You're the realm's greatest mathematician, but ever since you criticized the Emperor's tax laws,
你是王国里最伟大的数学家，但由于讽刺国王的税务法，
you've been locked in the dungeon with only a marker to count the days.
你被关进了大牢以笔划线记日。
But one day, you're suddenly brought before the Emperor who looks even angrier than usual.
但一天，你突然被带到国王面前，而国王看起来比往常更恼火。
One of his twelve governors has been convicted of paying his taxes with a counterfeit coin
他的十二个官员中的一个用了假金币来付了他的税，
which has already made its way into the treasury.
这枚加金币已经进到了宝库里。
As the kingdom's greatest mathematician, you've been granted a chance to earn your freedom by identifying the fake.
作为王国里最伟大的数学家，你只要能识别出假金币，就能重获自由。
Before you are the twelve identical looking coins and a balance scale.
在你面前是十二个看起来完全相同的金币和一个天平。
You know that the false coin will be very slightly lighter or heavier than the rest.
你知道假金币比其他的要重或轻一点点。
But the Emperor's not a patient man.
但国王没什么耐心。
You may only use the scale three times before you'll be thrown back into the dungeon.
你只能用天平测量三次，然后就会被关回大牢。
You look around for anything else you can use,
你向周围寻找可以用的工具，
but there's nothing in the room -- just the coins, the scale, and your trusty marker.
但屋子里什么也没有--只有金币、天平和你的笔。
How do you identify the counterfeit?
你该怎样找出假金币呢？
Obviously you can't weigh each coin against all of the others,
显然你不能把每个金币都和其他的比一次，
so you'll have to weigh several coins at the same time by splitting the stack into multiple piles
所以你必须每次把金币分成几组，几个金币一起称重，
then narrowing down where the false coin is.
缩小范围直到找到假金币。
Start by dividing the twelve coins into three equal piles of four.
先把十二枚金币分成三堆四枚的。
Placing two of these on the scale gives us two possible outcomes.
在天平上放两堆，有两种可能的情况。
If the two sides balance, all eight coins on the scale are real, and the fake must be among the remaining four.
如果天平两边平衡，则这八枚金币都是真的，假的一定在另外四枚里。
So how do you keep track of these results? That's where the marker comes in.
你怎么记录这些结论呢？就用你的笔。
Mark the eight authentic coins with a zero.
把八枚真金币标上圆圈。
Now, take three of them and weigh them against three unmarked coins.
取三枚真金币和三枚没有标记的金币称重。
If they balance, the remaining unmarked coin must be the fake.
如果平衡，那么剩下的那枚没有标记的金币一定是假的。

If they don't, draw a plus on the three unmarked coins if they're heavier or a minus if they're lighter.
如果不平衡而且三枚没有标记的金币比较重，就标上加号，如果它们比较轻，就标上减号。
Now, take two of the newly marked coins and weigh them against each other.
现在把刚称过的三枚金币中的其中两枚在天平上称重。
If they balance, the third coin is fake. Otherwise, look at their marks.
如果天平平衡，第三枚就是假的。否则，看这两枚金币的标记。
If they are plus coins, the heavier one is the imposter.
如果是加号，那么较重的那枚是假的。
If they are marked with minus, it's the lighter one.
如果是减号，那么较轻的那枚是假的。
But what if the first two piles you weigh don't balance?
但如果刚开始的两堆金币不平衡呢？
Mark the coins on the heavier side with a plus and those on the lighter side with a minus.
把较重一边的金币标上加号，较轻一边的金币标上减号。
You can also mark the remaining four coins with zeros since you know the fake one is already somewhere on the scale.
其他的四枚画圈，因为假金币一定在称过的八枚里。
Now, you'll need to think strategically so you can remove all remaining ambiguity in just two more weighings.
现在好好计划一下，才能在再称两次的情况下找出答案。
To do this, you'll need to reassemble the piles.
你需要重新组合这堆金币。
One method is to replace three of the plus coins with three of the minus coins,
一种方法是把三枚加号金币换成减号金币，
and replace those with three of the zero coins.
再把那三枚减号金币换成画圈的金币。
From here, you have three possibilities.
这时有三种情况。
If the previously heavier side of the scale is still heavier,
如果先前比较重的一边仍然较重，
that means either the remaining plus coin on that side is actually the heavier one,
那么要么是剩下的加号金币比较重，
or the remaining minus coin on the lighter side is actually the lighter one.
要么轻的那边的减号金币比较轻。
Choose either one of them, and weigh it against one of the regular coins to see which is true.
随便选一个，和画圈的金币比较来找到答案。
If the previously heavier side became lighter, that means one of the three minus coins you moved is actually the lighter one.
如果先前较重的一边变轻了，意味着被移动的三枚减号金币里有一枚比较轻。
Weigh two of them against each other.
把其中两枚相比即可。
If they balance, the third is counterfeit. If not, the lighter one is.
如果平衡，第三枚就是假的。如果不平衡，轻的那枚就是假的。
Similarly, if the two sides balanced after your substitution,
相似地，如果两边在替换后平衡，
then one of the three plus coins you removed must be the heavier one.
那么你移走的三枚加号金币里有一枚比较重。
Weigh two of them against each other.
把其中两枚相比。
If they balance, the third one is fake. If not, then it's the heavier one.
如果平衡，第三枚就是假的。如果不平衡，重的那枚就是假的。
The Emperor nods approvingly at your finding, and the counterfeiting Lord takes your place in the dungeon.
国王点头对你的智商表示赞许，而那名交了假金币的官员代替了你在地牢里的位置。