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
\boxed{12}
[THOUGHT] We are generating all partitions of 12 points on a circle into three non-empty subsets, each of size at least 3, and then accounting for rotations by finding the lexicographically smallest representation for each partition under rotation. This will give us the number of distinct figures up to rotation. [/THOUGHT] [PYTHON] from itertools import combinations
partitions_set = set()
For (3,3,6)
for subset1 in combinations(range(12), 3): remaining = set(range(12)) - set(subset1) for subset2 in combinations(remaining, 3): subset3 = tuple(remaining - set(subset2)) s1 = tuple(sorted(subset1)) s2 = tuple(sorted(subset2)) s3 = tuple(sorted(subset3)) part = tuple(sorted([s1, s2, s3])) min_rep = part for k in range(1, 12): s1_rot = tuple(sorted((x + k) % 12 for x in s1)) s2_rot = tuple(sorted((x + k) % 12 for x in s2)) s3_rot = tuple(sorted((x + k) % 12 for x in s3)) part_rot = tuple(sorted([s1_rot, s2_rot, s3_rot])) if part_rot < min_rep: min_rep = part_rot partitions_set.add(min_rep)
For (3,4,5)
for subset1 in combinations(range(12), 3): remaining = set(range(12)) - set(subset1) for subset2 in combinations(remaining, 4): subset3 = tuple(remaining - set(subset2)) s1 = tuple(sorted(subset1)) s2 = tuple(sorted(subset2)) s3 = tuple(sorted(subset3)) part = tuple(sorted([s1, s2, s3])) min_rep = part for k in range(1, 12): s1_rot = tuple(sorted((x + k) % 12 for x in s1)) s2_rot = tuple(sorted((x + k) % 12 for x in s2)) s3_rot = tuple(sorted((x + k) % 12 for x in s3)) part_rot = tuple(sorted([s1_rot, s2_rot, s3_rot])) if part_rot < min_rep: min_rep = part_rot partitions_set.add(min_rep)
For (4,4,4)
for subset1 in combinations(range(12), 4): remaining = set(range(12)) - set(subset1) for subset2 in combinations(remaining, 4): subset3 = tuple(remaining - set(subset2)) s1 = tuple(sorted(subset1)) s2 = tuple(sorted(subset2)) s3 = tuple(sorted(subset3)) part = tuple(sorted([s1, s2, s3])) min_rep = part for k in range(1, 12): s1_rot = tuple(sorted((x + k) % 12 for x in s1)) s2_rot = tuple(sorted((x + k) % 12 for x in s2)) s3_rot = tuple(sorted((x + k) % 12 for x in s3)) part_rot = tuple(sorted([s1_rot, s2_rot, s3_rot])) if part_rot < min_rep: min_rep = part_rot partitions_set.add(min_rep)
print(len(partitions_set)) [/PYTHON]
\boxed{3587}