Optimizing Chessboard Queen Placement with Python: A Step-by-Step Guide

In this article, we will demonstrate how to optimize the placement of queens on a chessboard using Python. This problem is a classic example of solving optimization problems with binary variables. By following our step-by-step guide, you'll be able to maximize the number of queens placed on the board without conflicts. Let's dive in! 🤓

Why Solve This Problem? 🤔

The N Queen problem has applications in various areas such as computer science, mathematics, and even physics. It helps us understand how to place multiple items on a limited space while ensuring they don't interfere with each other. Solving this problem can enhance your skills in problem-solving and optimization.

Tools Needed 🛠️

To solve the N Queen problem, we will use Python and packages such as pyomo and mathematical. These tools are essential for modeling the problem, defining constraints, and finding optimal solutions.

Solving the Problem 🔍

To start solving the N Queen problem, we need to create binary variables U_ij to represent if a queen exists on cell (i,j) or not. Our objective is to maximize this summation. Here are some rules:
1. For every given column j, there can't be more than one queen on the board. This rule ensures that each column only has one queen.
2. For every given row i, the same applies; there should be less than one queen on each row to avoid conflicts.
3. Diagonal rules: There can't be more than one queen on every diagonal or anti-diagonal. This ensures that queens don't attack each other.
4. Objective function: Our goal is to maximize the summation of U_ij for all i and j while satisfying these constraints.

Let's see how we can implement this in Python:
```python

Import required packages

import pyomo, mathematical

Define board size

n = 8 # 8x8 chessboard

Define sets of queens (i for rows and j for columns)

i = range(n)
j = range(n)

Define binary variable U_ij representing a queen on cell (i,j)

u = pyomo.Var(index=(i,j), domain=pyomo.BooleanDom dom, within=range(0,2))

Define constraints

pyomo.Constraint(j, rule=lambda j: sum(v[i][j] for i in range(n)) <= 1)
pyomo.Constraint(i, rule=lambda i: sum(v[i][j] for j in range(n)) <= 1)

Define diagonal constraints

diac = pyomo.Var(index=(range(n), range(n)), within=range(0,2))
pyomo.Constraint(range(len(diac)), lambda diac_ij: diac_ij[i - j] + diac_ij[i + j] <= 1)

Define objective function to maximize the sum of U_ij

obj = pyomo.Objective(rule=lambda v: sum(v[i][j] for i, j in zip(i, j)), sense=pyomo.maximize)

Solve the optimization problem

solver = pyomo.SolverFactory('glpk')
solver.solve(model)
print('Queen locations:', v.pprint())
```

Visualizing Results 🎨

After solving the optimization problem, we can visualize the queen placements on the board with a simple Python script.

Solving Larger Boards 🌐

You can easily solve this problem for larger boards by changing the board size and observing the results. For example, you can try a 16x16 chessboard to see how many queens can be placed without conflicts.

Conclusion 🎉

By following this step-by-step guide, you've learned how to solve the N Queen problem using Python and optimization techniques. This knowledge can help you tackle similar problems in various areas of computer science and mathematics. Keep exploring and improving your skills!💡