3分割問題を解く計算表を作った (n = 9 限定)
\(n=9\) の3分割問題を解く計算表を作った．
3分割問題って何？
3 分割問題 (英: 3-partition problem) とは，こんな問題です:
3分割問題とは、与えられた整数の多集合を，すべての和が同じである3つの集合に分割できるかどうかを決めるものである．より正確に言うと
1) この問題の入力は \(n=3m\) 個の正の整数の集合 \(S\) である．全ての整数の和は \(mT\)
2) 出力は， \(S\) を \(m\) 個の3つ組 \(p_{1}, p_{2}, …, p_{m}\) に分割し，それぞれの数の和が \(T\) になるような分割が存在するかどうか
3-partition problem – Wikipedia を改編
具体例を見てみましょう．例えば集合 \(S=\{ 20, 23, 25, 30, 49, 45, 27, 30, 30, 40, 22, 19 \}\) を考えることにすると，答えは「(条件を満たす分割は) ある」で，条件を満たす分割を具体的に示せば \(p_{1} = \{ 20, 25, 45 \}, p_{2} = \{ 23, 27, 40 \}, p_{3} = \{ 49, 22, 19 \}, p_{4} = \{ 30, 30, 30 \}\) ですね．
3分割問題は NP 完全な問題であることが知られている．NP 完全を簡単に説明すると「クラス NP に属する問題の中でも難しいもの」というような意味で，クラス NP に属する問題とは「解くのは難しいけど，検算するのは簡単な問題」という感じ．数式の見た目がかっこいいいからクラス NP の定義式だけ書いてみよう．
\(\displaystyle \mathrm{NP} = \bigcup_{k \in{\mathbb{N}}} {\mathrm{NTIME}(n^k)} \)
入力長 \(n=9\) の3分割問題を解きたい！
3分割問題は NP 完全問題なので，効率的な解法は発見されていない．すぐに思いつく解法は，条件を満たす分割が「ある」か「ない」か，全ての組み合わせを愚直に検証して解くこと (おそらく全探索よりマシな解法があると思いますが)．入力長 \(n=3m=9\) のとき，検証すべき全ての組み合わせの数を \(A\) とおくと，次の計算から \(A = 280\) 通りだと分かる．
\(\displaystyle A = \frac{ {}_9 \mathrm{C}_3 \cdot {}_6 \mathrm{C}_3 \cdot {}_3 \mathrm{C}_3}{ {}_3 \mathrm{P}_3} = \frac{n!}{m! \cdot (3!)^m} = \frac{362880}{6 \cdot 216} = 280 \)
入力 \(S\) に含まれる9つの整数を，入力の1番目，入力の2番目，…，入力の9番目のように区別することにして，入力の a 番目と b 番目と c 番目の合計値を S(a, b, c) と表記することにすれば，検証すべき280通りの \(p_{1}, p_{2}, p_{3}\) が書き下せる．リストはちょっと長いので記事末尾に掲載しますね．
後は計算するだけ
ここまで準備できれば後は簡単．計算表で =ROUND( RAND() * random_max ) とか書いて，ある数以下の整数の乱数を9つ生成して，280通りの \(p_{1}，p_{2}，p_{3}\) を計算して各パターンに対して =AND( p_1 = p_2, p_2 = p_3) を返せばいい．最終的な出力は =OR( 280個の真理値 ) だね．僕は，分割可能な場合にはチームごとに色分けしたり，入力数列の昇順整列とかもしてるけど，これは単なるオマケ．
RANDOM_MAX = 50 でも色々試しに計算した総当りで解くより少しはマシな解法ならありそうな気がする．でも3分割問題は NP 完全問題で，(もし P ≠ NP なら) これを「効率的に解くアルゴリズム」は存在しない．総当りすべき場合の数は 入力長 \(n = 3m\) を使って \( \displaystyle \frac{n!}{m! \cdot (3!)^m } \) と書けて，仮に総当りで解こうとすればその計算量は \(\mathcal{O}(n!)\) となってしまう．
今回は入力長 \(n=9\) という小さい数を扱ったので計算表でも現実的な時間で解けたけど，\(n=12\) なら場合の数は 15,400 通り，\(n=15\) なら場合の数は 1,401,400 通りというふうに，ものすごい速度で計算量が増えてしまう．総当り解法は明らかに「効率のいいアルゴリズム」ではないね．
考えうる 280通りの \(p_{1}, p_{2}, p_{3}\) のリスト
