Program to reset a polygon to its initial state in Python

A polygon can be transformed through rotations and flips along specific axes. This problem involves tracking these transformations and determining what operation is needed to reset the polygon to its initial state.

Understanding Polygon Transformations

For a polygon with n vertices, there are specific rules for transformation axes:

• If n is odd, each flipping axis passes through one vertex and the middle of the opposite side
• If n is even, half the axes pass through opposite vertices, half through opposite sides
• Adjacent axes have an angle of 360/(2n) degrees

The transformation operations work as follows:

• Rotation (type 1): A k-rotator rotates the polygon clockwise by (360 × k)/n degrees
• Flip (type 2): Flips the polygon along axis k

Algorithm

The solution tracks the polygon's state using two variables:

• position: Current rotational offset
• decision_var: Whether the polygon is flipped (boolean)

Example

Let's solve for a hexagon with transformations [[1, 2], [1, 4], [2, 3], [2, 5], [1, 6]]:

def solve(n, input_list):
    decision_var = False
    position = 0
    
    for item in input_list:
        x = item[0]  # Operation type (1=rotate, 2=flip)
        y = item[1]  # Axis/amount
        
        if x == 1:  # Rotation
            position += y
        else:       # Flip
            position = y - position
            decision_var = not decision_var
    
    position = position % n
    
    if decision_var:
        return (2, position)  # Need flip to reset
    else:
        return (1, n - position)  # Need rotation to reset

# Test with the example
n = 6
transformations = [[1, 2], [1, 4], [2, 3], [2, 5], [1, 6]]
result = solve(n, transformations)
print(f"Reset operation: {result}")

Reset operation: (1, 4)

Step-by-Step Execution

For the example input, here's how the algorithm processes each transformation:

def solve_with_trace(n, input_list):
    decision_var = False
    position = 0
    
    print(f"Initial state: position={position}, flipped={decision_var}")
    
    for i, item in enumerate(input_list):
        x, y = item[0], item[1]
        
        if x == 1:  # Rotation
            position += y
            print(f"Step {i+1}: Rotate by axis {y}, position={position}")
        else:       # Flip
            position = y - position
            decision_var = not decision_var
            print(f"Step {i+1}: Flip along axis {y}, position={position}, flipped={decision_var}")
    
    position = position % n
    print(f"Final: position={position} mod {n}, flipped={decision_var}")
    
    if decision_var:
        return (2, position)
    else:
        return (1, n - position)

result = solve_with_trace(6, [[1, 2], [1, 4], [2, 3], [2, 5], [1, 6]])
print(f"Reset needed: {result}")

Initial state: position=0, flipped=False
Step 1: Rotate by axis 2, position=2
Step 2: Rotate by axis 4, position=6
Step 3: Flip along axis 3, position=-3, flipped=True
Step 4: Flip along axis 5, position=8, flipped=False
Step 5: Rotate by axis 6, position=14
Final: position=2 mod 6, flipped=False
Reset needed: (1, 4)

How It Works

The algorithm maintains the polygon's state through transformations:

• Rotations add to the position counter
• Flips reflect the position and toggle the flip state
• The final position modulo n gives the net rotation
• If flipped, return a flip operation; otherwise return the inverse rotation

Conclusion

This solution efficiently tracks polygon transformations using position and flip state variables. The algorithm determines the exact inverse operation needed to reset any transformed polygon to its original orientation.