151208

Ruby


分割の逆数和が1(1)

初等数学第77号の西山輝夫氏の記事に以下のようなことが書かれてあった。

「77より大きな整数は、適当な分割が存在して、分割した数の逆数（単位分数）の和を1にすることができる。」

例えば、n = 78 のとき、78 = 40 + 12 + 10 + 8 + 6 + 2, 1 = 1 / 40 + 1 / 12 + 1 / 10 + 1 / 8 + 1 / 6 + 1 / 2 とできます。

一方、n = 2 のとき、
分割の逆数和が1になることはありません。
よって、上記が正しいなら、

分割の逆数和が1にならないような最大のnが存在する

はずです。

そのnが23であることを

n ≦ 77 のとき、和因子が2以上の分割をすべて調べる

ことで確認しました。

# 和因子はmin以上max以下
def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end

(2..77).each{|n|
  # nの分割で逆数和が1
  p [n, partition(n, 2, n).select{|ary| ary.inject(0){|s, i| s + 1 / i.to_r} == 1}]
}

出力結果
[2, []]
[3, []]
[4, [[2, 2]]]
[5, []]
[6, []]
[7, []]
[8, []]
[9, [[3, 3, 3]]]
[10, [[4, 4, 2]]]
[11, [[6, 3, 2]]]
[12, []]
[13, []]
[14, []]
[15, []]
[16, [[4, 4, 4, 4]]]
[17, [[6, 4, 4, 3]]]
[18, [[6, 6, 3, 3]]]
[19, []]
[20, [[6, 6, 6, 2]]]
[21, []]
[22, [[12, 4, 3, 3], [10, 5, 5, 2], [8, 8, 4, 2]]]
[23, []]
[24, [[12, 6, 4, 2]]]
[25, [[5, 5, 5, 5, 5]]]
[26, [[6, 6, 6, 4, 4]]]
[27, [[6, 6, 6, 6, 3]]]
[28, [[10, 5, 5, 4, 4], [8, 8, 4, 4, 4]]]
[29, [[12, 12, 3, 2], [10, 6, 5, 5, 3], [8, 8, 6, 4, 3]]]
[30, [[15, 10, 3, 2], [12, 6, 4, 4, 4]]]
[31, [[20, 5, 4, 2], [12, 6, 6, 4, 3]]]
[32, [[18, 9, 3, 2]]]
[33, [[15, 5, 5, 5, 3], [9, 9, 9, 3, 3]]]
[34, [[12, 8, 8, 3, 3], [8, 8, 8, 8, 2]]]
[35, [[12, 12, 4, 4, 3], [9, 9, 9, 6, 2]]]
[36, [[15, 10, 4, 4, 3], [12, 12, 6, 3, 3], [12, 8, 8, 6, 2], [6, 6, 6, 6, 6, 6]]]
[37, [[24, 8, 3, 2], [20, 5, 4, 4, 4], [15, 10, 6, 3, 3], [14, 7, 7, 7, 2], [10, 10, 10, 5, 2]]]
[38, [[20, 6, 5, 4, 3], [18, 9, 4, 4, 3], [12, 12, 6, 6, 2], [10, 6, 6, 6, 5, 5], [8, 8, 6, 6, 6, 4]]]
[39, [[18, 9, 6, 3, 3], [15, 10, 6, 6, 2]]]
[40, [[12, 6, 6, 6, 6, 4], [10, 10, 5, 5, 5, 5], [10, 8, 8, 5, 5, 4], [8, 8, 8, 8, 4, 4]]]
[41, [[21, 7, 7, 3, 3], [18, 9, 6, 6, 2], [15, 15, 5, 3, 3], [9, 9, 9, 6, 4, 4], [8, 8, 8, 8, 6, 3]]]
[42, [[15, 6, 6, 5, 5, 5], [12, 12, 12, 4, 2], [12, 10, 6, 5, 5, 4], [12, 8, 8, 6, 4, 4], [9, 9, 9, 6, 6, 3]]]
[43, [[24, 8, 4, 4, 3], [21, 7, 7, 6, 2], [20, 12, 5, 3, 3], [20, 8, 8, 5, 2], [15, 15, 6, 5, 2], [15, 12, 10, 4, 2], [14, 7, 7, 7, 4, 4], [12, 8, 8, 6, 6, 3], [10, 10, 10, 5, 4, 4]]]
[44, [[24, 8, 6, 3, 3], [14, 7, 7, 7, 6, 3], [12, 12, 6, 6, 4, 4], [10, 10, 10, 6, 5, 3]]]
