You are an expert visual puzzle solver with advanced pattern recognition capabilities. Your task is to systematically analyze a complex visual transformation puzzle, break it down into clear logical steps, and implement a robust code solution that can generalize to new test cases.
This is an Abstract Reasoning Challenge (ARC) puzzle that tests your ability to identify abstract patterns and transformations. The puzzle structure consists of:
- Multiple example pairs (typically 2-4 pairs) that demonstrate a consistent transformation rule
- Each pair contains:
- Input grid: A 2D array of colored pixels representing the initial state
- Output grid: A 2D array of colored pixels showing the result after applying the transformation
- The transformation rule is invariant across all training pairs - your job is to discover this hidden rule
- You are provided with one or more test input grids
- These grids follow the same underlying transformation rule as the training examples
- Your goal is to apply the discovered rule to generate the correct output grid(s)
- The test grids may have different dimensions or slight variations, but the core transformation logic remains consistent
- Pattern Recognition: Look for spatial relationships, color changes, shape manipulations, or geometric transformations
- Abstraction: The rule often involves abstract concepts like symmetry, repetition, conditional logic, or mathematical operations
- Generalization: Your solution must work not just for the training examples, but also for unseen test cases that follow the same rule
- Precision: Grid-based puzzles require exact pixel-level accuracy - small errors can invalidate the entire solution
The original problem is a 2D grid of pixels, each pixel value is a color. To make it easier to visualize, we use a color to represent the pixel value. The pixel value to color mapping is as follows:
-
black (0)
-
blue (1)
-
red (2)
-
green (3)
-
yellow (4)
-
grey (5)
-
pink (6)
-
orange (7)
-
light_blue (8)
-
brown (9)
-
white (10)
The training data is a few input-output pairs of the same puzzle. The following are the observations of the grids for each pair. You need to pay attention to the them and find out the USEFUL information:
Input Grid Color Statistics:
| Color (code) | Count |
|---|---|
| yellow (4) | 307 |
| blue (1) | 9 |
| grey (5) | 3 |
| green (3) | 3 |
| pink (6) | 2 |
Output Grid Color Statistics:
| Color (code) | Count |
|---|---|
| yellow (4) | 307 |
| blue (1) | 14 |
| green (3) | 2 |
| pink (6) | 1 |
Dot Counting Pattern: There are 1 isolated notable dots of pink color. There are 3 isolated notable dots of grey color. There are 2 isolated notable dots of green color.
Grid Structure Comparison: The two grids have the SAME SIZE. A few shapes are found in both input and output, consider them as ANCHOR SHAPES: A Single Pixel shape at (1, 12) A Single Pixel shape at (12, 3) A Single Pixel shape at (12, 14)
Shape Match: A Complex shape of size 3x2 at input (2, 2) also found in output starting from positions: (2, 11) A Complex shape of size 3x2 at input (10, 6) also found in output starting from positions: (10, 13), (10, 2)
Input Grid Color Statistics:
| Color (code) | Count |
|---|---|
| yellow (4) | 292 |
| blue (1) | 23 |
| red (2) | 3 |
| grey (5) | 2 |
| green (3) | 2 |
| light_blue (8) | 2 |
Output Grid Color Statistics:
| Color (code) | Count |
|---|---|
| yellow (4) | 289 |
| blue (1) | 31 |
| red (2) | 2 |
| green (3) | 1 |
| light_blue (8) | 1 |
Dot Counting Pattern: There are 2 isolated notable dots of red color. There are 2 isolated notable dots of grey color. There are 1 isolated notable dots of green color. There are 1 isolated notable dots of light_blue color.
Grid Structure Comparison: The two grids have the SAME SIZE. A few shapes are found in both input and output, consider them as ANCHOR SHAPES: A Single Pixel shape at (2, 7) A Single Pixel shape at (2, 13) A Single Pixel shape at (9, 9) A Single Pixel shape at (15, 3)
Shape Match: A Complex shape of size 3x3 at input (1, 1) also found in output starting from positions: (1, 6), (1, 12) A Complex shape of size 3x4 at input (7, 1) also found in output starting from positions: (7, 8) A Complex shape of size 5x3 at input (14, 12) also found in output starting from positions: (14, 1)
Now try your best to break down the current problem, I want you first to write a solution design, and then write the code to solve the puzzle. The test problem input and description are as follows:
Dot Counting Pattern: There are 3 isolated notable dots of pink color. There are 2 isolated notable dots of grey color. There are 1 isolated notable dots of green color. There are 2 isolated notable dots of light_blue color.
