2017年8月27日日曜日

170827

Ruby


j がn 以下のとき、Sum_{i=1..j} s_i | Product_{i=1..j} s_i を満たす1~n の順列

小学生のときから何度か考えた問題だったのですが、
昨日まで自明な[1] 以外に答えがあるかわかりませんでした。
実はこれ以外にも条件を満たすものがあります。

例えば、[10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 3, 7, 1, 9, 12] が条件を満たします。
Excel で簡単に確認できます。

10 10 10 1
15 25 150 6
5 30 750 25
6 36 4,500 125
4 40 18,000 450
8 48 144,000 3,000
2 50 288,000 5,760
14 64 4,032,000 63,000
11 75 44,352,000 591,360
13 88 576,576,000 6,552,000
3 91 1,729,728,000 19,008,000
7 98 12,108,096,000 123,552,000
1 99 12,108,096,000 122,304,000
9 108 108,972,864,000 1,009,008,000
12 120 1,307,674,368,000 10,897,286,400

n が17以下のとき、条件を満たすものを探してみた。

def search(a, prod, sum, size, num)
  if num == size + 1
    @ary << a.clone
  else
    (1..size).each{|i|
      p, s = prod * i, sum + i
      if a[i - 1] == 0 && p % s == 0
        a[i - 1] = num
        search(a, p, s, size, num + 1)
        a[i - 1] = 0
      end
    }
  end
end

def A(n)
  a = [0] * n
  @ary = []
  search(a, 1, 0, n, 1)
  @ary.map{|i| (1..n).map{|j| i.index(j) + 1}}
end