001: p_1 = S(1, 2, 3), p_2 = S(4, 5, 6), p_3 = S(7, 8, 9) 002: p_1 = S(1, 2, 3), p_2 = S(4, 5, 7), p_3 = S(6, 8, 9) 003: p_1 = S(1, 2, 3), p_2 = S(4, 5, 8), p_3 = S(6, 7, 9) 004: p_1 = S(1, 2, 3), p_2 = S(4, 5, 9), p_3 = S(6, 7, 8) 005: p_1 = S(1, 2, 3), p_2 = S(4, 6, 7), p_3 = S(5, 8, 9) 006: p_1 = S(1, 2, 3), p_2 = S(4, 6, 8), p_3 = S(5, 7, 9) 007: p_1 = S(1, 2, 3), p_2 = S(4, 6, 9), p_3 = S(5, 7, 8) 008: p_1 = S(1, 2, 3), p_2 = S(4, 7, 8), p_3 = S(5, 6, 9) 009: p_1 = S(1, 2, 3), p_2 = S(4, 7, 9), p_3 = S(5, 6, 8) 010: p_1 = S(1, 2, 3), p_2 = S(4, 8, 9), p_3 = S(5, 6, 7) 011: p_1 = S(1, 2, 4), p_2 = S(3, 5, 6), p_3 = S(7, 8, 9) 012: p_1 = S(1, 2, 4), p_2 = S(3, 5, 7), p_3 = S(6, 8, 9) 013: p_1 = S(1, 2, 4), p_2 = S(3, 5, 8), p_3 = S(6, 7, 9) 014: p_1 = S(1, 2, 4), p_2 = S(3, 5, 9), p_3 = S(6, 7, 8) 015: p_1 = S(1, 2, 4), p_2 = S(3, 6, 7), p_3 = S(5, 8, 9) 016: p_1 = S(1, 2, 4), p_2 = S(3, 6, 8), p_3 = S(5, 7, 9) 017: p_1 = S(1, 2, 4), p_2 = S(3, 6, 9), p_3 = S(5, 7, 8) 018: p_1 = S(1, 2, 4), p_2 = S(3, 7, 8), p_3 = S(5, 6, 9) 019: p_1 = S(1, 2, 4), p_2 = S(3, 7, 9), p_3 = S(5, 6, 8) 020: p_1 = S(1, 2, 4), p_2 = S(3, 8, 9), p_3 = S(5, 6, 7) 021: p_1 = S(1, 2, 5), p_2 = S(3, 4, 6), p_3 = S(7, 8, 9) 022: p_1 = S(1, 2, 5), p_2 = S(3, 4, 7), p_3 = S(6, 8, 9) 023: p_1 = S(1, 2, 5), p_2 = S(3, 4, 8), p_3 = S(6, 7, 9) 024: p_1 = S(1, 2, 5), p_2 = S(3, 4, 9), p_3 = S(6, 7, 8) 025: p_1 = S(1, 2, 5), p_2 = S(3, 6, 7), p_3 = S(4, 8, 9) 026: p_1 = S(1, 2, 5), p_2 = S(3, 6, 8), p_3 = S(4, 7, 9) 027: p_1 = S(1, 2, 5), p_2 = S(3, 6, 9), p_3 = S(4, 7, 8) 028: p_1 = S(1, 2, 5), p_2 = S(3, 7, 8), p_3 = S(4, 6, 9) 029: p_1 = S(1, 2, 5), p_2 = S(3, 7, 9), p_3 = S(4, 6, 8) 030: p_1 = S(1, 2, 5), p_2 = S(3, 8, 9), p_3 = S(4, 6, 7) 031: p_1 = S(1, 2, 6), p_2 = S(3, 4, 5), p_3 = S(7, 8, 9) 032: p_1 = S(1, 2, 6), p_2 = S(3, 4, 7), p_3 = S(5, 8, 9) 033: p_1 = S(1, 2, 6), p_2 = S(3, 4, 8), p_3 = S(5, 7, 9) 034: p_1 = S(1, 2, 6), p_2 = S(3, 4, 9), p_3 = S(5, 7, 8) 035: p_1 = S(1, 2, 6), p_2 = S(3, 5, 7), p_3 = S(4, 8, 9) 036: p_1 = S(1, 2, 6), p_2 = S(3, 5, 8), p_3 = S(4, 7, 9) 037: p_1 = S(1, 2, 6), p_2 = S(3, 5, 9), p_3 = S(4, 7, 8) 038: p_1 = S(1, 2, 6), p_2 = S(3, 7, 8), p_3 = S(4, 5, 9) 039: p_1 = S(1, 2, 6), p_2 = S(3, 7, 9), p_3 = S(4, 5, 8) 040: p_1 = S(1, 2, 6), p_2 = S(3, 8, 9), p_3 = S(4, 5, 7) 041: p_1 = S(1, 2, 7), p_2 = S(3, 4, 5), p_3 = S(6, 8, 9) 042: p_1 = S(1, 2, 7), p_2 = S(3, 4, 6), p_3 = S(5, 8, 9) 043: p_1 = S(1, 2, 7), p_2 = S(3, 4, 8), p_3 = S(5, 6, 9) 044: p_1 = S(1, 2, 7), p_2 = S(3, 4, 9), p_3 = S(5, 6, 8) 045: p_1 = S(1, 2, 7), p_2 = S(3, 5, 6), p_3 = S(4, 8, 9) 046: p_1 = S(1, 2, 7), p_2 = S(3, 5, 8), p_3 = S(4, 6, 9) 047: p_1 = S(1, 2, 7), p_2 = S(3, 5, 9), p_3 = S(4, 6, 8) 048: p_1 = S(1, 2, 7), p_2 = S(3, 6, 8), p_3 = S(4, 5, 9) 049: p_1 = S(1, 2, 7), p_2 = S(3, 6, 9), p_3 = S(4, 5, 8) 050: p_1 = S(1, 2, 7), p_2 = S(3, 8, 9), p_3 = S(4, 5, 6) 051: p_1 = S(1, 2, 8), p_2 = S(3, 4, 5), p_3 = S(6, 7, 9) 052: p_1 = S(1, 2, 8), p_2 = S(3, 4, 6), p_3 = S(5, 7, 9) 053: p_1 = S(1, 2, 8), p_2 = S(3, 4, 7), p_3 = S(5, 6, 9) 054: p_1 = S(1, 2, 8), p_2 = S(3, 4, 9), p_3 = S(5, 6, 7) 055: p_1 = S(1, 2, 8), p_2 = S(3, 5, 6), p_3 = S(4, 7, 9) 056: p_1 = S(1, 2, 8), p_2 = S(3, 5, 7), p_3 = S(4, 6, 9) 057: p_1 = S(1, 2, 8), p_2 = S(3, 5, 9), p_3 = S(4, 6, 7) 058: p_1 = S(1, 2, 8), p_2 = S(3, 6, 7), p_3 = S(4, 5, 9) 059: p_1 = S(1, 2, 8), p_2 = S(3, 6, 9), p_3 = S(4, 5, 7) 060: p_1 = S(1, 2, 8), p_2 = S(3, 7, 9), p_3 = S(4, 5, 6) 061: p_1 = S(1, 2, 9), p_2 = S(3, 4, 5), p_3 = S(6, 7, 8) 062: p_1 = S(1, 2, 9), p_2 = S(3, 4, 6), p_3 = S(5, 7, 8) 063: p_1 = S(1, 2, 9), p_2 = S(3, 4, 7), p_3 = S(5, 6, 8) 064: p_1 = S(1, 2, 9), p_2 = S(3, 4, 8), p_3 = S(5, 6, 7) 065: p_1 = S(1, 2, 9), p_2 = S(3, 5, 6), p_3 = S(4, 7, 8) 066: p_1 = S(1, 2, 9), p_2 = S(3, 5, 7), p_3 = S(4, 6, 8) 067: p_1 = S(1, 2, 9), p_2 = S(3, 5, 8), p_3 = S(4, 6, 7) 068: p_1 = S(1, 2, 9), p_2 = S(3, 6, 7), p_3 = S(4, 5, 8) 069: p_1 = S(1, 2, 9), p_2 = S(3, 6, 8), p_3 = S(4, 5, 7) 070: p_1 = S(1, 2, 9), p_2 = S(3, 7, 8), p_3 = S(4, 5, 6) 071: p_1 = S(1, 3, 4), p_2 = S(2, 5, 6), p_3 = S(7, 8, 9) 072: p_1 = S(1, 3, 4), p_2 = S(2, 5, 7), p_3 = S(6, 8, 9) 073: p_1 = S(1, 3, 4), p_2 = S(2, 5, 8), p_3 = S(6, 7, 9) 074: p_1 = S(1, 3, 4), p_2 = S(2, 5, 9), p_3 = S(6, 7, 8) 075: p_1 = S(1, 3, 4), p_2 = S(2, 6, 7), p_3 = S(5, 8, 9) 076: p_1 = S(1, 3, 4), p_2 = S(2, 6, 8), p_3 = S(5, 7, 9) 077: p_1 = S(1, 3, 4), p_2 = S(2, 6, 9), p_3 = S(5, 7, 8) 078: p_1 = S(1, 3, 4), p_2 = S(2, 7, 8), p_3 = S(5, 6, 9) 079: p_1 = S(1, 3, 4), p_2 = S(2, 7, 9), p_3 = S(5, 6, 8) 080: p_1 = S(1, 3, 4), p_2 = S(2, 8, 9), p_3 = S(5, 6, 7) 081: p_1 = S(1, 3, 5), p_2 = S(2, 4, 6), p_3 = S(7, 8, 9) 082: p_1 = S(1, 3, 5), p_2 = S(2, 4, 7), p_3 = S(6, 8, 9) 083: p_1 = S(1, 3, 5), p_2 = S(2, 4, 8), p_3 = S(6, 7, 9) 084: p_1 = S(1, 3, 5), p_2 = S(2, 4, 9), p_3 = S(6, 7, 8) 085: p_1 = S(1, 3, 5), p_2 = S(2, 6, 7), p_3 = S(4, 8, 9) 086: p_1 = S(1, 3, 5), p_2 = S(2, 6, 8), p_3 = S(4, 7, 9) 087: p_1 = S(1, 3, 5), p_2 = S(2, 6, 9), p_3 = S(4, 7, 8) 088: p_1 = S(1, 3, 5), p_2 = S(2, 7, 8), p_3 = S(4, 6, 9) 089: p_1 = S(1, 3, 5), p_2 = S(2, 7, 9), p_3 = S(4, 6, 8) 090: p_1 = S(1, 3, 5), p_2 = S(2, 8, 9), p_3 = S(4, 6, 7) 091: p_1 = S(1, 3, 6), p_2 = S(2, 4, 5), p_3 = S(7, 8, 9) 092: p_1 = S(1, 3, 6), p_2 = S(2, 4, 7), p_3 = S(5, 8, 9) 093: p_1 = S(1, 3, 6), p_2 = S(2, 4, 8), p_3 = S(5, 7, 9) 094: p_1 = S(1, 3, 6), p_2 = S(2, 4, 9), p_3 = S(5, 7, 8) 095: p_1 = S(1, 3, 6), p_2 = S(2, 5, 7), p_3 = S(4, 8, 9) 096: p_1 = S(1, 3, 6), p_2 = S(2, 5, 8), p_3 = S(4, 7, 9) 097: p_1 = S(1, 3, 6), p_2 = S(2, 5, 9), p_3 = S(4, 7, 8) 098: p_1 = S(1, 3, 6), p_2 = S(2, 7, 8), p_3 = S(4, 5, 9) 099: p_1 = S(1, 3, 6), p_2 = S(2, 7, 9), p_3 = S(4, 5, 8) 100: p_1 = S(1, 3, 6), p_2 = S(2, 8, 9), p_3 = S(4, 5, 7) 101: p_1 = S(1, 3, 7), p_2 = S(2, 4, 5), p_3 = S(6, 8, 9) 102: p_1 = S(1, 3, 7), p_2 = S(2, 4, 6), p_3 = S(5, 8, 9) 103: p_1 = S(1, 3, 7), p_2 = S(2, 4, 8), p_3 = S(5, 6, 9) 104: p_1 = S(1, 3, 7), p_2 = S(2, 4, 9), p_3 = S(5, 6, 8) 105: p_1 = S(1, 3, 7), p_2 = S(2, 5, 6), p_3 = S(4, 8, 9) 106: p_1 = S(1, 3, 7), p_2 = S(2, 5, 8), p_3 = S(4, 6, 9) 107: p_1 = S(1, 3, 7), p_2 = S(2, 5, 9), p_3 = S(4, 6, 8) 108: p_1 = S(1, 3, 7), p_2 = S(2, 6, 8), p_3 = S(4, 5, 9) 109: p_1 = S(1, 3, 7), p_2 = S(2, 6, 9), p_3 = S(4, 5, 8) 110: p_1 = S(1, 3, 7), p_2 = S(2, 8, 9), p_3 = S(4, 5, 6) 111: p_1 = S(1, 3, 8), p_2 = S(2, 4, 5), p_3 = S(6, 7, 9) 112: p_1 = S(1, 3, 8), p_2 = S(2, 4, 6), p_3 = S(5, 7, 9) 113: p_1 = S(1, 3, 8), p_2 = S(2, 4, 7), p_3 = S(5, 6, 9) 114: p_1 = S(1, 3, 8), p_2 = S(2, 4, 9), p_3 = S(5, 6, 7) 115: p_1 = S(1, 3, 8), p_2 = S(2, 5, 6), p_3 = S(4, 7, 9) 116: p_1 = S(1, 3, 8), p_2 = S(2, 5, 7), p_3 = S(4, 6, 9) 117: p_1 = S(1, 3, 8), p_2 = S(2, 5, 9), p_3 = S(4, 6, 7) 118: p_1 = S(1, 3, 8), p_2 = S(2, 6, 7), p_3 = S(4, 5, 9) 119: p_1 = S(1, 3, 8), p_2 = S(2, 6, 9), p_3 = S(4, 5, 7) 120: p_1 = S(1, 3, 8), p_2 = S(2, 7, 9), p_3 = S(4, 5, 6) 121: p_1 = S(1, 3, 9), p_2 = S(2, 4, 5), p_3 = S(6, 7, 8) 122: p_1 = S(1, 3, 9), p_2 = S(2, 4, 6), p_3 = S(5, 7, 8) 123: p_1 = S(1, 3, 9), p_2 = S(2, 4, 7), p_3 = S(5, 6, 8) 124: p_1 = S(1, 3, 9), p_2 = S(2, 4, 8), p_3 = S(5, 6, 7) 125: p_1 = S(1, 3, 9), p_2 = S(2, 5, 6), p_3 = S(4, 7, 8) 126: p_1 = S(1, 3, 9), p_2 = S(2, 5, 7), p_3 = S(4, 6, 8) 127: p_1 = S(1, 3, 9), p_2 = S(2, 5, 8), p_3 = S(4, 6, 7) 128: p_1 = S(1, 3, 9), p_2 = S(2, 6, 7), p_3 = S(4, 5, 8) 129: p_1 = S(1, 3, 9), p_2 = S(2, 6, 8), p_3 = S(4, 5, 7) 130: p_1 = S(1, 3, 9), p_2 = S(2, 7, 8), p_3 = S(4, 5, 6) 131: p_1 = S(1, 4, 5), p_2 = S(2, 3, 6), p_3 = S(7, 8, 9) 132: p_1 = S(1, 4, 5), p_2 = S(2, 3, 7), p_3 = S(6, 8, 9) 133: p_1 = S(1, 4, 5), p_2 = S(2, 3, 8), p_3 = S(6, 7, 9) 134: p_1 = S(1, 4, 5), p_2 = S(2, 3, 9), p_3 = S(6, 7, 8) 135: p_1 = S(1, 4, 5), p_2 = S(2, 6, 7), p_3 = S(3, 8, 9) 136: p_1 = S(1, 4, 5), p_2 = S(2, 6, 8), p_3 = S(3, 7, 9) 137: p_1 = S(1, 4, 5), p_2 = S(2, 6, 9), p_3 = S(3, 7, 8) 138: p_1 = S(1, 4, 5), p_2 = S(2, 7, 8), p_3 = S(3, 6, 9) 139: p_1 = S(1, 4, 5), p_2 = S(2, 7, 9), p_3 = S(3, 6, 8) 140: p_1 = S(1, 4, 5), p_2 = S(2, 8, 9), p_3 = S(3, 6, 7) 141: p_1 = S(1, 4, 6), p_2 = S(2, 3, 5), p_3 = S(7, 8, 9) 142: p_1 = S(1, 4, 6), p_2 = S(2, 3, 7), p_3 = S(5, 8, 9) 143: p_1 = S(1, 4, 6), p_2 = S(2, 3, 8), p_3 = S(5, 7, 9) 144: p_1 = S(1, 4, 6), p_2 = S(2, 3, 9), p_3 = S(5, 7, 8) 145: p_1 = S(1, 4, 6), p_2 = S(2, 5, 7), p_3 = S(3, 8, 9) 146: p_1 = S(1, 4, 6), p_2 = S(2, 5, 8), p_3 = S(3, 7, 9) 147: p_1 = S(1, 4, 6), p_2 = S(2, 5, 9), p_3 = S(3, 7, 8) 148: p_1 = S(1, 4, 6), p_2 = S(2, 7, 8), p_3 = S(3, 5, 9) 149: p_1 = S(1, 4, 6), p_2 = S(2, 7, 9), p_3 = S(3, 5, 8) 150: p_1 = S(1, 4, 6), p_2 = S(2, 8, 9), p_3 = S(3, 5, 7) 151: p_1 = S(1, 4, 7), p_2 = S(2, 3, 5), p_3 = S(6, 8, 9) 152: p_1 = S(1, 4, 7), p_2 = S(2, 3, 6), p_3 = S(5, 8, 9) 153: p_1 = S(1, 4, 7), p_2 = S(2, 3, 8), p_3 = S(5, 6, 9) 154: p_1 = S(1, 4, 7), p_2 = S(2, 3, 9), p_3 = S(5, 6, 8) 155: p_1 = S(1, 4, 7), p_2 = S(2, 5, 6), p_3 = S(3, 8, 9) 156: p_1 = S(1, 4, 7), p_2 = S(2, 5, 8), p_3 = S(3, 6, 9) 157: p_1 = S(1, 4, 7), p_2 = S(2, 5, 9), p_3 = S(3, 6, 8) 158: p_1 = S(1, 4, 7), p_2 = S(2, 6, 8), p_3 = S(3, 5, 9) 159: p_1 = S(1, 4, 7), p_2 = S(2, 6, 9), p_3 = S(3, 5, 8) 160: p_1 = S(1, 4, 7), p_2 = S(2, 8, 9), p_3 = S(3, 5, 6) 161: p_1 = S(1, 4, 8), p_2 = S(2, 3, 5), p_3 = S(6, 7, 9) 162: p_1 = S(1, 4, 8), p_2 = S(2, 3, 6), p_3 = S(5, 7, 9) 