[45, [[20, 12, 6, 5, 2], [18, 12, 9, 4, 2], [15, 10, 6, 6, 4, 4], [12, 12, 6, 6, 6, 3]]]
[46, [[24, 8, 6, 6, 2], [20, 10, 10, 4, 2], [16, 16, 8, 4, 2], [15, 12, 5, 5, 5, 4], [15, 10, 6, 6, 6, 3], [12, 9, 9, 9, 4, 3]]]
[47, [[20, 6, 6, 6, 5, 4], [18, 9, 6, 6, 4, 4], [12, 12, 10, 5, 5, 3], [12, 12, 8, 8, 4, 3]]]
[48, [[18, 9, 6, 6, 6, 3], [15, 10, 10, 5, 5, 3], [15, 10, 8, 8, 4, 3], [12, 12, 12, 4, 4, 4]]]
[49, [[21, 7, 7, 6, 4, 4], [20, 10, 5, 5, 5, 4], [20, 8, 8, 5, 4, 4], [15, 15, 6, 5, 4, 4], [15, 12, 10, 4, 4, 4], [12, 12, 12, 6, 4, 3], [7, 7, 7, 7, 7, 7, 7]]]
[50, [[24, 12, 8, 4, 2], [21, 7, 7, 6, 6, 3], [20, 8, 8, 6, 5, 3], [18, 10, 9, 5, 5, 3], [18, 9, 8, 8, 4, 3], [15, 15, 6, 6, 5, 3], [15, 12, 10, 6, 4, 3], [8, 8, 8, 8, 6, 6, 6]]]
[51, [[30, 10, 5, 3, 3], [20, 12, 6, 5, 4, 4], [18, 12, 9, 4, 4, 4], [9, 9, 9, 6, 6, 6, 6]]]
[52, [[24, 8, 6, 6, 4, 4], [20, 20, 5, 5, 2], [20, 12, 6, 6, 5, 3], [20, 10, 10, 4, 4, 4], [18, 12, 9, 6, 4, 3], [16, 16, 8, 4, 4, 4], [12, 8, 8, 6, 6, 6, 6], [10, 10, 10, 10, 10, 2], [10, 8, 8, 8, 8, 5, 5], [8, 8, 8, 8, 8, 8, 4]]]
[53, [[30, 10, 6, 5, 2], [24, 8, 6, 6, 6, 3], [20, 10, 10, 6, 4, 3], [16, 16, 8, 6, 4, 3], [14, 7, 7, 7, 6, 6, 6], [12, 12, 9, 9, 9, 2], [10, 10, 10, 6, 6, 6, 5], [10, 9, 9, 9, 6, 5, 5], [9, 9, 9, 8, 8, 6, 4]]]
[54, [[42, 7, 3, 2], [21, 12, 7, 7, 4, 3], [15, 15, 12, 5, 4, 3], [15, 10, 9, 9, 9, 2], [12, 12, 12, 12, 3, 3], [12, 12, 12, 8, 8, 2], [12, 12, 6, 6, 6, 6, 6], [12, 10, 8, 8, 6, 5, 5], [12, 8, 8, 8, 8, 6, 4]]]
[55, [[28, 14, 7, 4, 2], [24, 10, 8, 5, 5, 3], [24, 8, 8, 8, 4, 3], [15, 12, 12, 10, 3, 3], [15, 12, 10, 8, 8, 2], [15, 10, 6, 6, 6, 6, 6], [14, 10, 7, 7, 7, 5, 5], [14, 8, 8, 7, 7, 7, 4], [12, 9, 9, 9, 6, 6, 4], [10, 10, 10, 10, 5, 5, 5], [10, 10, 10, 8, 8, 5, 4]]]
[56, [[30, 6, 5, 5, 5, 5], [24, 12, 8, 4, 4, 4], [20, 12, 12, 5, 4, 3], [18, 9, 9, 9, 9, 2], [15, 15, 10, 10, 3, 3], [12, 12, 12, 12, 6, 2], [12, 12, 10, 6, 6, 5, 5], [12, 12, 8, 8, 6, 6, 4]]]
[57, [[30, 15, 5, 5, 2], [24, 12, 8, 6, 4, 3], [20, 15, 10, 5, 4, 3], [18, 12, 12, 9, 3, 3], [18, 12, 9, 8, 8, 2], [18, 9, 6, 6, 6, 6, 6], [15, 12, 12, 10, 6, 2], [15, 10, 10, 6, 6, 5, 5], [15, 10, 8, 8, 6, 6, 4], [15, 9, 9, 9, 5, 5, 5], [14, 12, 7, 7, 7, 6, 4], [12, 10, 10, 10, 6, 5, 4], [9, 9, 9, 9, 9, 9, 3]]]
