小學生算術題🧮紛爭第十彈🤕：Kaprekar constant

从 1, 2, 3, 4, 5, 6, 7, 8, 9 中随便挑 3 个数（如 1, 2, 3），然后用组成的最大三位数（321）减去最小三位数（123）……如此往复，最后一定会得到 495 。

小朋友问我为什么的时候，我故作镇定，∵第一反应是「难道是 3n + 1 猜想？完蛋😵💫」，但为了让小朋友们看到「碰到难题，最重要的是「坚持💪🏻」」，只得硬着头皮做。结果，发现不是 3n + 1 猜想，如释重负。证明其实不难，可能用了一刻钟吧：

1️⃣ 假设取的 3 个数分别是 $a_1$ , $a_2$ , $a_3$ ，且 $a_1 < a_2 < a_3$ 。那么，最大数是 $\overline{a_3 a_2 a_1}$ ，最小数是 $\overline{a_1 a_2 a_3}$ ，

$\begin{align} \Delta_1 &= \overline{a_3 a_2 a_1} - \overline{a_1 a_2 a_3} \\ &= (10^2 a_3+ 10 a_2 + a_1) - (10^2 a_1+ 10 a_2 + a_3) \\ &= 10^2 (a_3 - a_1) + (a_1 - a_3) \\ &= 10^2 (a_3 - a_1 -1) + 10 \times 9 + \big(10 - (a_3 - a_1) \big) \\ &= \overline{(a_3 - a_1 -1) 9 \big(10 - (a_3 - a_1) \big)} \end{align}$

看见，经过一轮变换， $a_2$ 没了，我们引入 $\delta_1 \overset{def}{=} a_3 - a_1$ 。

先来给 $\delta_1$ 定个界：

$\delta_1 = a_3 - a_1 \ge (a_2 + 1) - a_1 \ge \big( (a_1 + 1) + 1 \big) - a_1 = 2$ ，即 $\delta_1 \ge 2$ ；
$\delta_1 = a_3 - a_1 \le 9 - a_1 \le 9 - 1 = 8$ ，即 $\delta_1 \le 8$ ；

∴ $2 \le \delta_1 \le 8$ .

则新一轮的三个数变成了 $\delta_1 -1$ , 9, $10 - \delta_1$ ，且 $(\delta_1 -1) + (10 - \delta_1) = 9$ 。

如：321 - 123 = 198，百位上的 1 + 个位上的 8 = 9 。

2️⃣ 假设第二轮的 3 个数是 $b_1$ , $b_2$ , $b_3$ ， $b_1 < b_2 < b_3 = 9$ 。

$b_3 = 9$ ，而 $b_2$ 会被抵消掉，∴最关键的是 $b_1$ ：

$\begin{align} b_1 &= \min \begin{cases} \delta_1 - 1 \\ 10 - \delta_1 \le 8 \end{cases} \\ &= \begin{cases} \delta_1 - 1 , 若 \delta_1 < \frac{11}{2}, 即 \delta_1 \le 5 \\ 10 - \delta_1 , 若 \delta_1 > \frac{11}{2}, 即 \delta_1 \ge 6 \end{cases} \end{align}$

$\delta_2 = 9 - b_1$ ，而 $\Delta_2 = \overline{(\delta_2 - 1) 9 (10 - \delta_2)}$

3️⃣ 假设第三轮的 3 个数是 $c_1$ , $c_2$ , $c_3$ ， $c_1 < c_2 < c_3 = 9$ 。

$c_1 = \min \begin{cases} \delta_2 - 1 \\ 10 - \delta_2 \end{cases}$

正如 $123 \xrightarrow{321 - 123} 198 \xrightarrow{981 - 189} 792 \xrightarrow{972 - 279} 693 \xrightarrow{963 - 369} 594 \xrightarrow{954 - 459} 495 \xrightarrow{954 - 459} 495 \xrightarrow{954 - 459} \dots$ 。但我不能证明序列收敛： $a_1 < b_1 < c_1 < \dots , a_3 > b_3 > c_3 > \dots$ 😭。

然后我去问了 DeepSeek，它说：

$789 \xrightarrow{987 - 789} 198 \xrightarrow{981 - 189} 792 \xrightarrow{972 - 279} \dots$ ，7 → 1 → 7 → … ∴百位数并不严格递增。

真正收敛的是 $\{\delta_n\}$ 。

$\delta_n$	$\delta_{n + 1}$	$\delta_{n + 2} → \dots$
2	$9 - (\delta_n - 1) = 8$	→ 7 → 6 → 5
3	$9 - (\delta_n - 1) = 7$	→ 6 → 5
4	$9 - (\delta_n - 1) = 6$	→ 5
5	$9 - (\delta_n - 1) = 5$	→ 5
6	$9 - (10 - \delta_n) = 5$	→ 5
7	$9 - (10 - \delta_n) = 6$	→ 5
8	$9 - (10 - \delta_n) = 7$	→ 6 → 5
9	$9 - (10 - \delta_n) = 8$	→ 7 → 6 → 5

只要 $\delta_n = 5$ ，就掉进了不动点。

得证。把我累出一手汗😓。

附程序验证：共有 $9 \times 8 \times 7 = 504$ 种取法，∵会重排序，需扣掉位置的影响，即仅有 $\frac{504}{3 \times 2 \times 1} = 84$ 种实质不同的取法。

def reorder(ary)
  a = ary.min 
  c = ary.max 
  b = ((ary - [a]) - [c]).first
  # a < b < c
  [a, b, c]
end

def kaprekar(ary, level)
  print "#{ary.join} → "
  a, b, c = reorder(ary)
  delta = c - a
  u = delta - 1
  v = 9
  w = 10 - delta

  if [u, v, w] == ary
    print "... (#{level})\n"
    return
  else
    kaprekar([u, v, w], level + 1)
  end
end