163: p_1 = S(1, 4, 8), p_2 = S(2, 3, 7), p_3 = S(5, 6, 9) 164: p_1 = S(1, 4, 8), p_2 = S(2, 3, 9), p_3 = S(5, 6, 7) 165: p_1 = S(1, 4, 8), p_2 = S(2, 5, 6), p_3 = S(3, 7, 9) 166: p_1 = S(1, 4, 8), p_2 = S(2, 5, 7), p_3 = S(3, 6, 9) 167: p_1 = S(1, 4, 8), p_2 = S(2, 5, 9), p_3 = S(3, 6, 7) 168: p_1 = S(1, 4, 8), p_2 = S(2, 6, 7), p_3 = S(3, 5, 9) 169: p_1 = S(1, 4, 8), p_2 = S(2, 6, 9), p_3 = S(3, 5, 7) 170: p_1 = S(1, 4, 8), p_2 = S(2, 7, 9), p_3 = S(3, 5, 6) 171: p_1 = S(1, 4, 9), p_2 = S(2, 3, 5), p_3 = S(6, 7, 8) 172: p_1 = S(1, 4, 9), p_2 = S(2, 3, 6), p_3 = S(5, 7, 8) 173: p_1 = S(1, 4, 9), p_2 = S(2, 3, 7), p_3 = S(5, 6, 8) 174: p_1 = S(1, 4, 9), p_2 = S(2, 3, 8), p_3 = S(5, 6, 7) 175: p_1 = S(1, 4, 9), p_2 = S(2, 5, 6), p_3 = S(3, 7, 8) 176: p_1 = S(1, 4, 9), p_2 = S(2, 5, 7), p_3 = S(3, 6, 8) 177: p_1 = S(1, 4, 9), p_2 = S(2, 5, 8), p_3 = S(3, 6, 7) 178: p_1 = S(1, 4, 9), p_2 = S(2, 6, 7), p_3 = S(3, 5, 8) 179: p_1 = S(1, 4, 9), p_2 = S(2, 6, 8), p_3 = S(3, 5, 7) 180: p_1 = S(1, 4, 9), p_2 = S(2, 7, 8), p_3 = S(3, 5, 6) 181: p_1 = S(1, 5, 6), p_2 = S(2, 3, 4), p_3 = S(7, 8, 9) 182: p_1 = S(1, 5, 6), p_2 = S(2, 3, 7), p_3 = S(4, 8, 9) 183: p_1 = S(1, 5, 6), p_2 = S(2, 3, 8), p_3 = S(4, 7, 9) 184: p_1 = S(1, 5, 6), p_2 = S(2, 3, 9), p_3 = S(4, 7, 8) 185: p_1 = S(1, 5, 6), p_2 = S(2, 4, 7), p_3 = S(3, 8, 9) 186: p_1 = S(1, 5, 6), p_2 = S(2, 4, 8), p_3 = S(3, 7, 9) 187: p_1 = S(1, 5, 6), p_2 = S(2, 4, 9), p_3 = S(3, 7, 8) 188: p_1 = S(1, 5, 6), p_2 = S(2, 7, 8), p_3 = S(3, 4, 9) 189: p_1 = S(1, 5, 6), p_2 = S(2, 7, 9), p_3 = S(3, 4, 8) 190: p_1 = S(1, 5, 6), p_2 = S(2, 8, 9), p_3 = S(3, 4, 7) 191: p_1 = S(1, 5, 7), p_2 = S(2, 3, 4), p_3 = S(6, 8, 9) 192: p_1 = S(1, 5, 7), p_2 = S(2, 3, 6), p_3 = S(4, 8, 9) 193: p_1 = S(1, 5, 7), p_2 = S(2, 3, 8), p_3 = S(4, 6, 9) 194: p_1 = S(1, 5, 7), p_2 = S(2, 3, 9), p_3 = S(4, 6, 8) 195: p_1 = S(1, 5, 7), p_2 = S(2, 4, 6), p_3 = S(3, 8, 9) 196: p_1 = S(1, 5, 7), p_2 = S(2, 4, 8), p_3 = S(3, 6, 9) 197: p_1 = S(1, 5, 7), p_2 = S(2, 4, 9), p_3 = S(3, 6, 8) 198: p_1 = S(1, 5, 7), p_2 = S(2, 6, 8), p_3 = S(3, 4, 9) 199: p_1 = S(1, 5, 7), p_2 = S(2, 6, 9), p_3 = S(3, 4, 8) 200: p_1 = S(1, 5, 7), p_2 = S(2, 8, 9), p_3 = S(3, 4, 6) 201: p_1 = S(1, 5, 8), p_2 = S(2, 3, 4), p_3 = S(6, 7, 9) 202: p_1 = S(1, 5, 8), p_2 = S(2, 3, 