003 • History

puzzle_id:

1-3aa6fb7a

description:

example_summary

timestamp:

24.329.071439

call_count:

3

Begin puzzle: 1-3aa6fb7a

example_1

input

[[0 0 0 0 0 0 0]
 [0 8 0 0 0 0 0]
 [0 8 8 0 0 0 0]
 [0 0 0 0 8 8 0]
 [0 0 0 0 0 8 0]
 [0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0]]

output

[[0 0 0 0 0 0 0]
 [0 8 1 0 0 0 0]
 [0 8 8 0 0 0 0]
 [0 0 0 0 8 8 0]
 [0 0 0 0 1 8 0]
 [0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0]]

images

../../../../_images/000-example_1_input6.png ../../../../_images/000-example_1_output6.png

observations

This is an interesting puzzle. Let’s start the analysis.

In the input grid, we have two groups of azure (8) pixels. The first is a group of three forming an L shape in the top left quadrant. The second is a group of two azure (8) pixels forming a vertical line in the lower right quadrant. The remainder of the grid is white (0).

In the output grid, the azure (8) pixels are all the same position. The difference is that one white (0) pixel in each cluster has been changed to blue (1). In the first cluster, the top left of the L is changed to blue (1). In the second cluster, the bottom left pixel of the vertical line is changed to blue (1).

It seems like we might be looking for a corner or edge and changing the color based on that.

input:
  width: 7
  height: 7
  colors:
    - 0: 42
    - 8: 7
  objects:
    - 3: upper left quadrant, L shape, azure (8)
    - 2: lower right quadrant, vertical line, azure (8)

output:
  width: 7
  height: 7
  colors:
    - 0: 40
    - 1: 2
    - 8: 7
  objects:
    - 3: upper left quadrant, L shape, azure (8)
    - 2: lower right quadrant, vertical line, azure (8)
    - 1: top left position of the first cluster, blue
    - 1: bottom left position of the second cluster, blue

differences:
  cells_changed: 2
  colors_changed: white (0) to blue (1)
transformation:
  - identify clusters of azure (8) pixels
  - locate the top left pixel of each cluster
  - change the color of the top left pixel to blue (1)

code_execution:

import numpy as np

input_grid = np.array([[0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0], [0, 8, 8, 0, 0, 0, 0], [0, 0, 0, 0, 8, 8, 0], [0, 0, 0, 0, 0, 8, 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, 8, 1, 0, 0, 0, 0], [0, 8, 8, 0, 0, 0, 0], [0, 0, 0, 0, 8, 8, 0], [0, 0, 0, 0, 1, 8, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]])

def find_clusters(grid, color):
    clusters = []
    visited = np.zeros_like(grid, dtype=bool)

    def explore(row, col, cluster):
        if row < 0 or row >= grid.shape[0] or col < 0 or col >= grid.shape[1] or visited[row, col] or grid[row, col] != color:
            return
        visited[row, col] = True
        cluster.append((row, col))
        explore(row + 1, col, cluster)
        explore(row - 1, col, cluster)
        explore(row, col + 1, cluster)
        explore(row, col - 1, cluster)

    for r in range(grid.shape[0]):
        for c in range(grid.shape[1]):
            if grid[r, c] == color and not visited[r, c]:
                cluster = []
                explore(r, c, cluster)
                clusters.append(cluster)

    return clusters

def find_top_left(cluster):
    return min(cluster)

input_clusters = find_clusters(input_grid, 8)
output_clusters = find_clusters(output_grid, 8)

print("Input clusters:", input_clusters)
print("Output clusters:", output_clusters)

for cluster in input_clusters:
    top_left = find_top_left(cluster)
    print("Top left of input cluster:", top_left)

for cluster in output_clusters:
    top_left = find_top_left(cluster)
    print("Top left of output cluster:", top_left)

code_execution_result: 1

Input clusters: [[(1, 1), (2, 1), (2, 2)], [(3, 4), (3, 5), (4, 5)]]
Output clusters: [[(1, 1), (2, 1), (2, 2)], [(3, 4), (3, 5), (4, 5)]]
Top left of input cluster: (1, 1)
Top left of input cluster: (3, 4)
Top left of output cluster: (1, 1)
Top left of output cluster: (3, 4)

The code confirms our findings. The clusters of azure (8) pixels are the same in the input and output. The top left pixel of each cluster is identified correctly as (1,1) and (3,4).

However, the code did not identify the changed pixel as the top left of each cluster. We observed that the color of the top left pixel of each cluster in the input is changed from white (0) to blue (1) in the output.