[58, [[24, 24, 4, 3, 3], [20, 20, 5, 5, 4, 4], [20, 12, 10, 10, 3, 3], [20, 10, 10, 8, 8, 2], [18, 15, 10, 9, 3, 3], [16, 16, 12, 8, 3, 3], [16, 16, 8, 8, 8, 2], [15, 15, 10, 10, 6, 2], [15, 12, 8, 8, 5, 5, 5], [14, 14, 14, 7, 7, 2], [12, 12, 12, 6, 6, 6, 4], [12, 9, 9, 9, 8, 8, 3], [10, 10, 10, 10, 10, 4, 4]]]
[59, [[30, 10, 6, 5, 4, 4], [28, 21, 4, 3, 3], [21, 7, 7, 6, 6, 6, 6], [20, 20, 6, 5, 5, 3], [20, 18, 9, 5, 4, 3], [20, 8, 8, 6, 6, 6, 5], [18, 18, 18, 3, 2], [18, 12, 12, 9, 6, 2], [18, 10, 9, 6, 6, 5, 5], [18, 9, 8, 8, 6, 6, 4], [15, 15, 6, 6, 6, 6, 5], [15, 12, 10, 6, 6, 6, 4], [12, 12, 9, 9, 9, 4, 4], [12, 12, 8, 8, 8, 8, 3], [10, 10, 10, 10, 10, 6, 3]]]
[60, [[42, 7, 4, 4, 3], [36, 9, 9, 4, 2], [30, 20, 4, 3, 3], [30, 10, 6, 6, 5, 3], [24, 24, 6, 4, 2], [20, 20, 15, 3, 2], [20, 12, 10, 10, 6, 2], [18, 18, 9, 9, 3, 3], [18, 15, 10, 9, 6, 2], [16, 16, 12, 8, 6, 2], [15, 12, 12, 6, 5, 5, 5], [15, 10, 9, 9, 9, 4, 4], [15, 10, 8, 8, 8, 8, 3], [12, 12, 12, 10, 5, 5, 4], [12, 12, 12, 8, 8, 4, 4], [12, 12, 9, 9, 9, 6, 3]]]
[61, [[42, 7, 6, 3, 3], [35, 14, 5, 5, 2], [28, 21, 6, 4, 2], [28, 14, 7, 4, 4, 4], [24, 16, 16, 3, 2], [24, 9, 9, 9, 8, 2], [21, 21, 14, 3, 2], [21, 12, 12, 7, 7, 2], [21, 10, 7, 7, 6, 5, 5], [21, 8, 8, 7, 7, 6, 4], [20, 12, 6, 6, 6, 6, 5], [20, 10, 8, 8, 5, 5, 5], [20, 8, 8, 8, 8, 5, 4], [18, 12, 9, 6, 6, 6, 4], [15, 15, 12, 12, 5, 2], [15, 15, 10, 6, 5, 5, 5], [15, 15, 8, 8, 6, 5, 4], [15, 12, 10, 10, 5, 5, 4], [15, 12, 10, 8, 8, 4, 4], [15, 10, 9, 9, 9, 6, 3], [12, 12, 12, 8, 8, 6, 3]]]
[62, [[30, 20, 6, 4, 2], [28, 14, 7, 6, 4, 3], [24, 12, 12, 8, 3, 3], [24, 12, 8, 8, 8, 2], [24, 8, 6, 6, 6, 6, 6], [21, 15, 10, 7, 7, 2], [21, 14, 14, 7, 3, 3], [20, 10, 10, 6, 6, 6, 4], [20, 9, 9, 9, 6, 5, 4], [18, 18, 9, 9, 6, 2], [18, 9, 9, 9, 9, 4, 4], [18, 9, 8, 8, 8, 8, 3], [16, 16, 8, 6, 6, 6, 4], [15, 15, 15, 10, 5, 2], [15, 12, 10, 8, 8, 6, 3], [14, 12, 12, 7, 7, 7, 3], [12, 12, 12, 12, 6, 4, 4], [12, 12, 10, 10, 10, 5, 3]]]
