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August 13, 2005

几道迷题

在Fayolle家住的三个星期里头，Lucie时不时给我出迷题，幸运的是，我都解出了，她说一定要找出一个能难倒我的。其实最难的部分是让她把题干解释清楚，因为我的法语词汇量很小。我这里写出几个我还记得的迷题：

1。一个大房子里头有两个房间，房间A里头有三个开关，其中一个控制着房间B的电灯，你要找出是哪一个，但是现在你在房间A，看不到房间B的任何变化。问题是，你该怎么做，只需去一次房间B就可以知道是哪个开关控制着房间B的电灯？

2。下一个符号是什么？

3。据改编，抗日战争期间，在一个日本监狱里头关了100个中国士兵，一天他们被告知，第二天要对他们进行处决，但是会以一个游戏的方式：他们排成一列，面向同一个列向，给每人头上戴一顶帽子，红色，蓝色，或者白色，这样，他们都只能看到前面所有人的帽子颜色，看不到自己的，也看不到后面的，每一个士兵会被问到他帽子的颜色，如果他猜错，他就会被立即杀掉，从队伍的最后一个士兵开始问起，依次向前问过去，士兵的回答，其他的士兵都可以听到。好了，这个夜里，这100个士兵就聚在一起商量对策，结果智慧勇敢的中国士兵就想出了一个好办法，只牺牲了一位战士。请问他们的对策是怎样的？（原题是问怎么做可以使得伤亡最小）（因为有同学问我，所以我补充一句，士兵当然只能说一个颜色，不能说其他废话）(事实上，这位牺牲的战士仍然有1/3的存活几率，只是这个策略可以保证99个战士都能活下来，至少在这个游戏里头。）

4。法国人发明了断头台这样的刑罚（据Lucie说）。现在有一个囚犯，他面前有两扇门，一扇通往自由，另一扇通往断头台，每一扇门前面有一个守卫，这两个守卫里头，一个只说真话，一个只说假话，囚犯不知道哪扇门通往自由，哪个守卫是说真话那个，但是两个守卫都知道，现在这个囚犯只能对其中一人问一个问题。请问，这个囚犯问什么问题，可以获得自由？

5。一个寺庙里的和尚每天白天聚到一个房间里念经。每年固定的某一天，总会有一些和尚得一种病，叫做Diablo Rouge(红色魔鬼），得病后，额头上会立即出现一个明显的红点，一旦知道自己得病，晚上回到房间，他们会立即自杀，这些和尚只能看到别人额头上是否有红点，但是他们又不可以互相交流。在那天以后的第七天，这些得病的和尚都猜出自己得病，然后都自杀了，请问得病和尚的数量是多少？

Updated at August 20, 2005 08:00 AM：提示
Updated at August 22, 2005 12:39 AM：答案

提示：
1。电灯是既可以发光，也可以发热的。
2。跟数学关系比较密切。
3。最后一个战士牺牲自己，给大家发出了一个信号，用来形容前面99个战士的一个整体状态。而前面这99个战士既可以听到所有后面战士的声音，可以看到前面战士的帽子颜色，所以他们可以根据这个信号，推测自己的帽子颜色。而这个描述99个颜色的信号，归根到底是个余数问题。
4。想办法消除两个守卫之间的差异。
5。和尚是很聪明的，他们可以想别人是怎么想的。还有跟数学归纳法有点关系。假设有一个和尚得病，他看到其他的所有和尚都是没有红点的，他就知道自己得病了，因为根据题干，第一天是必定有和尚得病的。接着，再假设有两个和尚得病...

答案：

1。先打开开关1，等一段时间，再关上，打开开关2，马上跑到房间B，看灯泡是否亮，摸灯泡是否烫手。不多解释了。
2。这5个符号分别是数字1,2,3,4,5跟其镜象的组合。所以下一个就是...
3。假设红色＝0，蓝色＝1，白色＝2，最后那位战士A可以看到前面所有99顶帽子，如果，这99顶帽子颜色的和，除以3，得到的余数为0，那么战士A就说红色，以此类推其他两个余数。前面99位战士中任何一个可以凭这样的一个信号推断自己的帽子颜色，因为他可以听到所有后面战士说的颜色，也可以看到所有前面战士的帽子颜色，他的帽子是蓝，是红，还是白，影响着总数（99个颜色之和）的特征（所谓余数）。
4。对任何一个守卫这样问：“哪扇是自由之门，请问他会怎样回答这个问题？”
5。答案是7个和尚在第七天晚上同时自杀。
命题：得病和尚数为n，则得病和尚在第n天晚上自杀。
论证：
设，x个和尚得病，则他们在第f(x)天晚上自杀。
分命题1：x=1，则f(x)=1。
论证1：得病和尚书为1，则他在第一天白天看到所有其他和尚正常，于是他推断得病的是自己，于是在第1天晚上自杀。
分命题2：f(x+1)=f(x)+1。
论证2：得病和尚数为x+1，则任意分析其中一个得病和尚，我们叫他A，他看到x个得病和尚，他假设自己没有得病，那么得病和尚数为x，则他们应该在第f(x)天晚上自杀，于是他等待到f(x)+1天的白天，但是他又看到这x个和尚活着，（为什么？因为他们的情况跟A一样，同样在等），这个时候，A知道自己的假设是错误的，那么就是说自己是得病的。当然，其他的得病和尚跟A一样聪明，他们也猜到了，于是都在晚上自杀了。
根据分命题1，分命题2，数学归纳得f(x)=x，x属于自然数，排除一年365天这些复杂因素。

Posted by 西瓜 at August 13, 2005 03:25 PM

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