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

2017年8月26日土曜日

170826

New game(Cubes)(2)

1~n のカードを使った場合を考える。
最初の人が1 とn 以外を引き、n 番目の人が1 とn 以外で終わる
場合があるか考えてみた。

今のところ以下の場合が見つかった。
n = 6
[[2, 1, 6, 5, 4, 3], [2, 6, 1, 5, 4, 3]]
n = 7
[[2, 1, 6, 5, 7, 3, 4], [2, 6, 1, 5, 7, 3, 4], [3, 1, 5, 7, 2, 6, 4], [3, 5, 1, 7, 2, 6, 4], [3, 5, 7, 1, 2, 6, 4], [3, 7, 5, 1, 2, 6, 4], [5, 1, 3, 7, 2, 6, 4], [5, 3, 1, 7, 2, 6, 4], [5, 3, 7, 1, 2, 6, 4], [5, 7, 3, 1, 2, 6, 4], [6, 1, 2, 5, 7, 3, 4], [6, 2, 1, 5, 7, 3, 4]]

例えば、
2 | 2^3 (2 | 8),
2 + 1 | 2^3 + 1^3 (3 | 9),
2 + 1 + 6 | 2^3 + 1^3 + 6^3 (9 | 225),
2 + 1 + 6 + 5 | 2^3 + 1^3 + 6^3 + 5^3 (14 | 350),
2 + 1 + 6 + 5 + 4 | 2^3 + 1^3 + 6^3 + 5^3 + 4^3 (18 | 414),
2 + 1 + 6 + 5 + 4 + 3 | 2^3 + 1^3 + 6^3 + 5^3 + 4^3 + 3^3 (21 | 441)
なので、[2, 1, 6, 5, 4, 3] は条件を満たす。

2017年8月24日木曜日

170824

New game(Cubes)(1)

2~4人用のゲームを考えてみました。

①最初の人が1~13 のカードから一枚引きます。
sum1 は引いたカードから始めます。
sum3 は引いたカードの3乗から始めます。
例えば、引いたカードが5 とします。
sum1 は5、sum3 は125からとなります。

②次の人は以下の条件を満たす数字を言わないと負けです。
ア)1~13 でまだ使われていない数字 (x とする) である。
sum1 += x, sum3 += x^3 とし、
イ)sum3 がsum1 で割り切れる。

③まだ使われていない数字が一つになっても勝負が決まらなかったら、
負けていない人全員が勝ちです。

2017年8月13日日曜日

170813

PARI


A159463

3848000000 までは条件を満たすものがないことを確認のうえで、2*10^10 までの範囲で探してみた。

for (i=3848000000, 2*10^10, if (sumdigits(i^3)==216, print1(i, ", ")))