Detailed code in lib llm_libs.shapes.py is as follows, those code are useful for you to solve the puzzle.
import numpy as np
from visual_tools.vis_grid import draw_grid
from llm_libs.colors import cmap, EMPTY
from enum import Enum
from functools import reduce
class Pixel:
def __init__(self, x, y, color):
self.x = x
self.y = y
self.color = color
class BBox:
def __init__(self, min_r, min_c, max_r, max_c):
self.x = min_r
self.y = min_c
self.h = max_r - min_r + 1
self.w = max_c - min_c + 1
class ShapeType(Enum):
"""Enumeration of different shape types that can be detected."""
SINGLE_PIXEL = "Single Pixel"
LINE_HORIZONTAL = "Horizontal Line"
LINE_VERTICAL = "Vertical Line"
LINE_DIAGONAL = "Diagonal Line"
RECTANGLE = "Rectangle"
SQUARE = "Square"
COMPLEX = "Complex"
class Shape:
def __init__(self, pixels:list[Pixel], bb_box:BBox):
'''
pixels: a list of Pixel objects
bb_box: the bounding box of the shape
'''
self.pixels = pixels
self.bb_box = bb_box
self.pos = (self.bb_box.x, self.bb_box.y)
# Set color if all non-background pixels have the same color, otherwise None
self.color_name, self.color_code = self._describe_color()
self.array = self.to_grid()
self.mask = self.get_background_mask()
self.shape_type = self._describe_shape()
def __hash__(self):
# Hash based on the shape's array and color
# Convert array to bytes for hashing
return hash((self.pos, self.color_code))
def to_grid(self):
# use border color to represent the background
array = np.ones((self.bb_box.h, self.bb_box.w), dtype=np.int64) * EMPTY
for pixel in self.pixels:
array[pixel.x-self.bb_box.x, pixel.y-self.bb_box.y] = pixel.color
return array
def _describe_shape(self)->ShapeType:
if len(self.pixels) == 1:
return ShapeType.SINGLE_PIXEL
# check for line
if all(pixel.x == self.pixels[0].x for pixel in self.pixels):
return ShapeType.LINE_HORIZONTAL
elif all(pixel.y == self.pixels[0].y for pixel in self.pixels):
return ShapeType.LINE_VERTICAL
elif all(pixel.x - pixel.y == self.pixels[0].x - self.pixels[0].y for pixel in self.pixels):
return ShapeType.LINE_DIAGONAL
# check for rectangle and square
min_x = min(pixel.x for pixel in self.pixels)
max_x = max(pixel.x for pixel in self.pixels)
min_y = min(pixel.y for pixel in self.pixels)
max_y = max(pixel.y for pixel in self.pixels)
if (max_x - min_x) * (max_y - min_y) == len(self.pixels):
return ShapeType.SQUARE if max_x - min_x == max_y - min_y else ShapeType.RECTANGLE
return ShapeType.COMPLEX
def _describe_color(self)->tuple[str, str]:
colors = set([p.color for p in self.pixels])
if len(colors) == 1 :
color_code = colors.pop()
return cmap.get_name(color_code), color_code
else:
return "mixed", None
def get_background_mask(self):
return self.array == EMPTY
def to_image(self):
return draw_grid(self.array)
def __eq__(self, other: 'Shape'):
# exact shape and color match
return np.array_equal(self.array, other.array)
def __repr__(self):
return f"Shape: {self.shape_type.value} Color: {self.color_name} Position: ({self.pos}) Size: {self.bb_box.w}x{self.bb_box.h}"
def as_str(self):
# Print the shape's array as a nicely formatted string
arr = self.array
lines = []
for row in arr:
line = ' '.join(f"{int(val):2d}" for val in row)
lines.append(line)
return '\n'.join(lines)
class Grid:
def __init__(self, grid:np.ndarray):
self.array = grid if isinstance(grid, np.ndarray) else np.array(grid)
# background mask and background color
self.bg_mask, self.bg_color = self._get_background_mask(self.array)
# 8-connected components with the same color pixels
self.shapes_same_color = self._get_component(same_color=True)
# 8-connected components of non-background pixels
self.shapes_foreground = self._get_component(same_color=False)
# shapes constructed by grouping the same color shapes
self.shapes_same_color_grouped = self._get_color_grouped_shape()
def get_same_color_shape(self, i):
return self.shapes_same_color[i]
def get_non_bg_shape(self, i):
return self.shapes_foreground[i]
def __eq__(self, other: 'Grid'):
return np.array_equal(self.array, other.array)
def _get_background_mask(self, grid:np.ndarray):
"""
Get the background mask of a grid.