n = 17
(1..n).each{|i| p [i, A(i)]}

出力結果
[1, [[1]]]
[2, []]
[3, []]
[4, []]
[5, []]
[6, []]
[7, []]
[8, []]
[9, []]
[10, []]
[11, []]
[12, []]
[13, []]
[14, []]
[15, [[10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 3, 7, 1, 9, 12], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 7, 1, 9, 4, 8], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 9, 4, 1, 7, 8], [10, 15, 5, 6, 14, 13, 2, 7, 8, 11, 9, 4, 1, 3, 12], [10, 15, 5, 6, 14, 13, 2, 7, 8, 11, 9, 4, 1, 12, 3], [10, 15, 5, 6, 14, 13, 7, 2, 8, 11, 9, 4, 1, 3, 12], [10, 15, 5, 6, 14, 13, 7, 2, 8, 11, 9, 4, 1, 12, 3], [10, 15, 5, 6, 14, 13, 7, 8, 2, 11, 9, 4, 1, 3, 12], [10, 15, 5, 6, 14, 13, 7, 8, 2, 11, 9, 4, 1, 12, 3], [10, 15, 5, 6, 14, 13, 9, 8, 11, 7, 2, 4, 1, 3, 12], [10, 15, 5, 6, 14, 13, 9, 8, 11, 7, 2, 4, 1, 12, 3], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 7, 1, 9, 4, 8], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 9, 4, 1, 7, 8], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 3, 7, 1, 9, 12], [15, 10, 5, 6, 12, 2, 14, 11, 13, 3, 7, 1, 9, 4, 8], [15, 10, 5, 6, 12, 2, 14, 11, 13, 3, 9, 4, 1, 7, 8], [15, 10, 5, 6, 14, 13, 2, 7, 8, 11, 9, 4, 1, 3, 12], [15, 10, 5, 6, 14, 13, 2, 7, 8, 11, 9, 4, 1, 12, 3], [15, 10, 5, 6, 14, 13, 7, 2, 8, 11, 9, 4, 1, 3, 12], [15, 10, 5, 6, 14, 13, 7, 2, 8, 11, 9, 4, 1, 12, 3], [15, 10, 5, 6, 14, 13, 7, 8, 2, 11, 9, 4, 1, 3, 12], [15, 10, 5, 6, 14, 13, 7, 8, 2, 11, 9, 4, 1, 12, 3], [15, 10, 5, 6, 14, 13, 9, 8, 11, 7, 2, 4, 1, 3, 12], [15, 10, 5, 6, 14, 13, 9, 8, 11, 7, 2, 4, 1, 12, 3], [15, 10, 5, 6, 14, 13, 12, 3, 2, 11, 7, 1, 9, 4, 8], [15, 10, 5, 6, 14, 13, 12, 3, 2, 11, 9, 4, 1, 7, 8]]]
[16, []]
[17, [[10, 15, 5, 6, 4, 8, 2, 14, 11, 9, 12, 16, 13, 1, 17, 7, 3], [10, 15, 5, 6, 4, 8, 2, 14, 11, 9, 16, 12, 13, 1, 17, 7, 3], [10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 3, 7, 12, 16, 17, 1, 9], [10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 3, 17, 12, 16, 7, 1, 9], [10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 12, 17, 3, 16, 7, 1, 9], [10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 16, 1, 3, 17, 7, 12, 9], [10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 16, 1, 12, 9, 17, 7, 3], [10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 17, 3, 12, 16, 7, 1, 9], [10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 17, 7, 16, 12, 3, 1, 9], [10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 17, 12, 3, 16, 7, 1, 9], [10, 15, 5, 6, 4, 8, 16, 11, 13, 2, 14, 1, 3, 17, 7, 12, 9], [10, 15, 5, 6, 4, 8, 16, 11, 13, 2, 14, 1, 12, 9, 17, 7, 3], [10, 15, 5, 6, 12, 2, 14, 11, 9, 4, 8, 16, 13, 1, 17, 7, 3], [10, 15, 5, 6, 12, 2, 14, 11, 9, 16, 8, 4, 13, 1, 17, 7, 3], [10, 15, 5, 6, 12, 2, 14, 11, 9, 16, 8, 17, 3, 7, 1, 4, 13], [10, 15, 5, 6, 12, 2, 14, 11, 9, 16, 8, 17, 7, 3, 1, 4, 13], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 17, 4, 7, 1, 8, 16, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 17, 4, 7, 1, 16, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 17, 4, 7, 16, 1, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 17, 4, 7, 16, 8, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 17, 4, 8, 16, 7, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 17, 4, 16, 7, 1, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 17, 4, 16, 7, 8, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 17, 4, 16, 8, 7, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 8, 4, 17, 3, 16, 7, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 1, 3, 17, 7, 4, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 1, 3, 17, 7, 8, 4, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 4, 9, 8, 1, 17, 7, 3], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 4, 17, 1, 9, 8, 7, 3], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 4, 17, 3, 7, 1, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 4, 17, 3, 7, 8, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 4, 17, 3, 8, 7, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 4, 17, 7, 3, 1, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 4, 17, 7, 3, 8, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 16, 4, 17, 7, 8, 3, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 3, 4, 7, 1, 8, 16, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 3, 4, 7, 1, 16, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 3, 4, 7, 16, 1, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 3, 4, 7, 16, 8, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 3, 4, 8, 16, 7, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 3, 4, 16, 7, 1, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 3, 4, 16, 7, 8, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 3, 4, 16, 8, 7, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 7, 8, 16, 4, 3, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 7, 16, 4, 3, 1, 8, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 7, 16, 4, 3, 8, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 7, 16, 4, 8, 3, 1, 9], [10, 15, 5, 6, 12, 2, 14, 11, 13, 17, 7, 16, 8, 4, 3, 1, 9], [10, 15, 5, 6, 12, 16, 11, 13, 2, 9, 1, 8, 17, 7, 4, 14, 3], [10, 15, 5, 6, 12, 16, 11, 13, 2, 9, 1, 17, 8, 7, 4, 14, 3], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 1, 3, 17, 7, 4, 8, 9], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 1, 3, 17, 7, 8, 4, 9], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 4, 9, 8, 1, 17, 7, 3], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 4, 17, 1, 9, 8, 7, 3], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 4, 17, 3, 7, 1, 8, 9], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 4, 17, 3, 7, 8, 1, 9], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 4, 17, 3, 8, 7, 1, 9], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 4, 17, 7, 3, 1, 8, 9], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 4, 17, 7, 3, 8, 1, 9], [10, 15, 5, 6, 12, 16, 11, 13, 2, 14, 4, 17, 7, 8, 3, 1, 9], [10, 15, 5, 6, 14, 13, 2, 7, 3, 16, 17, 11, 1, 8, 4, 12, 