6), p_3 = S(4, 7, 9) 203: p_1 = S(1, 5, 8), p_2 = S(2, 3, 7), p_3 = S(4, 6, 9) 204: p_1 = S(1, 5, 8), p_2 = S(2, 3, 9), p_3 = S(4, 6, 7) 205: p_1 = S(1, 5, 8), p_2 = S(2, 4, 6), p_3 = S(3, 7, 9) 206: p_1 = S(1, 5, 8), p_2 = S(2, 4, 7), p_3 = S(3, 6, 9) 207: p_1 = S(1, 5, 8), p_2 = S(2, 4, 9), p_3 = S(3, 6, 7) 208: p_1 = S(1, 5, 8), p_2 = S(2, 6, 7), p_3 = S(3, 4, 9) 209: p_1 = S(1, 5, 8), p_2 = S(2, 6, 9), p_3 = S(3, 4, 7) 210: p_1 = S(1, 5, 8), p_2 = S(2, 7, 9), p_3 = S(3, 4, 6) 211: p_1 = S(1, 5, 9), p_2 = S(2, 3, 4), p_3 = S(6, 7, 8) 212: p_1 = S(1, 5, 9), p_2 = S(2, 3, 6), p_3 = S(4, 7, 8) 213: p_1 = S(1, 5, 9), p_2 = S(2, 3, 7), p_3 = S(4, 6, 8) 214: p_1 = S(1, 5, 9), p_2 = S(2, 3, 8), p_3 = S(4, 6, 7) 215: p_1 = S(1, 5, 9), p_2 = S(2, 4, 6), p_3 = S(3, 7, 8) 216: p_1 = S(1, 5, 9), p_2 = S(2, 4, 7), p_3 = S(3, 6, 8) 217: p_1 = S(1, 5, 9), p_2 = S(2, 4, 8), p_3 = S(3, 6, 7) 218: p_1 = S(1, 5, 9), p_2 = S(2, 6, 7), p_3 = S(3, 4, 8) 219: p_1 = S(1, 5, 9), p_2 = S(2, 6, 8), p_3 = S(3, 4, 7) 220: p_1 = S(1, 5, 9), p_2 = S(2, 7, 8), p_3 = S(3, 4, 6) 221: p_1 = S(1, 6, 7), p_2 = S(2, 3, 4), p_3 = S(5, 8, 9) 222: p_1 = S(1, 6, 7), p_2 = S(2, 3, 5), p_3 = S(4, 8, 9) 223: p_1 = S(1, 6, 7), p_2 = S(2, 3, 8), p_3 = S(4, 5, 9) 224: p_1 = S(1, 6, 7), p_2 = S(2, 3, 9), p_3 = S(4, 5, 8) 225: p_1 = S(1, 6, 7), p_2 = S(2, 4, 5), p_3 = S(3, 8, 9) 226: p_1 = S(1, 6, 7), p_2 = S(2, 4, 8), p_3 = S(3, 5, 9) 227: p_1 = S(1, 6, 7), p_2 = S(2, 4, 9), p_3 = S(3, 5, 8) 228: p_1 = S(1, 6, 7), p_2 = S(2, 5, 8), p_3 = S(3, 4, 9) 229: p_1 = S(1, 6, 7), p_2 = S(2, 5, 9), p_3 = S(3, 4, 8) 230: p_1 = S(1, 6, 7), p_2 = S(2, 8, 9), p_3 = S(3, 4, 5) 231: p_1 = S(1, 6, 8), p_2 = S(2, 3, 4), p_3 = S(5, 7, 9) 232: p_1 = S(1, 6, 8), p_2 = S(2, 3, 5), p_3 = S(4, 7, 9) 233: p_1 = S(1, 6, 8), p_2 = S(2, 3, 7), p_3 = S(4, 5, 9) 234: p_1 = S(1, 6, 8), p_2 = S(2, 3, 9), p_3 = S(4, 5, 7) 235: p_1 = S(1, 6, 8), p_2 = S(2, 4, 5), p_3 = S(3, 7, 9) 236: p_1 = S(1, 6, 8), p_2 = S(2, 4, 7), p_3 = S(3, 5, 9) 237: p_1 = S(1, 6, 8), p_2 = S(2, 4, 9), p_3 = S(3, 5, 7) 238: p_1 = S(1, 6, 8), p_2 = S(2, 5, 7), p_3 = S(3, 4, 9) 239: p_1 = S(1, 6, 8), p_2 = S(2, 5, 9), p_3 = S(3, 4, 7) 240: p_1 = S(1, 6, 8), p_2 = S(2, 7, 9), p_3 = S(3, 4, 5) 241: p_1 = S(1, 6, 9), p_2 = S(2, 3, 4), p_3 = S(5, 7, 8) 242: p_1 = S(1, 