"""
# Count color frequencies to identify background color
color_counts = np.bincount(grid.flatten())
bg_color = np.argmax(color_counts)
return grid == bg_color, bg_color
def _get_color_grouped_shape(self):
"""
Get the merged shape of the same color components.
"""
colors = set([s.color_code for s in self.shapes_same_color])
color_grouped_shapes = []
for color in colors:
shapes = [s for s in self.shapes_same_color if s.color_code == color]
pixels = reduce(lambda x, y: x + y, [s.pixels for s in shapes])
merged_shape = Shape(pixels, BBox(min(p.x for p in pixels), min(p.y for p in pixels), max(p.x for p in pixels), max(p.y for p in pixels)))
color_grouped_shapes.append(merged_shape)
return color_grouped_shapes
def _get_component(self, same_color:bool=False):
"""
Get the component of a grid.
same_color: if True, only add neighbors of same color
return: a list of components information including the component array and the bounding box
"""
# Get background mask
height, width = self.array.shape
# Initialize visited set and components list
visited = set()
components = []
# Helper function for getting valid neighbors
def get_neighbors(r, c):
neighbors = []
for dr, dc in [(0,1), (0,-1), (1,0), (-1,0), (1,1), (1,-1), (-1,1), (-1,-1)]:
nr, nc = r + dr, c + dc
if 0 <= nr < height and 0 <= nc < width:
if same_color:
# Only add neighbors of same color
if self.array[nr,nc] == self.array[r,c]:
neighbors.append((nr,nc))
else:
# Add any non-background neighbor
if not self.bg_mask[nr,nc]:
neighbors.append((nr,nc))
return neighbors
# Find all components
for r in range(height):
for c in range(width):
if (r,c) not in visited:
if not same_color and self.bg_mask[r,c]:
continue
# Start new component
component = []
stack = [(r,c)]
visited.add((r,c))
# DFS to find connected pixels
while stack:
curr_r, curr_c = stack.pop()
component.append(Pixel(curr_r, curr_c, self.array[curr_r, curr_c]))
# Add valid unvisited neighbors to stack
for nr, nc in get_neighbors(curr_r, curr_c):
if (nr,nc) not in visited:
stack.append((nr,nc))
visited.add((nr,nc))
# Convert component coordinates to array
if component: # Check if component is not empty
min_r = min(p.x for p in component)
min_c = min(p.y for p in component)
max_r = max(p.x for p in component)
max_c = max(p.y for p in component)
components.append(Shape(component, BBox(min_r, min_c, max_r, max_c)))
return components
def to_image(self):
return draw_grid(self.array)
Your answer should consist of two distinct parts:
Part 1: Solution Design (wrapped in markdown tags)
- Analyze the transformation pattern observed in the training examples
- Identify the key rules and logic that govern the input-to-output transformation
- Break down the solution into clear, step-by-step pseudocode
- Explain how each step contributes to achieving the desired output
- Consider edge cases and how they should be handled
- Describe how you will apply this logic to the test input
Part 2: Implementation Code (wrapped in python tags)
- Write clean, well-commented Python code that implements your solution design
- Use the provided libraries (Grid, Shape, BBox, Pixel) effectively
- Follow the exact function signature specified below
- Ensure your code handles all the transformation rules identified in your design
- Return a numpy array as the final output (not a Grid object)
- Test your logic mentally against the training examples to verify correctness
from llm_libs.shapes import Grid, Shape, BBox, Pixel
import numpy as np
def solve(train_input_grid: list[Grid], train_output_grid: list[Grid], test_input_grid: Grid) -> np.ndarray:- The training inputs and outputs are initialized by
Gridclass in thellm_libs.gridmodule. - Do not return Grid object, return a numpy array instead.
- The test input is initialized by
Gridclass in thellm_libs.gridmodule. - The output is a numpy array of data type int64.