出力結果
3848163483, 4462569999, 4479677412, 4586158119, 4594661259, 4594665192, 4594700889, 4625720379, 4641588459, 5644008999, 5828410842, 5833034823, 5838252576, 5848025709, 6453471192, 6617331999, 6619097067, 6686657169, 7107126942, 7230291999, 7277907183, 7298954529, 7361104599, 7655249499, 7696321113, 7752292599, 7763935413, 7807648032, 7849790355, 7861079679, 7870842513, 7882662024, 7899769239, 7928513619, 7936309899, 8040925644, 8087527866, 8168254083, 8188243593, 8217313053, 8242024149, 8280652632, 8290998999, 8324036319, 8329419948, 8329894083, 8371920453, 8385953229, 8386243995, 8386623423, 8427714576, 8429586192, 8434259829, 8508456957, 8608298259, 8702125599, 8749417494, 8750034276, 8767580826, 8813357679, 8814253116, 8827927632, 8831296395, 8831814675, 8861606859, 8870574612, 8878954773, 8887441362, 8950297629, 8999835345, 9061217229, 9099512259, 9105748326, 9111328683, 9164461683, 9200791413, 9200826942, 9201177342, 9203865459, 9205040826, 9232148829, 9236117229, 9236435532, 9236514492, 9239456457, 9243860229, 9259475979, 9265691559, 9271519599, 9271524144, 9274580826, 9275392599, 9278832576, 9279153159, 9282750192, 9282751383, 9282766863, 9283020159, 9316934409, 9320930229, 9386451357, 9457795719, 9464078862, 9491142213, 9497838099, 9537601263, 9545158146, 9545221113, 9545999622, 9563558259, 9574843299, 9575174463, 9578073729, 9582330753, 9596601429, 9607458999, 9610159842, 9613846716, 9614711529, 9618097326, 9618778272, 9618968892, 9631726989, 9636575532, 9642246894, 9643724112, 9650346753, 9650922513, 9652586844, 9654353742, 9654406353, 9718439997, 9776365719, 9791995299, 9823536192, 9826299999, 9847688529, 9854520498, 9857965089, 9857992413, 9862414998, 9887976423, 9895235574, 9895341129, 9895486122, 9898595148, 9898982259, 9912163032, 9918939999, 9919211679, 9924690132, 9926069772, 9928621557, 9929468529, 9932542299, 9952642494, 9953055627, 9955426599, 9956248755, 9959838219, 9960709155, 9961179132, 9961850913, 9962116599, 9962791716, 9962818149, 9963523692, 9964500795, 9965882259, 9966167163, 9966507279, 9966509223, 9966539799, 9972758859, 9976527882, 9983271999, 9986144136, 9989218299, 9989955213, 9991252299, 9993261123, 9993291123, 9993528429, 9996275175, 9996293619, 9999285879, 9999322254, 9999622299, 9999884994, 10256240559, 10321795953, 10620675192, 10828176813, 10857602286, 10880214366, 10897084083, 10910782389, 11186324622, 11186886192, 11266225857, 11286808683, 11315902632, 11370297999, 11417020842, 11423631999, 11446868529, 11588777229, 11616830229, 11617298529, 11634560229, 11637268389, 11642213649, 11643937533, 11647125372, 11647130139, 11664062862, 11665725999, 11691171129, 11695093323, 11756672592, 11807411529, 11811377229, 11811758529, 11826153186, 11840423865, 11877919353, 11880333729, 11899222857, 11906656866, 11924760159, 11927673195, 11930343132, 11934524994, 12050665413, 12092110596, 12093909459, 12096216942, 12096420816, 12116617536, 12118932999, 12135047592, 12136133616, 12137074998, 12141696999, 12141817155, 12153012849, 12154928829, 12156965616, 12161694954, 12162601362, 12163498326, 12163694355, 12163800192, 12163950216, 12168826599, 12200108142, 12215592999, 12229380942, 12270919353, 12275892966, 12293590839, 12298067397, 12310594782, 12311001435, 12317862159, 12317862429, 12319645383, 12330836523, 12336976389, 12337172799, 12339601116, 12340675899, 12357812076, 12359468529, 12363309999, 12363853557, 12376026519, 12376424739, 12383229366, 12383430192, 12383443866, 12383662689, 12385098882, 12385353327, 12385401435, 12400520031, 12492838116, 12493322682, 12503951559, 12507937353, 12511815999, 12518904513, 12529101195, 12531377157, 12533690229, 12533768316, 12545179413, 12547763706, 12550680426, 12552833442, 12556368759, 12557048889, 12561210129, 12566893413, 12567405213, 12569572581, 12570578595, 12571176813, 12571753299, 12573497349, 12573535539, 12573725376, 12575803692, 12576022635, 12576046053, 12576657459, 12577544616, 12577744782, 12577852866, 12577957554, 12582127995, 12591835413, 12592266582, 12592876188, 12594343854, 12595805667, 12596593413, 12596866599, 12596895132, 12597947679, 12598345293, 12598555299, 12598759599, 12598919442, 12599183199, 12599199513, 12599206299, 12944118675, 12980221383, 12997431423, 13180703316, 13199314683, 13199955999, 13332161559, 13351296966, 13360520529, 13368260229, 13384936749, 13517391942, 13535392599, 13535773413, 13548322389, 13568440716, 13678009899, 13693242576, 13715265999, 13731044169, 13747156299, 13750448382, 13750688529, 13783206099, 13838219124, 13839978936, 13852012632, 13852880649, 13853848383, 13853919312, 13864160349, 13864329999, 13898065716, 13903732413, 13905828999, 13907328999, 13914442122, 13922177037, 13922665383, 13922695329, 13923047142, 14022070593, 14027154423, 14057062476, 14058882759, 14060891169, 14070898329, 14073877809, 14074362627, 14074377759, 14075769423, 14084497149, 14087191389, 14092747926, 14092831683, 14094160731, 14094161169, 14094592926, 14094597123, 14158022394, 14192741775, 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2017年8月7日月曜日

