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数珠順列の問題の解法 | 夢を叶える塾
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数珠順列の問題の解法

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数珠型になる順列を求める問題です。

基本問題

6つの石を使って腕輪を作るとき、何通りの腕輪を作ることができるか求めなさい。

解き方

数珠順列の考え方を使います。
まず円順列の様に回転して同じ並びがあります。
ABCDEFとBCDEFA,…が等しいという考え方です。
さらに回転ではなく「ひっくり返しても同じ腕輪になる」という事を考慮する必要があります。
腕輪はひっくり返して使っても良いわけですよね。
ABCDEFとFEDBCAが同じという考え方です。

解説

まず円順列の求め方で円形に並べます。
更にひっくり返したものが同じになります。

6つの石をABCDEFとします。
ABCDEFとFEDBCAが同じという観点です。
BCDEFAは円順列の考えでABCDEFと同じです。
このBCDEFAもひっくり返してAFEDCBになります。
ABCDEFに対してBCDEFA,CDEFAB,DEFABC,EFABCD,FABCDEが円順列の考えて同じ。
ABCDEFに対してFEDCBAが、BCDEFAに対してAFEDCBが、CDEFABに対してBAFEDCが、
DEFABCに対してCBAFEDが、EFABCDに対してDCBAFEが、FABCDEに対してEDCBAFが数珠順列の考えで同じになります。
つまり12で割ることになります。
これは、6で割って、さらに2で割ると考えると良いでしょう。

6!\div 12=60通りになります。

終わりに

ひっくり返して同じという考えに至れるかどうかです。
空間的にひっくり返せるものは数珠順列の考えとするのかどうか吟味してください。

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