9], [10, 15, 5, 6, 14, 13, 2, 7, 3, 16, 17, 11, 1, 8, 12, 4, 9], [10, 15, 5, 6, 14, 13, 2, 7, 3, 16, 17, 11, 1, 12, 4, 8, 9], [10, 15, 5, 6, 14, 13, 2, 7, 3, 16, 17, 11, 1, 12, 8, 4, 9], [10, 15, 5, 6, 14, 13, 2, 7, 8, 4, 16, 17, 11, 12, 3, 1, 9], [10, 15, 5, 6, 14, 13, 2, 7, 8, 16, 4, 17, 11, 12, 3, 1, 9], [10, 15, 5, 6, 14, 13, 2, 7, 8, 16, 12, 17, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 2, 7, 12, 16, 4, 1, 3, 17, 11, 8, 9], [10, 15, 5, 6, 14, 13, 2, 7, 12, 16, 8, 17, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 2, 7, 12, 16, 17, 8, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 2, 7, 12, 16, 17, 11, 4, 3, 1, 8, 9], [10, 15, 5, 6, 14, 13, 2, 7, 12, 16, 17, 11, 4, 3, 8, 1, 9], [10, 15, 5, 6, 14, 13, 2, 7, 12, 16, 17, 11, 4, 8, 3, 1, 9], [10, 15, 5, 6, 14, 13, 2, 7, 12, 16, 17, 11, 8, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 2, 3, 16, 17, 11, 1, 8, 4, 12, 9], [10, 15, 5, 6, 14, 13, 7, 2, 3, 16, 17, 11, 1, 8, 12, 4, 9], [10, 15, 5, 6, 14, 13, 7, 2, 3, 16, 17, 11, 1, 12, 4, 8, 9], [10, 15, 5, 6, 14, 13, 7, 2, 3, 16, 17, 11, 1, 12, 8, 4, 9], [10, 15, 5, 6, 14, 13, 7, 2, 8, 4, 16, 17, 11, 12, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 2, 8, 16, 4, 17, 11, 12, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 2, 8, 16, 12, 17, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 2, 12, 16, 4, 1, 3, 17, 11, 8, 9], [10, 15, 5, 6, 14, 13, 7, 2, 12, 16, 8, 17, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 2, 12, 16, 17, 8, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 2, 12, 16, 17, 11, 4, 3, 1, 8, 9], [10, 15, 5, 6, 14, 13, 7, 2, 12, 16, 17, 11, 4, 3, 8, 1, 9], [10, 15, 5, 6, 14, 13, 7, 2, 12, 16, 17, 11, 4, 8, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 2, 12, 16, 17, 11, 8, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 8, 2, 4, 16, 17, 11, 12, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 8, 2, 16, 4, 17, 11, 12, 3, 1, 9], [10, 15, 5, 6, 14, 13, 7, 8, 2, 16, 12, 17, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 7, 17, 11, 1, 8, 16, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 7, 17, 11, 1, 16, 8, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 7, 17, 11, 16, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 7, 17, 11, 16, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 16, 8, 17, 11, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 16, 17, 8, 11, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 16, 17, 11, 7, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 16, 17, 11, 7, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 4, 16, 17, 11, 8, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 17, 4, 7, 1, 8, 16, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 17, 4, 7, 1, 16, 8, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 17, 4, 7, 16, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 17, 4, 7, 16, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 17, 4, 8, 16, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 17, 4, 16, 7, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 17, 4, 16, 7, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 17, 4, 16, 8, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 16, 4, 8, 17, 11, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 16, 4, 17, 8, 11, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 16, 4, 17, 11, 7, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 16, 4, 17, 11, 7, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 16, 4, 17, 11, 8, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 3, 2, 16, 8, 4, 17, 11, 7, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 7, 2, 4, 1, 3, 17, 11, 8, 9], [10, 15, 5, 6, 14, 13, 12, 16, 7, 2, 8, 17, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 7, 2, 17, 8, 11, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 7, 2, 17, 11, 4, 3, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 16, 7, 2, 17, 11, 4, 3, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 7, 2, 17, 11, 4, 8, 3, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 7, 2, 17, 11, 8, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 4, 7, 11, 2, 3, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 4, 7, 11, 2, 3, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 4, 7, 11, 2, 8, 3, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 11, 7, 2, 4, 3, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 11, 7, 2, 4, 3, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 11, 7, 2, 4, 8, 3, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 11, 7, 2, 8, 4, 3, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 11, 7, 4, 2, 3, 1, 8, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 11, 7, 4, 2, 3, 8, 1, 9], [10, 15, 5, 6, 14, 13, 12, 16, 17, 11, 7, 4, 2, 8, 3, 1, 9], [15, 10, 5, 6, 4, 8, 2, 14, 11, 9, 12, 16, 13, 1, 17, 7, 3], [15, 10, 5, 6, 4, 8, 2, 14, 11, 9, 16, 12, 13, 1, 17, 7, 3], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 3, 7, 12, 16, 17, 1, 9], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 3, 17, 12, 16, 7, 1, 9], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 12, 17, 3, 16, 7, 1, 9], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 16, 1, 3, 17, 7, 12, 9], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 16, 1, 12, 9, 17, 7, 3], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 17, 3, 12, 16, 7, 1, 9], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 17, 7, 16, 12, 3, 1, 9], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 17, 12, 3, 16, 7, 1, 9], [15, 10, 5, 6, 4, 8, 16, 11, 13, 2, 14, 1, 3, 17, 7, 12, 9], [15, 10, 5, 6, 4, 8, 16, 11, 13, 2, 14, 1, 12, 9, 17, 7, 3], [15, 10, 5, 6, 12, 2, 14, 11, 9, 4, 8, 16, 13, 1, 17, 7, 3], [15, 10, 5, 6, 12, 2, 14, 11, 9, 16, 8, 4, 13, 1, 17, 7, 3], [15, 10, 5, 6, 12, 2, 14, 11, 9, 16, 8, 17, 3, 7, 1, 4, 13], [15, 10, 5, 6, 12, 2, 14, 11, 9, 16, 8, 17, 7, 3, 1, 4, 13], [15, 10, 5, 6, 12, 2, 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