# Problem Description
This is a geometry problem that determines whether two line segments intersect within a circle. The circle has 8 points numbered 1-8 arranged around its circumference. Each line segment is defined by its endpoints, which are numbered points on the circle. The function needs to return True if the lines intersect and False if they don't.

# Visual Facts
1. The circle contains 8 points numbered 1 through 8
2. Points are evenly spaced around the circumference
3. Points are arranged clockwise in the order: 8, 1, 3, 2, 5, 4, 7, 6
4. Two lines are shown:
   - A blue line connecting points 8 and 2
   - An orange line connecting points 1 and 5
5. The example shows these lines intersecting inside the circle
6. Each point has a fixed position on the circle's circumference
7. The points form a perfect octagon if connected sequentially

# Visual Patterns
1. Line Intersection Pattern:
   - Lines intersect when they cross each other's path inside the circle

2. Point Position Pattern:
   - Each point's position can be thought of as having an angular value (45° apart)
   - 8 is at the top, 1 is at the top-right, 3 is at the right, and so on
   - The relative positions of points determine whether lines will intersect

3. Intersection Rules:
   suppose two lines are defined by points A, B and C, D
   - if A or B is between C and D, then the lines intersect
   - For example, the orange line connets 1 and 5, where 1 (index 1) is between 8 (index 0) and 2 (index 3) according to the point order, then the blue line intersects the orange line
   - sort the index, not the value before comparison

# Solution
from typing import Tuple

def solution(line1: Tuple[int, int], line2: Tuple[int, int]) -> bool:
    """
    Determines if two line segments intersect within the circle.
    
    Parameters:
    line1: A tuple containing the numbers at both ends of the first line segment.
    line2: A tuple containing the numbers at both ends of the second line segment.
    
    Returns:
    bool: True if the line segments intersect, False otherwise.
    """
    mapping = {
        1: 1,
        2: 3,
        3: 2,
        4: 5,
        5: 4,
        6: 7,
        7: 6,
        8: 8
    }
    line1 = tuple(mapping[x] for x in line1)
    line2 = tuple(mapping[x] for x in line2)
    a, b = sorted(line1)
    c, d = sorted(line2)
    
    # Check if the points (a, b) and (c, d) intersect on the circle
    return (a < c < b < d) or (c < a < d < b)


# Test Script
def test():
    test_cases = [
        ((3, 7), (1, 5), True),
        ((2, 8), (1, 3), False),
        ((2, 8), (5, 8), False),
        ((3, 6), (1, 8), False),
        ((1, 4), (5, 8), True),
        ((1, 6), (2, 8), True),
        ((3, 7), (2, 4), False),
        ((1, 7), (3, 2), False),
        ((1, 8), (3, 2), False),
        ((1, 6), (3, 8), True)
    ]

    for line1, line2, expected_output in test_cases:
        try:
            output = solution(line1, line2)
        except Exception as e:
            error_message = (
                f"An exception occurred while running the test case:\n"
                f"`line1`: {line1}\n"
                f"`line2`: {line2}\n\n"
                f"Exception: '{str(e)}'"
            )
            assert False, error_message

        if output != expected_output:
            error_message = (
                f"The following Test Case failed:\n"
                f"`line1`: {line1}\n"
                f"`line2`: {line2}\n\n"
                f"Actual Output: {output}\n"
                f"Expected Output: {expected_output}"
            )
            assert False, error_message

test()