the working output grid:

INSTRUCTIONS:

use function_call to initialize the working output grid:

  • initialize_output_from_input: good when examples show few differences between input and output

  • initialize_output_by_size: create a fresh grid from size and color

See also