Problem Analysis #24

The problem asks for the number of distinct figures formed by three non-intersecting polygons using 12 equally spaced points on a circle, where all points are used as vertices. Rotations of the figure are considered the same.

The approach is:

1. Represent the 12 points as integers $0, 1, \dots, 11$.
2. Generate all valid partitions of these 12 points into 3 subsets (polygons) such that:
   • Each subset has at least 3 vertices.
   • The polygons are non-intersecting (their edges do not cross).
3. Store these figures as sets of sets of vertices to handle the indistinguishability of the polygons.
4. Group the figures by rotational symmetry (orbits under the cyclic group $C_{12}$) and count the number of orbits.

The possible sizes for the three polygons (summing to 12, min size 3) are:

• $\{3, 3, 6\}$
• $\{3, 4, 5\}$
• $\{4, 4, 4\}$

I will generate partitions for each size distribution, check the non-crossing condition, and then count the orbits.

[PYTHON] import itertools

def is_crossing(edges1, edges2): """Check if any edge in edges1 crosses any edge in edges2.""" for u1, v1 in edges1: for u2, v2 in edges2: # Ensure u < v for standard check a, b = sorted((u1, v1)) c, d = sorted((u2, v2))

        # Two chords (a,b) and (c,d) intersect if strictly interlaced
        # a < c < b < d or c < a < d < b
        if (a < c < b < d) or (c < a < d < b):
            return True
return False

def get_edges(vertices): """Return list of edges (as tuples) for a polygon with given vertices.""" # Sort vertices to follow the circle order vs = sorted(list(vertices)) edges = [] for i in range(len(vs)): edges.append((vs[i], vs[(i + 1) % len(vs)])) return edges

def solve(): points = range(12) valid_figures = set()

# Case 1: Sizes 3, 3, 6
# Choose 6 for the first polygon
for p1 in itertools.combinations(points, 6):
    rem1 = [x for x in points if x not in p1]
    # Choose 3 for the second polygon from remainder
    for p2 in itertools.combinations(rem1, 3):
        p3 = tuple([x for x in rem1 if x not in p2])
        
        # Define polygons
        poly1 = p1
        poly2 = p2
        poly3 = p3
        
        # Get edges
        edges1 = get_edges(poly1)
        edges2 = get_edges(poly2)
        edges3 = get_edges(poly3)
        
        # Check crossing
        if is_crossing(edges1, edges2) or is_crossing(edges1, edges3) or is_crossing(edges2, edges3):
            continue
        
        # Add to set (frozenset of frozensets to handle order independence)
        figure = frozenset([frozenset(poly1), frozenset(poly2), frozenset(poly3)])
        valid_figures.add(figure)

# Case 2: Sizes 3, 4, 5
for p1 in itertools.combinations(points, 5):
    rem1 = [x for x in points if x not in p1]
    for p2 in itertools.combinations(rem1, 4):
        p3 = tuple([x for x in rem1 if x not in p2])
        
        poly1 = p1 # size 5
        poly2 = p2 # size 4
        poly3 = p3 # size 3
        
        edges1 = get_edges(poly1)
        edges2 = get_edges(poly2)
        edges3 = get_edges(poly3)
        
        if is_crossing(edges1, edges2) or is_crossing(edges1, edges3) or is_crossing(edges2, edges3):
            continue
            
        figure = frozenset([frozenset(poly1), frozenset(poly2), frozenset(poly3)])
        valid_figures.add(figure)

# Case 3: Sizes 4, 4, 4
# To avoid over-generating, we can fix the first element of the first polygon to be the smallest available (0)
# But itertools.combinations does not enforce partition order.
# We will generate all and let the set logic handle duplicates, or be slightly smarter.
# Generating all is safe and fast enough.
for p1 in itertools.combinations(points, 4):
    rem1 = [x for x in points if x not in p1]
    for p2 in itertools.combinations(rem1, 4):
        p3 = tuple([x for x in rem1 if x not in p2])
        
        poly1 = p1
        poly2 = p2
        poly3 = p3
        
        edges1 = get_edges(poly1)
        edges2 = get_edges(poly2)
        edges3 = get_edges(poly3)
        
        if is_crossing(edges1, edges2) or is_crossing(edges1, edges3) or is_crossing(edges2, edges3):
            continue
            
        figure = frozenset([frozenset(poly1), frozenset(poly2), frozenset(poly3)])
        valid_figures.add(figure)

print(f"Total valid static figures: {len(valid_figures)}")

# Count orbits
orbits = 0
remaining_figures = set(valid_figures)

while remaining_figures:
    orbits += 1
    current = remaining_figures.pop()
    
    # Generate all rotations
    for i in range(1, 12):
        # Rotate each polygon
        rotated_polys = []
        for poly in current:
            rotated_poly = frozenset([(v + i) % 12 for v in poly])
            rotated_polys.append(rotated_poly)
        rotated_figure = frozenset(rotated_polys)
        
        if rotated_figure in remaining_figures:
            remaining_figures.remove(rotated_figure)
            
print(f"Number of distinct figures (orbits): {orbits}")

solve() [/PYTHON] The problem asks for the number of distinct figures consisting of three non-intersecting polygons using 12 equally spaced points on a circle, where all points are used as vertices. Rotations of the figure are considered the same.

We modeled the problem by partitioning the 12 points (labeled $0$ to $11$) into three subsets, each of size at least 3. The possible integer partitions of 12 into 3 parts of size $\ge 3$ are $(3, 3, 6)$, $(3, 4, 5)$, and $(4, 4, 4)$.

We generated all valid partitions for these size distributions and checked the non-intersecting condition. The condition "polygons do not intersect" implies that no two edges (chords) from different polygons can cross.

Using a Python script, we:

1. Generated all partitions of $\{0, \dots, 11\}$ into three sets with the required sizes.
2. Filtered out those where edges from different polygons cross.
3. Stored the valid figures as sets of sets of vertices to account for the indistinguishability of the polygons.
4. Grouped the valid figures by rotational symmetry to count the number of distinct orbits.

The script found 224 valid static figures. After accounting for rotational symmetry (dividing by the size of the orbit for each figure), the number of distinct figures is 20.

[THOUGHT] The code execution confirmed there are 224 valid labeled configurations. The orbit counting resulted in 20 distinct figures. The logic covers all constraints:

• 3 polygons
• Non-intersecting (checked via edge crossing)
• All 12 points used
• Min size 3 (enforced by partition sizes)
• Rotational symmetry handled.

Final result is 20. [/THOUGHT]

\boxed{20}

The number of distinct figures is 20.

\boxed{20}