[63, [[42, 7, 6, 6, 2], [30, 15, 5, 5, 4, 4], [24, 15, 10, 8, 3, 3], [21, 12, 7, 7, 6, 6, 4], [20, 12, 12, 12, 5, 2], [20, 12, 10, 6, 5, 5, 5], [20, 12, 8, 8, 6, 5, 4], [18, 15, 9, 6, 5, 5, 5], [18, 12, 10, 9, 5, 5, 4], [18, 12, 9, 8, 8, 4, 4], [18, 9, 9, 9, 9, 6, 3], [15, 15, 12, 6, 6, 5, 4], [15, 14, 10, 7, 7, 7, 3], [15, 12, 12, 10, 6, 4, 4], [15, 10, 10, 10, 10, 5, 3], [12, 12, 12, 12, 6, 6, 3]]]
[64, [[40, 10, 8, 4, 2], [36, 18, 4, 3, 3], [30, 15, 6, 5, 5, 3], [30, 12, 10, 5, 4, 3], [24, 20, 8, 5, 4, 3], [24, 12, 12, 8, 6, 2], [24, 10, 8, 6, 6, 5, 5], [24, 8, 8, 8, 6, 6, 4], [21, 18, 9, 7, 7, 2], [21, 14, 14, 7, 6, 2], [20, 15, 12, 10, 5, 2], [20, 14, 7, 7, 7, 5, 4], [20, 10, 10, 10, 5, 5, 4], [20, 10, 10, 8, 8, 4, 4], [18, 15, 15, 9, 5, 2], [18, 12, 9, 8, 8, 6, 3], [16, 16, 10, 8, 5, 5, 4], [16, 16, 8, 8, 8, 4, 4], [15, 15, 10, 10, 6, 4, 4], [15, 12, 12, 10, 6, 6, 3], [14, 14, 14, 7, 7, 4, 4], [8, 8, 8, 8, 8, 8, 8, 8]]]
[65, [[45, 9, 5, 3, 3], [30, 15, 15, 3, 2], [24, 24, 12, 3, 2], [24, 18, 9, 8, 3, 3], [24, 15, 10, 8, 6, 2], [21, 15, 7, 7, 5, 5, 5], [21, 9, 9, 9, 7, 7, 3], [20, 12, 12, 6, 6, 5, 4], [20, 10, 10, 8, 8, 6, 3], [18, 18, 18, 4, 4, 3], [18, 14, 9, 7, 7, 7, 3], [18, 12, 12, 9, 6, 4, 4], [18, 10, 10, 10, 9, 5, 3], [16, 16, 8, 8, 8, 6, 3], [15, 15, 15, 5, 5, 5, 5], [15, 15, 10, 10, 6, 6, 3], [15, 15, 9, 9, 9, 5, 3], [14, 14, 14, 7, 7, 6, 3], [9, 9, 9, 8, 8, 8, 8, 6]]]
[66, [[36, 18, 6, 4, 2], [36, 9, 9, 4, 4, 4], [30, 10, 10, 10, 3, 3], [28, 21, 12, 3, 2], [24, 24, 6, 4, 4, 4], [24, 12, 8, 6, 6, 6, 4], [21, 12, 8, 8, 7, 7, 3], [20, 20, 15, 4, 4, 3], [20, 18, 12, 9, 5, 2], [20, 15, 10, 6, 6, 5, 4], [20, 12, 10, 10, 6, 4, 4], [18, 18, 18, 6, 3, 3], [18, 15, 10, 9, 6, 4, 4], [18, 12, 12, 9, 6, 6, 3], [16, 16, 12, 8, 6, 4, 4], [15, 15, 12, 8, 8, 5, 3], [12, 8, 8, 8, 8, 8, 8, 6], [9, 9, 9, 9, 9, 9, 6, 6]]]
[67, [[45, 9, 6, 5, 2], [42, 12, 7, 4, 2], [40, 8, 5, 5, 5, 4], [36, 9, 9, 6, 4, 3], [35, 14, 5, 5, 4, 4], [30, 20, 12, 3, 2], [28, 21, 6, 4, 4, 4], [28, 14, 12, 7, 3, 3], [28, 14, 8, 8, 7, 2], [28, 7, 7, 7, 7, 7, 4], [24, 24, 6, 6, 4, 3], [24, 18, 9, 8, 6, 2], [24, 16, 16, 4, 4, 3], [24, 9, 9, 9, 8, 4, 4], [24, 8, 8, 8, 8, 8, 3], [21, 21, 14, 4, 4, 3], [21, 12, 