趣题:为什么偏偏是 6 格?

无穷多个相同大小的正方形格子排成一排,向左右两边无限地延伸。每个格子里都有 0 个、 1 个或多个原子。每一次,你可以对它们做下面两种操作之一:

  • 选择某个格子,保证该格子内至少含有 1 个原子。将该格子内的其中 1 个原子分裂为 2 个,从而使得该格子内的原子数量减 1 ,两边的邻格里的原子数量分别加 1。
  • 选择某个格子,保证两边的邻格里均至少含有 1 个原子。从两边的邻格里各取 1 个原子聚合起来,从而使得两边的邻格里的原子数量分别减 1 ,该格子内的原子数量加 1。

初始时,某个格子里有 1 个原子。现在,你需要在若干次操作之后,让它右移 6 格。也就是说,你需要用若干次操作把下面的第一个图变成第二个图(其中,数字 1 表示该格内的原子数为 1 )。继续阅读下去之前,你不妨自己先试一试。你可以在纸上画好格子,用硬币、大米、巧克力豆等物体代替原子。

 

 

 

 

 

 

 

 

 

其中一种方法如下。

你或许会觉得这里面隐藏着一些规律。我们似乎可以对上面的方法进行扩展,把初始时的那个原子移到它右边的任意一格里,比方说把初始时的那个原子移到它右边第 2 格,或者第 3 格,或者第 7 格的位置。然而,简单地试试,你会发现,这还真的不行。上述方法只适用于右移 6 格的情况。事实上,我们可以证明,如果某个格子里的 1 个原子最终可以变为另外一个格子里的 1 个原子,这两个格子的位置一定相差 6 或者 6 的整倍数。

让我们像下面这样,为每一个格子都赋一个值,其中 ω = (-1)1/3 ,即 -1 在复数范围内的一个立方根。注意到 ω 满足 ω3 + 1 = 0 ,即 (ω + 1)(ω2 – ω + 1) = 0 ;而 ω + 1 ≠ 0 ,于是只能是 ω2 – ω + 1 = 0 。

对于整个棋盘上的任意一种原子排布,我们都可以把每个放有原子的格子里的值加在一起,一个格子里有几个原子,它对应的值就加几次。我们把如此得到的总和叫做此时棋盘上原子排布的特征值。例如,如果 ω-2 上有 2 个原子, ω2 上有 1 个原子, ω3 上有 4 个原子,其他格子都是空的,那么此时棋盘上原子排布的特征值就是 2 · ω-2 + 1 · ω2 + 4 · ω3

当位于 ωn 位置上的原子分裂时,特征值会减小一个 ωn ,但会增加一个 ωn-1 和一个 ωn+1 ,所以特征值的变化量为 – ωn + ωn-1 + ωn+1 = ωn-1 · (- ω + 1 + ω2) 。然而,之前我们说过, ω 的取值满足 ω2 – ω + 1 = 0 。因此,任何一次原子分裂操作都不会改变原子排布的特征值。

类似地,原子聚合导致的特征值的变化量为 – ωn-1 – ωn+1 + ωn = ωn-1 · (- 1 – ω2 + ω) = 0 。因此,任何一次原子聚合操作也都不会改变原子排布的特征值。

如果原来整个棋盘上只有 ωn 位置上有 1 个原子,后来变得整个棋盘上只有 ωm 位置上有 1 个原子,这就意味着 ωn = ωm ,即 ωm / ωn = 1 ,即 ωm-n = 1 。考虑到 ω = (-1)1/3 ,因此为了让 ωm-n = 1 ,只有可能 m – n 是 6 的整倍数。这就解答了前面提出的问题。

我最初是在这里看见的这个问题: http://blog.sigfpe.com/2007/09/arboreal-isomorphisms-from-nuclear.html 。如果你对这个问题感兴趣的话,你一定会喜欢下面这个类似的但是更经典、更精彩的问题: http://www.matrix67.com/blog/archives/4595

63 条评论

  • q68257962

    一变二,二变一,还是比较容易朝着x^3=1的三个根方向思考的。

  • F_XZ

    跟着去看了旧的那篇, Conway大神是厉害呀…

  • Rholin

    第一次这么前排

  • fanyer

    taste good!!!

  • fdksao

    我很好奇中间的动图是用什么软件画的?

  • dreamylife

    我也很好奇整个的排版是不是Markdown或者LaTeX之类的标记语言

  • 0x3b

    写了篇小文讨论了这个计算特征值的方法。有一个有意思的结论,就是任意非零的特征值都对应一组可以互相转换的棋盘状态。特征值为零的状态,除去没有原子的情况,也是可以互相转换的。这就是说可以有一个快速的方法判断两个棋盘状态是否能够互相转换。

  • 石櫻燈籠

    仅仅是ω = (-1)1/3是怎么来的我就搞不懂QAQ

  • 小柿子

    华丽!

  • C程序语言

    ω = (-1)1/3 从这开始就懵逼了

  • 32805433

    这样也可以:赋值… 2 6 3 1/2 1/6 1/3 2 6 3 1/2 1/6 ..然后每次操作积不变.

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    alert(‘hi boys’);

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    看完感觉好厉害啊!!!

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  • rohinb97

    I’m not sure how intuitive substituting positions with \omega felt, but my initial approach was like this and I can move the atom/coin/whatever by any number of spaces required. Here is how I approached this problem:
    Lets take the entire thing as a number (that is what my intuition said) and use the decimal point as a marker. Hence a split from 1 is 10.1
    Now no matter how many times you split the leftmost atom you can always come back to the original configuration. So the trick of shifting felt like splitting the rightmost atom and my initial approach was to clean up the mess I make on the left side. So I split the rightmost atom and whenever I get the chance I combine the leftmost atom. Here is the game:
    Move 0: 1
    Move 1: 10.1 (Initial split)
    Move 2: 11.01 (Rightmost split)
    Move 3: 11.101 (Another rightmost split)
    Move 4: 2.011 (Leftmost recombine)
    Move 5: 1.101 (Leftmost recombine, notice how this is a shifted version of move 2)
    Move 6: 1.01 (Did the rightmost recombine, just to show the shift)
    Move 7: 0.1 (Atom shifted by one)

    Did I do something wrong here? It seems like the same approach

  • 破壁人五号

    将格子赋值为 x, y, y-x, -x, -y, x-y, x … 也能推出来,同样要求每次操作和不变

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