Therefore, the natural language program is:

Identify clusters of azure (8) pixels. For each cluster, identify the top left pixel position. Change the color of the pixel at that position from white (0) to blue (1).

example_2

input

[[0 0 0 0 8 8 0]
 [0 0 0 0 0 8 0]
 [0 0 8 0 0 0 0]
 [0 0 8 8 0 0 0]
 [0 0 0 0 0 0 0]
 [0 0 0 0 8 0 0]
 [0 0 0 8 8 0 0]]

output

[[0 0 0 0 8 8 0]
 [0 0 0 0 1 8 0]
 [0 0 8 1 0 0 0]
 [0 0 8 8 0 0 0]
 [0 0 0 0 0 0 0]
 [0 0 0 1 8 0 0]
 [0 0 0 8 8 0 0]]

images

../../../../_images/001-example_2_input6.png ../../../../_images/001-example_2_output6.png

observations

In this example, we again see clusters of azure (8) pixels in the input and output. There are three clusters this time. In the first, two azure (8) pixels form a horizontal line. In the second, two azure (8) pixels also form a horizontal line. In the third, two azure (8) pixels also form a horizontal line.

In the output, the positions of the azure (8) pixels remain unchanged. However, the white (0) pixel at the top left corner of each cluster is changed to blue (1).

This aligns with our previous observation about the top left of each cluster being the target for change.

input:
  width: 7
  height: 7
  colors:
    - 0: 37
    - 8: 12
  objects:
    - 2: top row, horizontal pair, azure (8)
    - 2: row 3, horizontal pair, azure (8)
    - 2: bottom row, horizontal pair, azure (8)

output:
  width: 7
  height: 7
  colors:
    - 0: 34
    - 1: 3
    - 8: 12
  objects:
    - 2: top row, horizontal pair, azure (8)
    - 2: row 3, horizontal pair, azure (8)
    - 2: bottom row, horizontal pair, azure (8)
    - 1: top left of first cluster, blue (1)
    - 1: top left of second cluster, blue (1)
    - 1: top left of third cluster, blue (1)

differences:
  cells_changed: 3
  colors_changed: white (0) to blue (1)
transformation:
  - identify clusters of azure (8) pixels
  - locate the top left pixel of each cluster
  - change the color of the top left pixel to blue (1)

code_execution:

import numpy as np

input_grid = np.array([[0, 0, 0, 0, 8, 8, 0], [0, 0, 0, 0, 0, 8, 0], [0, 0, 8, 0, 0, 0, 0], [0, 0, 8, 8, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 8, 0, 0], [0, 0, 0, 8, 8, 0, 0]])
output_grid = np.array([[0, 0, 0, 0, 8, 8, 0], [0, 0, 0, 0, 1, 8, 0], [0, 0, 8, 1, 0, 0, 0], [0, 0, 8, 8, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 8, 0, 0], [0, 0, 0, 8, 8, 0, 0]])

def find_clusters(grid, color):
    clusters = []
    visited = np.zeros_like(grid, dtype=bool)

    def explore(row, col, cluster):
        if row < 0 or row >= grid.shape[0] or col < 0 or col >= grid.shape[1] or visited[row, col] or grid[row, col] != color:
            return
        visited[row, col] = True
        cluster.append((row, col))
        explore(row + 1, col, cluster)
        explore(row - 1, col, cluster)
        explore(row, col + 1, cluster)
        explore(row, col - 1, cluster)

    for r in range(grid.shape[0]):
        for c in range(grid.shape[1]):
            if grid[r, c] == color and not visited[r, c]:
                cluster = []
                explore(r, c, cluster)
                clusters.append(cluster)

    return clusters

def find_top_left(cluster):
    return min(cluster)

input_clusters = find_clusters(input_grid, 8)
output_clusters = find_clusters(output_grid, 8)

print("Input clusters:", input_clusters)
print("Output clusters:", output_clusters)

for cluster in input_clusters:
    top_left = find_top_left(cluster)
    print("Top left of input cluster:", top_left)

for cluster in output_clusters:
    top_left = find_top_left(cluster)
    print("Top left of output cluster:", top_left)

code_execution_result: 1

Input clusters: [[(0, 4), (0, 5), (1, 5)], [(2, 2), (3, 2), (3, 3)], [(5, 4), (6, 3), (6, 4)]]
Output clusters: [[(0, 4), (0, 5), (1, 5)], [(2, 2), (3, 2), (3, 3)], [(5, 4), (6, 3), (6, 4)]]
Top left of input cluster: (0, 4)
Top left of input cluster: (2, 2)
Top left of input cluster: (5, 4)
Top left of output cluster: (0, 4)
Top left of output cluster: (2, 2)
Top left of output cluster: (5, 4)

The code output is correct again. However, I notice the program has an error when determining the top left. The function used is min(). This will return the minimum based on the first coordinate - the row. It is not considering the column if the row values are equal.

To correct this, the function for find_top_left should be min(cluster, key = lambda x: (x[0], x[1])).

The code confirms the clusters are the same and correctly identifies the top left of each cluster in the input.

The Natural Language Program for transformation remains: Identify clusters of azure (8) pixels. For each cluster, identify the top left pixel position. Change the color of the pixel at that position from white (0) to blue (1).

examples summary

INSTRUCTIONS:

  • summarize your observations to explain the transformation of the input to output

  • use code_execution to investigate properties, patterns and differences in the grids

See also