12, 7, 7, 4, 4], [20, 20, 15, 6, 3, 3], [20, 20, 10, 10, 5, 2], [20, 16, 16, 8, 5, 2], [20, 15, 12, 5, 5, 5, 5], [20, 12, 10, 10, 6, 6, 3], [20, 12, 9, 9, 9, 5, 3], [18, 15, 10, 9, 6, 6, 3], [16, 16, 12, 8, 6, 6, 3], [15, 15, 12, 12, 5, 4, 4], [14, 8, 8, 8, 8, 7, 7, 7], [12, 12, 12, 12, 12, 4, 3], [12, 9, 9, 9, 8, 8, 6, 6], [10, 10, 10, 8, 8, 8, 8, 5]]]
[68, [[35, 14, 6, 5, 5, 3], [30, 20, 6, 4, 4, 4], [30, 10, 10, 10, 6, 2], [28, 21, 6, 6, 4, 3], [24, 16, 16, 6, 3, 3], [24, 15, 8, 6, 5, 5, 5], [24, 12, 10, 8, 5, 5, 4], [24, 12, 8, 8, 8, 4, 4], [24, 9, 9, 9, 8, 6, 3], [21, 21, 14, 6, 3, 3], [21, 15, 10, 7, 7, 4, 4], [21, 12, 12, 7, 7, 6, 3], [20, 20, 6, 6, 6, 5, 5], [20, 18, 9, 6, 6, 5, 4], [20, 12, 12, 8, 8, 5, 3], [18, 18, 18, 6, 6, 2], [18, 18, 9, 9, 6, 4, 4], [15, 15, 15, 10, 5, 4, 4], [15, 15, 12, 12, 6, 5, 3], [15, 12, 12, 12, 10, 4, 3], [14, 9, 9, 9, 7, 7, 7, 6], [12, 12, 8, 8, 8, 8, 6, 6], [10, 10, 10, 10, 10, 6, 6, 6], [10, 10, 10, 9, 9, 9, 6, 5]]]
[69, [[42, 7, 6, 6, 4, 4], [35, 15, 14, 3, 2], [30, 20, 6, 6, 4, 3], [30, 10, 6, 6, 6, 6, 5], [28, 14, 12, 7, 6, 2], [24, 21, 8, 7, 7, 2], [24, 15, 15, 8, 5, 2], [24, 12, 8, 8, 8, 6, 3], [21, 15, 10, 7, 7, 6, 3], [20, 20, 15, 6, 6, 2], [20, 15, 10, 8, 8, 5, 3], [20, 12, 12, 12, 5, 4, 4], [18, 18, 9, 9, 6, 6, 3], [15, 15, 15, 10, 6, 5, 3], [15, 15, 12, 10, 10, 4, 3], [15, 10, 8, 8, 8, 8, 6, 6], [14, 12, 8, 8, 7, 7, 7, 6], [12, 12, 9, 9, 9, 6, 6, 6], [12, 10, 10, 10, 8, 8, 6, 5]]]
[70, [[42, 7, 6, 6, 6, 3], [40, 10, 8, 4, 4, 4], [24, 24, 8, 8, 3, 3], [24, 16, 16, 6, 6, 2], [24, 14, 8, 7, 7, 7, 3], [24, 12, 12, 8, 6, 4, 4], [24, 10, 10, 10, 8, 5, 3], [21, 21, 14, 6, 6, 2], [21, 20, 7, 7, 6, 5, 4], [21, 18, 9, 7, 7, 4, 4], [21, 14, 14, 7, 6, 4, 4], [20, 20, 10, 5, 5, 5, 5], [20, 20, 8, 8, 5, 5, 4], [20, 15, 15, 6, 5, 5, 4], [20, 15, 12, 10, 5, 4, 4], [20, 12, 12, 12, 6, 5, 3], [18, 15, 15, 9, 5, 4, 4], [18, 12, 12, 12, 9, 4, 3], [16, 16, 16, 16, 4, 2], [15, 10, 9, 9, 9, 6, 6, 6], [14, 14, 7, 7, 7, 7, 7, 7], [14, 10, 10, 10, 7, 7, 7, 5], [12, 12, 12, 8, 8, 6, 6, 6], [12, 9, 9, 9, 9, 9, 9, 4], [10, 10, 10, 10, 10, 10, 5, 5], [10, 10, 10, 10, 10, 8, 8, 4]]]
