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Solving the Jumping Julia Maze

#puzzles
Table of Contents

Introduction

The Jumping Julia Maze from the amazing Julia Robinson Maths Festival is an intriguing puzzle where players must navigate from a starting position to a goal by making jumps based on numbers in a grid. In this post, we’ll break down the problem and implement a solution using Python, NetworkX, and Matplotlib.

Problem Description

Solution Approach

1. Graph Representation

The key insight is to model this as a graph problem:

  • Each square in the grid becomes a node.
  • Valid jumps become directed edges between nodes.
  • Finding a solution becomes a shortest path problem.

Why a graph?

  • Natural representation of connected positions.
  • Well-established algorithms for path finding.
  • Easy visualization capabilities.

2. Implementation Components

a. The JumpingMaze Class

class JumpingMaze:
    def __init__(self, grid):
        self.grid = np.array(grid)
        self.rows, self.cols = self.grid.shape
        self.G = self._create_graph()

b. Graph Creation

The critical part is creating edges for valid jumps:

def _create_graph(self):
    G = nx.DiGraph()
    
    # Add nodes
    for i in range(self.rows):
        for j in range(self.cols):
            G.add_node((i, j), pos=(j, -i))
    
    # Add edges for valid jumps
    for i in range(self.rows):
        for j in range(self.cols):
            jumps = self.grid[i, j]
            if jumps == 0:  # Skip goal node
                continue
            
            # Horizontal jumps
            for dx in [-jumps, jumps]:
                new_j = j + dx
                if 0 <= new_j < self.cols:
                    G.add_edge((i, j), (i, new_j))
            
            # Vertical jumps
            for dy in [-jumps, jumps]:
                new_i = i + dy
                if 0 <= new_i < self.rows:
                    G.add_edge((i, j), (new_i, j))

Key Points:

  • Each node is represented by its (row, col) coordinates.
  • Edges are only created for exact jump distances.
  • The goal node (0 value) doesn’t need outgoing edges.

3. Finding the Solution

We use NetworkX’s shortest_path algorithm:

def find_solution(self, start=(0, 0)):
    goal = (self.rows - 1, self.cols - 1)
    try:
        return nx.shortest_path(self.G, start, goal)
    except nx.NetworkXNoPath:
        return None

This efficiently finds:

  • The shortest sequence of jumps to reach the goal.
  • Or determines that no solution exists.

4. Visualizing the solution

Our visualization includes several key features:

Node colors:

  • Light blue for regular squares.
  • Light green for squares in the solution path.

Arrows:

  • Gray curved arrows for all possible jumps.
  • Bold red curved arrows for the solution path.

Labels:

  • Grid numbers in each square.
  • Sequential numbers for solution steps.
  • Special labels for Start and Goal.

How to Use the Solution

  1. Create your grid:
grid = [
    [3, 2, 2, 1],
    [3, 2, 1, 3],
    [1, 1, 1, 1],
    [1, 3, 1, 0]
]
  1. Initialize the maze:
maze = JumpingMaze(grid)
  1. Find and visualize the solution:
path = maze.find_solution()
if path:
    maze.visualize(path)
else:
    print("No solution exists!")

A hard maze example

Here is a hard maze along with the solution:

Conclusion

The Jumping Julia Maze demonstrates how graph theory can elegantly solve path-finding puzzles. By converting the grid into a graph, we can leverage existing algorithms to find solutions efficiently and create clear visualizations to understand the path.

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