170807

Ruby


Matiyasevich's Identity

ベルヌーイ数に関する恒等式を確認してみた。
三つ目が1997年にMatiyasevich が見つけた恒等式です。

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    ary << a[0] # Bn = a[0]
  }
  ary
end

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

n = 20
p b_ary = bernoulli(2 * n)
p ''
(2..n).each{|i|
  x = i + 1
  p [i, x == (0..i).inject(0){|s, j| s + ncr(i + 1, j) * b_ary[j]}, x]
}
p ''
(2..n).each{|i|
  x = -(i + 1) * b_ary[i]
  p [i, x == (2..i - 2).inject(0){|s, j| s + ncr(i, j) * b_ary[j] * b_ary[i - j]}, x]
}
p ''
(2..n).each{|i|
  x = i * (i + 1) * b_ary[i]
  p [i, x == (i + 2) * (2..i - 2).inject(0){|s, j| s + b_ary[j] * b_ary[i - j]} - 2 * (2..i - 2).inject(0){|s, j| s += ncr(i + 2, j) * b_ary[j] * b_ary[i - j]}, x]
}

出力結果
[(1/1), (1/2), (1/6), (0/1), (-1/30), (0/1), (1/42), (0/1), (-1/30), (0/1), (5/66), (0/1), (-691/2730), (0/1), (7/6), (0/1), (-3617/510), (0/1), (43867/798), (0/1), (-174611/330), (0/1), (854513/138), (0/1), (-236364091/2730), (0/1), (8553103/6), (0/1), (-23749461029/870), (0/1), (8615841276005/14322), (0/1), (-7709321041217/510), (0/1), (2577687858367/6), (0/1), (-26315271553053477373/1919190), (0/1), (2929993913841559/6), (0/1), (-261082718496449122051/13530)]
""
[2, true, 3]
[3, true, 4]
[4, true, 5]
[5, true, 6]
[6, true, 7]
[7, true, 8]
[8, true, 9]
[9, true, 10]
[10, true, 11]
[11, true, 12]
[12, true, 13]
[13, true, 14]
[14, true, 15]
[15, true, 16]
[16, true, 17]
[17, true, 18]
[18, true, 19]
[19, true, 20]
[20, true, 21]
""
[2, false, (-1/2)]
[3, true, (0/1)]
[4, true, (1/6)]
[5, true, (0/1)]
[6, true, (-1/6)]
[7, true, (0/1)]
[8, true, (3/10)]
[9, true, (0/1)]
[10, true, (-5/6)]
[11, true, (0/1)]
[12, true, (691/210)]
[13, true, (0/1)]
[14, true, (-35/2)]
[15, true, (0/1)]
[16, true, (3617/30)]
[17, true, (0/1)]
[18, true, (-43867/42)]
[19, true, (0/1)]
[20, true, (1222277/110)]
""
[2, false, (1/1)]
[3, true, (0/1)]
[4, true, (-2/3)]
[5, true, (0/1)]
[6, true, (1/1)]
[7, true, (0/1)]
[8, true, (-12/5)]
[9, true, (0/1)]
[10, true, (25/3)]
[11, true, (0/1)]
[12, true, (-1382/35)]
[13, true, (0/1)]
[14, true, (245/1)]
[15, true, (0/1)]
[16, true, (-28936/15)]
[17, true, (0/1)]
[18, true, (131601/7)]
[19, true, (0/1)]
[20, true, (-2444554/11)]