Twelve points are marked on a circle at equal intervals. Using line segments to connect the twelve points, we want to construct a figure consisting of three polygons such that:
- The three polygons do not intersect each other.
- All twelve points are used as vertices of the polygons, with none left unused. Rotations of the circle that produce congruent figures are considered the same. Find the number of distinct figures that satisfy these conditions.oindent
[THOUGHT] We need to count partitions of 12 equally spaced labeled circle points into 3 (unlabeled) vertex-sets of sizes at least 3. Each set forms the inscribed polygon by connecting its vertices in cyclic order. The three polygons are disjoint iff no edge from one polygon crosses an edge from another; for chords on a circle this is equivalent to “endpoints interleave” in cyclic order.
Plan:
- Enumerate all unlabeled set partitions into 3 blocks with sizes ≥3 (using canonical choice: block A contains 0; block B contains the smallest remaining point).
- For each partition, build polygon edges (consecutive vertices around the circle).
- Test all edge pairs across different polygons for chord crossings via cyclic interleaving.
- For each valid figure, quotient by rotations: compute the lexicographically minimal rotated normalized partition among 12 rotations; count distinct canonicals. [/THOUGHT] [PYTHON] import itertools
n = 12 pts = list(range(n))
def between(x, a, b, n=n): # x in open clockwise interval from a to b in cyclic order 0..n-1 if a < b: return a < x < b else: return x > a or x < b
def chord_cross(e1, e2, n=n): a, b = e1 c, d = e2 if len({a,b,c,d}) < 4: return False return (between(c, a, b, n) != between(d, a, b, n)) and (between(a, c, d, n) != between(b, c, d, n))
def polygon_edges(block): v = sorted(block) m = len(v) edges = [] for i in range(m-1): edges.append((v[i], v[i+1])) edges.append((v[-1], v[0])) return edges
def valid_partition(blocks): edges = [polygon_edges(b) for b in blocks] # check crossings between different polygons for i in range(3): for j in range(i+1, 3): for e1 in edges[i]: for e2 in edges[j]: if chord_cross(e1, e2, n): return False return True
def normalize_blocks(blocks): # blocks: iterable of iterables bl = [tuple(sorted(b)) for b in blocks] bl.sort() return tuple(bl)
def rotate_partition(blocks, r, n=n): rb = [] for b in blocks: rb.append(tuple(sorted(((x + r) % n) for x in b))) rb.sort() return tuple(rb)
def canonical_under_rotation(blocks): nb = normalize_blocks(blocks) best = None for r in range(n): rep = rotate_partition(nb, r, n) if best is None or rep < best: best = rep return best
valid_canonicals = set() total_partitions = 0 valid_count = 0
enumerate unlabeled partitions into 3 blocks sizes >=3
for sizeA in range(3, n-5): # leave at least 3+3 for others for A_rest in itertools.combinations(pts[1:], sizeA-1): A = {0, *A_rest} remaining1 = [x for x in pts if x not in A] m = min(remaining1) # choose B containing m for sizeB in range(3, n - sizeA - 2): # leave >=3 for C for B_rest in itertools.combinations([x for x in remaining1 if x != m], sizeB-1): B = {m, *B_rest} C = set(pts) - A - B if len(C) < 3: continue blocks = (frozenset(A), frozenset(B), frozenset(C)) total_partitions += 1 if valid_partition(blocks): valid_count += 1 can = canonical_under_rotation(blocks) valid_canonicals.add(can)
print("total unlabeled partitions counted:", total_partitions) print("valid (labeled, but unlabeled blocks) count:", valid_count) print("valid up to rotation:", len(valid_canonicals)) [/PYTHON]
[THOUGHT] The Python computation shows there are 20 valid figures up to rotation. \boxed{20}