004 • History

puzzle_id:

2-0ca9ddb6

description:

example_summary

timestamp:

24.329.071439

call_count:

4

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_input7.png ../../../../_images/000-example_1_output7.png

observations

The input grid appears to be a 9x9 grid with a sparse population of colored cells. There is a single red cell (color 2) and a single blue cell (color 1). The output grid is the same size but introduces new yellow cells (color 4) and orange cells (color 7). The original red and blue cells remain in place.

The new yellow cells appear in a specific pattern around the red cell. There are four yellow cells, two cells away in each cardinal direction from the red cell. The orange cells form a cross shape around the blue cell, one cell away in the cardinal directions and two diagonal cells adjacent to the blue.

Analysis

I will use Python to analyze the grid to confirm and detail the observations made.

code_execution:

import numpy as np

# Define the input and output grids
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]
])

# Find the positions of the non-zero cells in the input grid
input_positions = np.argwhere(input_grid != 0)
print("Non-zero positions in input grid:", input_positions)

# Find the positions of the new cells in the output grid
new_cells_positions = np.argwhere((output_grid != 0) & (input_grid == 0))
print("New cells in output grid:", new_cells_positions)
print("Colors of new cells:", output_grid[tuple(new_cells_positions.T)])

code_execution_result: 1

Non-zero positions in input grid: [[3 2]
 [6 6]]
New cells in output grid: [[2 1]
 [2 3]
 [4 1]
 [4 3]
 [5 6]
 [6 5]
 [6 7]
 [7 6]]
Colors of new cells: [4 4 4 4 7 7 7 7]

Observations

The code confirms the positions of the original non-zero cells. The new yellow cells (color 4) are located at positions (2,1), (2,3), (4,1), (4,3) and the orange cells (color 7) are at (5,6), (6,5), (6,7), (7,6).

Input

input:
  width: 9
  height: 9
  colors:
    - 0: 79
    - 1: 1
    - 2: 1
  objects:
    - 1x1, position (3,2), color red - single red cell
    - 1x1, position (6,6), color blue - single blue cell

Output

output:
  width: 9
  height: 9
  colors:
    - 0: 71
    - 1: 1
    - 2: 1
    - 4: 4
    - 7: 4
  objects:
    - 1x1, position (3,2), color red - original red cell
    - 1x1, position (6,6), color blue - original blue cell
    - 1x1, position (2,1), (2,3), (4,1), (4,3), color yellow - new yellow cells
    - 1x1, position (5,6), (6,5), (6,7), (7,6), color orange - new orange cells

Differences

differences:
  cells_changed: 8
  colors_changed: added colors 4 and 7 (yellow and orange)
  transformation:
    - new yellow cells appear in cross shape around red cell with distance 2
    - new orange cells appear in cross shape around blue cell with distance 1

Transformation

The transformation involves placing new cells of specific colors at fixed distances around the original colored cells. Yellow cells appear two cells away from the red cell in the cardinal directions. Orange cells appear one cell away from the blue cell in the cardinal directions. This is a pattern of expansion from the original colored cells, with different distances and colors for each.

Therefore, the natural language program for this transformation is:

For each red cell, place yellow cells two cells away in the north, south, east, and west directions. For each blue cell, place orange cells one cell away in the north, south, east, and west directions.

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_input7.png ../../../../_images/001-example_2_output7.png

observations

This example shows a more complex scenario with multiple instances of red (color 2) and blue (color 1) cells, as well as an additional color, azure (color 8) in the input. The output grid introduces yellow (color 4) and orange (color 7) cells in a similar pattern to the previous example, but with some adaptations for the multiple input cells.

Specifically, the yellow cells still appear two cells away from the red cells in the cardinal directions, and the orange cells appear one cell away from the blue cells. However, there are interactions where these patterns overlap. The blue cell at (6,6) follows the same pattern as in the first example. The blue cell at (3,2) also follows this pattern, but the orange cells overlap with the yellow cells generated from the red cells. The red cells generate yellow cells as expected, but they overlap with existing elements. The azure cell (8) remains unchanged.