(1..9).to_a.combination(3) do |v|
  v.permutation do |w|
    # p w
    kaprekar(w, 0)
  end
end

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# 845 → 396 → 594 → 495 → ... (3)
# 854 → 396 → 594 → 495 → ... (3)
# 459 → 495 → ... (1)
# 495 → ... (0)
# 549 → 495 → ... (1)
# 594 → 495 → ... (1)
# 945 → 495 → ... (1)
# 954 → 495 → ... (1)
# 467 → 297 → 693 → 594 → 495 → ... (4)
# 476 → 297 → 693 → 594 → 495 → ... (4)
# 647 → 297 → 693 → 594 → 495 → ... (4)
# 674 → 297 → 693 → 594 → 495 → ... (4)
# 746 → 297 → 693 → 594 → 495 → ... (4)
# 764 → 297 → 693 → 594 → 495 → ... (4)
# 468 → 396 → 594 → 495 → ... (3)
# 486 → 396 → 594 → 495 → ... (3)
# 648 → 396 → 594 → 495 → ... (3)
# 684 → 396 → 594 → 495 → ... (3)
# 846 → 396 → 594 → 495 → ... (3)
# 864 → 396 → 594 → 495 → ... (3)
# 469 → 495 → ... (1)
# 496 → 495 → ... (1)
# 649 → 495 → ... (1)
# 694 → 495 → ... (1)
# 946 → 495 → ... (1)
# 964 → 495 → ... (1)
# 478 → 396 → 594 → 495 → ... (3)
# 487 → 396 → 594 → 495 → ... (3)
# 748 → 396 → 594 → 495 → ... (3)
# 784 → 396 → 594 → 495 → ... (3)
# 847 → 396 → 594 → 495 → ... (3)
# 874 → 396 → 594 → 495 → ... (3)
# 479 → 495 → ... (1)
# 497 → 495 → ... (1)
# 749 → 495 → ... (1)
# 794 → 495 → ... (1)
# 947 → 495 → ... (1)
# 974 → 495 → ... (1)
# 489 → 495 → ... (1)
# 498 → 495 → ... (1)
# 849 → 495 → ... (1)
# 894 → 495 → ... (1)
# 948 → 495 → ... (1)
# 984 → 495 → ... (1)
# 567 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 576 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 657 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 675 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 756 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 765 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 568 → 297 → 693 → 594 → 495 → ... (4)
# 586 → 297 → 693 → 594 → 495 → ... (4)
# 658 → 297 → 693 → 594 → 495 → ... (4)
# 685 → 297 → 693 → 594 → 495 → ... (4)
# 856 → 297 → 693 → 594 → 495 → ... (4)
# 865 → 297 → 693 → 594 → 495 → ... (4)
# 569 → 396 → 594 → 495 → ... (3)
# 596 → 396 → 594 → 495 → ... (3)
# 659 → 396 → 594 → 495 → ... (3)
# 695 → 396 → 594 → 495 → ... (3)
# 956 → 396 → 594 → 495 → ... (3)
# 965 → 396 → 594 → 495 → ... (3)
# 578 → 297 → 693 → 594 → 495 → ... (4)
# 587 → 297 → 693 → 594 → 495 → ... (4)
# 758 → 297 → 693 → 594 → 495 → ... (4)
# 785 → 297 → 693 → 594 → 495 → ... (4)
# 857 → 297 → 693 → 594 → 495 → ... (4)
# 875 → 297 → 693 → 594 → 495 → ... (4)
# 579 → 396 → 594 → 495 → ... (3)
# 597 → 396 → 594 → 495 → ... (3)
# 759 → 396 → 594 → 495 → ... (3)
# 795 → 396 → 594 → 495 → ... (3)
# 957 → 396 → 594 → 495 → ... (3)
# 975 → 396 → 594 → 495 → ... (3)
# 589 → 396 → 594 → 495 → ... (3)
# 598 → 396 → 594 → 495 → ... (3)
# 859 → 396 → 594 → 495 → ... (3)
# 895 → 396 → 594 → 495 → ... (3)
# 958 → 396 → 594 → 495 → ... (3)
# 985 → 396 → 594 → 495 → ... (3)
# 678 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 687 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 768 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 786 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 867 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 876 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 679 → 297 → 693 → 594 → 495 → ... (4)
# 697 → 297 → 693 → 594 → 495 → ... (4)
# 769 → 297 → 693 → 594 → 495 → ... (4)
# 796 → 297 → 693 → 594 → 495 → ... (4)
# 967 → 297 → 693 → 594 → 495 → ... (4)
# 976 → 297 → 693 → 594 → 495 → ... (4)
# 689 → 297 → 693 → 594 → 495 → ... (4)
# 698 → 297 → 693 → 594 → 495 → ... (4)
# 869 → 297 → 693 → 594 → 495 → ... (4)
# 896 → 297 → 693 → 594 → 495 → ... (4)
# 968 → 297 → 693 → 594 → 495 → ... (4)
# 986 → 297 → 693 → 594 → 495 → ... (4)
# 789 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 798 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 879 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 897 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 978 → 198 → 792 → 693 → 594 → 495 → ... (5)
# 987 → 198 → 792 → 693 → 594 → 495 → ... (5)

Ref:

小學生算術題🧮紛爭第十彈🤕：Kaprekar constant

小學生算術題🧮紛爭第十彈🤕：Kaprekar constant

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