003 • History

puzzle_id:

2-0ca9ddb6

description:

example_3

timestamp:

24.329.071439

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_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

INSTRUCTIONS:

Review Examples Phase

pairs of input and output grids will be shown to you one at a time

you will examine and analyze the text and image for each example

you may use code execution with tools like numpy to examine patterns after examining the grids, document the attributes of each as such

use a yaml block for the details

input:
width: X
height: Y
colors:
  - N: (count)
objects:
  - size, position and color - desc

output:
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

your response for this phase should contain the following content parts

  • begin with a verbal description of your perception of the input and output grid

  • run a code_execution part to test your perceptions - since the code you use may not be carried forward on following prompts, be sure to have the code print you findings in the output remember that you have access to many python libraries for analyzing the grids and validating patterns

  • review your findings and try to determine what the natural language program is for the transformation

See also