Lights Out

An interactive Python implementation of the classic Lights Out puzzle, extended with multiple algebraic state spaces and nontrivial topological grids.

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Overview

This project implements a fully playable version of Lights Out with several enhancements:

• Variable grid sizes from 2×2 up to 15×15
• Multiple state spaces over Z_p for p in {2,3,5,7}
• Support for nontrivial topologies:
  • Simple grid
  • Cylinder
  • Möbius strip
  • Torus
  • Klein bottle
  • Projective plane
• Built-in solver using linear algebra over finite fields
• Graphical user interface with topology visualization

Each game is initialized in a way that guarantees it is solvable.

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How It Works

Each board configuration is modeled as a vector over Z_p, and button presses correspond to adding columns of an adjacency matrix A.

To solve the puzzle, we seek a press vector x that sends the current board state b back to the zero state. This means solving the linear system

A x ≡ -b (mod p)

where:

• b is the current board state
• x is the solution vector of button presses

A reduced form of the adjacency matrix is computed along with a solver matrix A⁺, allowing solutions to be generated efficiently during gameplay.

If the system has nontrivial nullity, multiple solutions may exist; the program returns one valid solution.

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Features

• Topology-aware neighbor structure
  Button interactions depend on the selected surface, including wraparound and twist identifications on spaces such as the torus, Möbius strip, Klein bottle, and projective plane.

• Guaranteed solvability
  Initial states are generated from valid press vectors, ensuring that every puzzle can be completed.

• Hint system
  Generates a valid sequence of button presses to solve the current board.

• Dynamic UI

  • Adjustable board size and state space
  • Visual preview of each topology
  • Responsive grid layout

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Requirements

• Python 3.x
• numpy
• Pillow

Install dependencies with:

pip install numpy pillow

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Running the Program

From the project directory:

python Lights_Out.py

Notes

• The appearance of the application icon may vary depending on the operating system and desktop environment.
• Image paths are resolved relative to the script location for portability.