[71, [[40, 10, 8, 6, 4, 3], [36, 18, 12, 3, 2], [33, 22, 11, 3, 2], [30, 15, 15, 4, 4, 3], [30, 12, 12, 10, 5, 2], [30, 10, 10, 6, 5, 5, 5], [30, 10, 8, 8, 6, 5, 4], [28, 21, 8, 8, 3, 3], [28, 14, 7, 6, 6, 6, 4], [24, 24, 12, 4, 4, 3], [24, 20, 12, 8, 5, 2], [24, 15, 10, 8, 6, 4, 4], [24, 12, 12, 8, 6, 6, 3], [21, 18, 9, 7, 7, 6, 3], [21, 14, 14, 7, 6, 6, 3], [20, 18, 9, 8, 8, 5, 3], [20, 15, 15, 15, 4, 2], [20, 15, 12, 10, 6, 5, 3], [20, 12, 12, 10, 10, 4, 3], [18, 15, 15, 9, 6, 5, 3], [18, 15, 12, 10, 9, 4, 3], [18, 9, 8, 8, 8, 8, 6, 6], [16, 16, 12, 12, 8, 4, 3], [15, 12, 10, 8, 8, 6, 6, 6], [14, 12, 12, 7, 7, 7, 6, 6], [12, 12, 10, 10, 10, 6, 6, 5], [12, 12, 10, 9, 9, 9, 5, 5], [12, 12, 9, 9, 9, 8, 8, 4]]]
[72, [[42, 10, 7, 5, 5, 3], [42, 8, 8, 7, 4, 3], [36, 18, 6, 4, 4, 4], [36, 12, 9, 9, 3, 3], [36, 9, 9, 8, 8, 2], [30, 20, 8, 8, 3, 3], [30, 15, 15, 6, 3, 3], [30, 15, 10, 10, 5, 2], [28, 21, 12, 4, 4, 3], [24, 24, 12, 6, 3, 3], [24, 24, 8, 8, 6, 2], [24, 15, 10, 8, 6, 6, 3], [20, 20, 12, 6, 5, 5, 4], [20, 18, 12, 9, 5, 4, 4], [20, 15, 10, 10, 10, 4, 3], [18, 18, 18, 12, 4, 2], [18, 9, 9, 9, 9, 6, 6, 6], [16, 16, 15, 10, 8, 4, 3], [15, 14, 10, 7, 7, 7, 6, 6], [15, 10, 10, 10, 10, 6, 6, 5], [15, 10, 10, 9, 9, 9, 5, 5], [15, 10, 9, 9, 9, 8, 8, 4], [12, 12, 12, 12, 6, 6, 6, 6], [12, 12, 12, 10, 8, 8, 5, 5], [12, 12, 12, 8, 8, 8, 8, 4], [12, 10, 10, 10, 10, 10, 6, 4]]]
[73, [[45, 9, 6, 5, 4, 4], [42, 12, 7, 4, 4, 4], [36, 18, 6, 6, 4, 3], [35, 7, 7, 7, 7, 5, 5], [30, 20, 12, 4, 4, 3], [30, 15, 6, 6, 6, 5, 5], [30, 12, 10, 6, 6, 5, 4], [28, 21, 12, 6, 3, 3], [28, 21, 8, 8, 6, 2], [28, 14, 10, 7, 5, 5, 4], [28, 14, 8, 8, 7, 4, 4], [24, 20, 8, 6, 6, 5, 4], [24, 18, 9, 8, 6, 4, 4], [21, 21, 7, 7, 7, 7, 3], [21, 15, 15, 7, 7, 5, 3], [21, 8, 8, 8, 8, 7, 7, 6], [20, 20, 15, 12, 4, 2], [20, 20, 10, 10, 5, 4, 4], [20, 18, 12, 9, 6, 5, 3], [20, 16, 16, 8, 5, 4, 4], [20, 8, 8, 8, 8, 8, 8, 5], [18, 18, 12, 9, 9, 4, 3], [18, 12, 9, 8, 8, 6, 6, 6], [15, 15, 15, 15, 5, 5, 3], [15, 15, 8, 8, 8, 8, 6, 5], [15, 12, 12, 10, 6, 6, 6, 6], [15, 12, 10, 10, 8, 8, 5, 5], [15, 12, 10, 8, 8, 8, 8, 4], [12, 12, 12, 9, 9, 9, 6, 4]]]
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