002 • Response

puzzle_id:: 2-0ca9ddb6
description:: example_2
timestamp:: 24.329.071439
call_count:: 2
model:: models/gemini-exp-1114

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.

time
response time	total elapsed
62.128	130.353

usage
token type	call count	total used
Prompt	4905	7370
Response	1987	2557
Total	6892	9927
Cached	0	0