Analysis

I will use Python to confirm and analyze the positions of the cells and the patterns of new cell placement.

```python
import numpy as np

# Define the input and output grids
input_grid_2 = 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_2 = 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]
])

# Find the positions of non-zero cells in the input grid
input_positions_2 = np.argwhere(input_grid_2 != 0)
print("Non-zero positions in input grid 2:", input_positions_2)

# Find the positions of the new cells in the output grid
new_cells_positions_2 = np.argwhere((output_grid_2 != 0) & (input_grid_2 == 0))
print("New cells in output grid 2:", new_cells_positions_2)
print("Colors of new cells:", output_grid_2[tuple(new_cells_positions_2.T)])

Non-zero positions in input grid 2: [[0 3]
 [2 6]
 [3 2]
 [6 6]
 [7 1]]
New cells in output grid 2: [[1 5]
 [1 7]
 [2 2]
 [3 1]
 [3 3]
 [3 5]
 [3 7]
 [4 2]
 [5 6]
 [6 0]
 [6 2]
 [6 5]
 [6 7]
 [7 6]
 [8 0]
 [8 2]]
Colors of new cells: [4 4 7 7 7 4 4 7 7 4 4 7 7 7 4 4]

Observations

The code confirms the positions of the original cells and identifies the new cells. The new cells’ positions align with the visual observations, with yellow cells (4) around red cells (2) and orange cells (7) around blue cells (1). Overlapping patterns result in some of the cells being combined.

Input

input:
  width: 9
  height: 9
  colors:
    - 0: 76
    - 1: 2
    - 2: 2
    - 8: 1
  objects:
    - 1x1, position (0,3), color azure - single azure cell
    - 1x1, position (2,6), color red
    - 1x1, position (3,2), color blue
    - 1x1, position (6,6), color blue
    - 1x1, position (7,1), color red

Output

output:
  width: 9
  height: 9
  colors:
    - 0: 57
    - 1: 2
    - 2: 2
    - 4: 8
    - 7: 8
    - 8: 1
  objects:
    - 1x1, position (0,3), color azure - original azure cell unchanged
    - 1x1, position (2,6), color red - original red cell
    - 1x1, position (3,2), color blue - original blue cell
    - 1x1, position (6,6), color blue - original blue cell
    - 1x1, position (7,1), color red - original red cell
    - 1x1, positions (1,5), (1,7), (3,5), (3,7), (6,0), (6,2), (8,0), (8,2), color yellow - new yellow cells
    - 1x1, positions (2,2), (3,1), (3,3), (4,2), (5,6), (6,5), (6,7), (7,6), color orange - new orange cells

Differences

differences:
  cells_changed: 16
  colors_changed: added colors 4 and 7 (yellow and orange)
  transformation:
    - new yellow cells appear in cross shape around red cells with distance 2
    - new orange cells appear in cross shape around blue cells with distance 1
    - overlapping patterns result in merged cells

Transformation

The transformation is similar to the previous example, but with adjustments for multiple cells. Each red cell generates yellow cells two cells away in the cardinal directions, and each blue cell generates orange cells one cell away in the cardinal directions. Overlapping of these patterns leads to merged cells of the same color.

The transformation rules still follow:

For each red cell, place yellow cells two cells away in the north, south, east, and west directions. For each blue cell, place orange cells one cell away in the north, south, east, and west directions.

However, if a cell location is already occupied, it will retain the new color if it coincides with the pattern from another cell. If multiple colors would coincide, the cell remains unchanged.

Therefore, the natural language program for this transformation is refined as:

For each red cell, place yellow cells two cells away in the north, south, east, and west directions unless the position is already occupied by a non-zero cell. For each blue cell, place orange cells one cell away in the north, south, east, and west directions unless the position is already occupied by a non-zero cell. If cells overlap, maintain the new color if it matches the pattern, otherwise, keep the existing non-zero cell.

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_input5.png ../../../../_images/002-example_3_output5.png

observations

observations This example introduces a magenta cell (color 6), which, like the azure cell in the previous example, remains unchanged in the output. The red (2) and blue (1) cells continue to generate yellow (4) and orange (7) cells respectively, following the established pattern. The magenta cell doesn’t seem to influence the placement of any new cells.

Analysis

I will use Python to confirm the positions and counts of each color and the changes between the input and output grids.

code_execution:

import numpy as np

input_grid_3 = np.array([
    [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_grid_3 = np.array([
    [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]
])


input_positions_3 = np.argwhere(input_grid_3 != 0)
print("Non-zero positions in input grid 3:", input_positions_3)

new_cells_positions_3 = np.argwhere((output_grid_3 != 0) & (input_grid_3 == 0))
print("New cells in output grid 3:", new_cells_positions_3)
print("Colors of new cells 3:", output_grid_3[tuple(new_cells_positions_3.T)])

code_execution_result: 1

Non-zero positions in input grid 3: [[2 2]
 [5 6]
 [7 3]]
New cells in output grid 3: [[1 1]
 [1 3]
 [3 1]
 [3 3]
 [6 3]
 [7 2]
 [7 4]
 [8 3]]
Colors of new cells 3: [4 4 4 4 7 7 7 7]

Input

input:
  width: 9
  height: 9
  colors:
    - 0: 78
    - 1: 1
    - 2: 1
    - 6: 1
  objects:
    - 1x1, position (2,2), color red
    - 1x1, position (5,6), color magenta
    - 1x1, position (7,3), color blue

Output

output:
  width: 9
  height: 9
  colors:
    - 0: 70
    - 1: 1
    - 2: 1
    - 4: 4
    - 6: 1
    - 7: 4
  objects:
    - 1x1, position (2,2), color red
    - 1x1, position (5,6), color magenta
    - 1x1, position (7,3), color blue
    - 1x1, positions (1,1), (1,3), (3,1), (3,3), color yellow
    - 1x1, positions (6,3), (7,2), (7,4), (8,3), color orange

Differences

differences:
  cells_changed: 8
  colors_changed: added colors 4 and 7 (yellow and orange)
transformation:
  - new yellow cells appear in cross shape around red cell with distance 2
  - new orange cells appear in cross shape around blue cell with distance 1
  - magenta cell remains unchanged and doesn't influence new cell placement

Transformation

This example reinforces the previously observed pattern. Red cells generate yellow cells two cells away in the cardinal directions, and blue cells generate orange cells one cell away in the cardinal directions. Cells of other colors (magenta in this case) are unaffected by the transformation and do not influence the placement of new cells.

The natural language program remains consistent:

For each red cell, place yellow cells two cells away in the north, south, east, and west directions unless the position is already occupied by a non-zero cell. For each blue cell, place orange cells one cell away in the north, south, east, and west directions unless the position is already occupied by a non-zero cell.

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