本周问题：
三个数学家被囚禁，并被要求玩一个游戏：他们每个人的额头上都会被涂上一个绿色或蓝色的斑点，每种颜色的概率是50%，相互独立。他们必须同时猜测自己斑点的颜色，或者保持沉默。如果没有人猜测，或者有人猜错，他们就会输掉游戏。
问题：
他们如何制定计划，以确保50%的获胜概率？ 他们如何将获胜概率提高到75%？ 狱卒偷听了数学家的讨论并修改了规则：现在斑点的颜色由狱卒选择（而不是随机分配）。如果每个囚犯都可以抛一次公平硬币，他们能否利用硬币制定修改后的策略，使成功率达到75%？（数学家可以看到所有三枚硬币的结果，但狱卒看不到任何硬币结果。）
(来源：New Scientist，BrainTwister #41)
原文：
Three mathematicians are held prisoner and made to play a game in which they each have a blob painted on their forehead in green or blue, with an independent 50 per cent chance of either colour for each prisoner. They must all simultaneously guess their own blob colour or say nothing. They lose the game if nobody guesses or if someone guesses wrong.
How can they plan to play so they guarantee a 50 per cent chance of winning?
How can they increase their chances of winning to 75 per cent?
The prison guards overhear the mathematicians plotting. To thwart the prisoners’ strategy, the guards change the rules so they now choose the blob colours. If each prisoner has a fair coin they can flip once, can they use them to come up with a modified strategy with a 75 per cent success rate? The prisoners can see all three coins, but the guards can’t see any of them.
上期问题：四灯两按钮
一台机器上有四个指示灯，分别标记为 A 到 D，任意时刻只有一个灯亮 。它还有一个红色按钮和一个蓝色按钮。按下按钮会根据箭头所示，改变哪个灯亮着。（从某个字母指向其自身的箭头表示按下按钮时灯不会改变。）
如果 D 灯亮着，有多少种不同的按三次按钮的方式可以使 A 灯亮？ 如果最后按下的是蓝色按钮，哪个灯不可能亮着？ 如果看不到灯光，你可以按下哪三个按钮的组合来确保最终亮的是 A 或 D？ 按下哪五个按钮的顺序可以保证最终亮的是 A 灯？
答案：
从灯D到灯A有四种路径：蓝-蓝-红，蓝-红-蓝，红-蓝-蓝，以及红-红-蓝。
按下蓝色后无法直接点亮灯B，因为没有蓝色箭头指向它。
按下“蓝-红-蓝”可以确保灯A或灯D被点亮。第一次按蓝色可以点亮灯A、灯C或灯D。接着按下红色会导致灯D、灯A或灯B被点亮，而最后的蓝色不会改变灯D或灯A的状态，但会将灯B切换为灯A。
按下“蓝-红-蓝-红-蓝”可以确保灯A被点亮。我们知道在按下“蓝-红-蓝”后，灯A或灯D会被点亮。接着按下红色会使点亮状态变为灯A或灯B，最后一次按下蓝色会将灯B切换为灯A，或保持灯A的状态不变。
原文
There are four ways to get from light D to light A: blue-blue-red, blue-red-blue, red-blue-blue and red-red-blue.
Light B cannot be lit directly after blue is pressed, as no blue arrow points to it.
Pressing blue-red-blue guarantees that either A or D is lit. Pressing blue first ensures that A, C or D is lit. Pressing red then causes D, A or B to be lit, and the final blue doesn’t change D or A, but will turn B to A.
Pressing blue-red-blue-red-blue guarantees that A is lit. We know after blue-red-blue that A or D is lit. Pressing red makes that A or B, then the final blue turns B to A or leaves A unchanged.
