"""
Construct a 7-day schedule using triangle pairing.
24 people → 8 triples of 3 → pair triples into boats of 6.
"""
import itertools, math, json

N = 24
TOTAL = math.comb(N, 2)

def norm(p,q): return (min(p,q), max(p,q))

def count_pairs(sched):
    pairs = set()
    for day in sched:
        for b in day:
            pairs.update(norm(p,q) for p,q in itertools.combinations(b, 2))
    return len(pairs)

def missing_pairs(sched):
    pairs = set()
    for day in sched:
        for b in day:
            pairs.update(norm(p,q) for p,q in itertools.combinations(b, 2))
    return set(norm(i,j) for i in range(N) for j in range(i+1,N)) - pairs

# 8 triples (groups of 3). Each person p is in triple t = p // 3.
# Person p is in triple t = p // 3, at position pos = p % 3.
triples = [[3*t + j for j in range(3)] for t in range(8)]

# A 1-factorization of K_8 (vertices 0..7 representing the triples)
# Using the standard construction: vertices are elements of Z_7 ∪ {∞}
# For round r = 0..6: pair (∞, r) and (r+i, r-i) for i = 1..3
one_factorization = []
vertices = list(range(7))  # 0..6
for r in range(7):
    matching = [(r, 7)]  # (triple r, triple 7)  [7 represents ∞, i.e., triple 7]
    for i in range(1, 4):
        a = (r + i) % 7
        b = (r - i) % 7
        matching.append((a, b))
    one_factorization.append(matching)

print("1-factorization of K_8:")
for i, matching in enumerate(one_factorization):
    print(f"  Factor {i+1}: {matching}")

# Build schedule
def matching_to_boats(matching):
    boats = []
    for (t1, t2) in matching:
        boat = sorted(triples[t1] + triples[t2])
        boats.append(tuple(boat))
    return boats

sched = [matching_to_boats(m) for m in one_factorization]

cov = count_pairs(sched)
print(f"\n7-day coverage: {cov}/{TOTAL}")

if cov == TOTAL:
    print("SUCCESS! All 276 pairs covered.")
    print("\nSchedule:")
    for i, day in enumerate(sched):
        print(f"  Day {i+1}: {day}")
    print(f"\nJSON format:\n{json.dumps(sched)}")
else:
    print(f"Missing: {len(missing_pairs(sched))} pairs")
    print(f"Missing pairs: {sorted(missing_pairs(sched))[:20]}...")