13枚のコインの中に1枚だけ偽のコインがあります。
偽物は本物と重さが違いますが、本物より軽いか重いかは解っていません。
天秤を使って偽物のコインを見つけてください。
(おいらの)解答です。見ないでおくのも手。
めちゃくちゃ難しいわけでもないし、探せば見つけられるものなのでさっさと出しちゃっとく^^;。
1. 1回目
4,4,5 個のグループに分け、4,4 を天秤にかける。
○○○○ ●●●● ◎◎◎◎◎
 ̄ ̄ ̄ ̄U ̄ ̄ ̄ ̄
1-1. 1回目がつりあった場合
2回目
◎に偽者がある。3,2 に分けて、本物と 3 を天秤にかける。
○○○ ◎◎◎ ◎◎
 ̄ ̄ ̄U ̄ ̄ ̄
(○や●は本物であることが確定している)
1-1-1. 2回目がつりあった場合
3回目
残った 2個の◎に偽物がある。本物と偽物候補一つを天秤にかける。
○ ◎ ◎
 ̄U ̄
つりあったら量らなかった◎が偽物
つりあわなかったら天秤にかけた◎が偽物
1-1-2. 2回目がつりあわなかった場合
3回目
◎が上がったか下がったかにより、偽物が本物より重いか軽いかが確定している。
◎を 1,1,1 に分け、1,1 を天秤にかける。
◎ ◎ ◎
 ̄U ̄
つりあったら残った◎が偽物。
つりあわなかったら2回目の結果に応じて上がったほうまたは下がったほうが偽物。
1-2. 1回目がつりあわなかった場合
2回目
天秤にかけた8個の中に偽物がある。◎は本物。
以下のように天秤にかける。
○●● ○●● ○○
 ̄ ̄ ̄U ̄ ̄ ̄
(●は1回目で下がったほうとする)
1-2-1. 2回目がつりあった場合
3回目
残った 2個の○に偽物がある。本物と偽物候補一つを天秤にかける。
○ ◎ ○
 ̄U ̄
つりあったら残した○が偽物
つりあわなかったら天秤にかけた○が偽物
1-2-2. 2回目がつりあわなかった場合
3回目
●は1回目で下がったほうなので、2回目で下がったほうの●のいずれかか、上がったほうの○が偽物。
下がったほうの●同士を天秤にかける。
● ● ○
 ̄U ̄
つりあったら○が偽物
つりあわなかったら下がったほうの●が偽物
鍵は、最初のコインの数が5枚の場合は3回必要だけど、本物と分かっているものが別途存在する場合は2回でいけるというところですね。
1-1-1 と 1-1-2 でつりあったケースだけ、つまり一度も量らなかったコインが偽物だった場合だけ、偽物が重いか軽いかが不明のままです。それ以外は全て偽物が重いか軽いかも判明します。
1-2.では以下のように量ることもできます。
○●●● ●◎◎◎ ○○○
 ̄ ̄ ̄ ̄U ̄ ̄ ̄ ̄
●が重い(○が軽い)ことが既に判明しているので、この結果に応じて後一回量ればいいわけですね。
リンク元の ELLE さん(mixi 内)の方法だと、ここでこう量ります。
○○●●● ◎◎◎◎◎ ○○●
 ̄ ̄ ̄ ̄ ̄U ̄ ̄ ̄ ̄ ̄
なるほどなぁ。と、いうことは以下でもいいわけですね。
○○●●● ●◎◎◎◎ ○○◎
 ̄ ̄ ̄ ̄ ̄U ̄ ̄ ̄ ̄ ̄
○○●●◎ ●◎◎◎◎ ○○●
 ̄ ̄ ̄ ̄ ̄U ̄ ̄ ̄ ̄ ̄
また、1-1. では以下でもいけます。(○が本物)
◎○○ ◎◎○ ◎◎
 ̄ ̄ ̄U ̄ ̄ ̄
つりあわなかったら最後に右に乗せていた◎◎を量るわけです。2回目の結果に応じて求まります。
とまぁ考えましたが、14枚を3回で求める方法はわかんないです。さすがに無理ですかね。(証明されてたと思うから当たり前なんだけど)
実はこれ 1-2 の量り方が結構ミソで、確定本物◎を使わないで行くとおそらくこの量り方しかありません。
◎混じりの量り方は別解に記したとおり、何通りもありますね。
面倒なので、片方の天秤に一枚おき、もう片方の天秤のコインを一枚づつ交換。これなら、レジスタが1つだけで命令をパイプラインに順番に流し込むだけなので、高速に処理することができます(とか言ってみたり(^^;;)
「処理」してどうすんねんっ!!と言わねばなるまい:p。そいえば、最小サイクルで求めよって問題にしたらどうなるんだろう…。
ってかそういうのはありそうですねぇ。昔は命令ごとに使用クロックが違ってたからなんかいろいろ考えてたような気が…。今は命令セットすらおぼえる気になれないので全然だめですが…。
詳しい参考情報。
http://www.hi-ho.ne.jp/yoshik-y/mathematics/m035.html
一回目
○○○○○ ○○○○○ (○○○測らない)
二回目(パターン1:5個のほうに偽物)
○○ ○○ (○測らない)
二回目(パターン2:3個のほうに偽物)
○ ○ (○測らない)
三回目(パターン1の後)
○ ○
この方法じゃ答えにならないでしょうか?
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