6, 9), p_2 = S(2, 3, 5), p_3 = S(4, 7, 8) 243: p_1 = S(1, 6, 9), p_2 = S(2, 3, 7), p_3 = S(4, 5, 8) 244: p_1 = S(1, 6, 9), p_2 = S(2, 3, 8), p_3 = S(4, 5, 7) 245: p_1 = S(1, 6, 9), p_2 = S(2, 4, 5), p_3 = S(3, 7, 8) 246: p_1 = S(1, 6, 9), p_2 = S(2, 4, 7), p_3 = S(3, 5, 8) 247: p_1 = S(1, 6, 9), p_2 = S(2, 4, 8), p_3 = S(3, 5, 7) 248: p_1 = S(1, 6, 9), p_2 = S(2, 5, 7), p_3 = S(3, 4, 8) 249: p_1 = S(1, 6, 9), p_2 = S(2, 5, 8), p_3 = S(3, 4, 7) 250: p_1 = S(1, 6, 9), p_2 = S(2, 7, 8), p_3 = S(3, 4, 5) 251: p_1 = S(1, 7, 8), p_2 = S(2, 3, 4), p_3 = S(5, 6, 9) 252: p_1 = S(1, 7, 8), p_2 = S(2, 3, 5), p_3 = S(4, 6, 9) 253: p_1 = S(1, 7, 8), p_2 = S(2, 3, 6), p_3 = S(4, 5, 9) 254: p_1 = S(1, 7, 8), p_2 = S(2, 3, 9), p_3 = S(4, 5, 6) 255: p_1 = S(1, 7, 8), p_2 = S(2, 4, 5), p_3 = S(3, 6, 9) 256: p_1 = S(1, 7, 8), p_2 = S(2, 4, 6), p_3 = S(3, 5, 9) 257: p_1 = S(1, 7, 8), p_2 = S(2, 4, 9), p_3 = S(3, 5, 6) 258: p_1 = S(1, 7, 8), p_2 = S(2, 5, 6), p_3 = S(3, 4, 9) 259: p_1 = S(1, 7, 8), p_2 = S(2, 5, 9), p_3 = S(3, 4, 6) 260: p_1 = S(1, 7, 8), p_2 = S(2, 6, 9), p_3 = S(3, 4, 5) 261: p_1 = S(1, 7, 9), p_2 = S(2, 3, 4), p_3 = S(5, 6, 8) 262: p_1 = S(1, 7, 9), p_2 = S(2, 3, 5), p_3 = S(4, 6, 8) 263: p_1 = S(1, 7, 9), p_2 = S(2, 3, 6), p_3 = S(4, 5, 8) 264: p_1 = S(1, 7, 9), p_2 = S(2, 3, 8), p_3 = S(4, 5, 6) 265: p_1 = S(1, 7, 9), p_2 = S(2, 4, 5), p_3 = S(3, 6, 8) 266: p_1 = S(1, 7, 9), p_2 = S(2, 4, 6), p_3 = S(3, 5, 8) 267: p_1 = S(1, 7, 9), p_2 = S(2, 4, 8), p_3 = S(3, 5, 6) 268: p_1 = S(1, 7, 9), p_2 = S(2, 5, 6), p_3 = S(3, 4, 8) 269: p_1 = S(1, 7, 9), p_2 = S(2, 5, 8), p_3 = S(3, 4, 6) 270: p_1 = S(1, 7, 9), p_2 = S(2, 6, 8), p_3 = S(3, 4, 5) 271: p_1 = S(1, 8, 9), p_2 = S(2, 3, 4), p_3 = S(5, 6, 7) 272: p_1 = S(1, 8, 9), p_2 = S(2, 3, 5), p_3 = S(4, 6, 7) 273: p_1 = S(1, 8, 9), p_2 = S(2, 3, 6), p_3 = S(4, 5, 7) 274: p_1 = S(1, 8, 9), p_2 = S(2, 3, 7), p_3 = S(4, 5, 6) 275: p_1 = S(1, 8, 9), p_2 = S(2, 4, 5), p_3 = S(3, 6, 7) 276: p_1 = S(1, 8, 9), p_2 = S(2, 4, 6), p_3 = S(3, 5, 7) 277: p_1 = S(1, 8, 9), p_2 = S(2, 4, 7), p_3 = S(3, 5, 6) 278: p_1 = S(1, 8, 9), p_2 = S(2, 5, 6), p_3 = S(3, 4, 7) 279: p_1 = S(1, 8, 9), p_2 = S(2, 5, 7), p_3 = S(3, 4, 6) 280: p_1 = S(1, 8, 9), p_2 = S(2, 6, 7), p_3 = S(3, 4, 5)
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