这座城市刚落成一座独一无二的法贝热彩蛋博物馆
The city has just opened its one-of-a-kind Fabergé Egg Museum
这座100层的大楼 每层都展出1个蛋
with a single egg displayed on each floor of a 100-story building.
世界上最臭名昭著的珠宝大盗已经盯上了这些蛋
And the world’s most notorious jewel thief already has her eyes on the prize.
因为安防很严 蛋又很大
Because security is tight and the eggs are so large,
她只有机会 通过把彩蛋从楼上丢下
she’ll only get the chance to steal one
落到准备好的卡车里来偷一个
by dropping it out the window into her waiting truck
并在警察赶到之前撤退
and repelling down before the police can arrive.
所有蛋的重量和构造都一样
All eggs are identical in weight and construction,
但每层的蛋都比下层的更稀有和贵重
but each floor’s egg is more rare and valuable than the one below it.
大盗自然想要顶层最贵重的蛋
While the thief would naturally like to take the priceless egg at the top,
但她也觉得彩蛋从100楼掉下来会摔碎
she suspects it won’t survive a 100-story drop.
稳妥起见 她决定偷自己能拿到的最贵的蛋
Being pragmatic, she decides to settle for the most expensive egg she can get.
在博物馆礼品店里 她找到了2枚纪念品蛋
In the museum’s gift shop, she finds two souvenir eggs,
它们是真蛋的完美复制品 但完全不值钱
perfect replicas that are perfectly worthless.
计划是 把这2枚蛋扔下来
The plan is to test drop them
以此试出蛋掉下来不会摔碎的
to find the highest floor at which an egg will survive the fall
最高楼层
without breaking.
当然 实验只能重复到
Of course, the experiment can only be repeated
两个复制品蛋都摔碎
until both replica eggs are smashed.
而且从窗户扔纪念品太多次
And throwing souvenirs out the window too many times
可能也会引起守卫的注意
is probably going to draw the guards’ attention.
为了保证她找到正确的楼层
What’s the least number of tries it would take
最少需要试几次呢？
to guarantee that she find the right floor?
如果你想自己找出答案 那就先别往下看
Pause here if you want to figure it out for yourself!
3秒后揭晓答案
Answer in: 3
2秒后揭晓答案
Answer in: 2
1秒后揭晓答案
Answer in: 1
如果你对解决这个问题没有头绪
If you’re having trouble getting started on the solution,
从一个简单的思路开始可能更有帮助
it might help to start with a simpler scenario.
想象大盗只有1个复制品蛋
Imagine our thief only had one replica egg.
那么她只有1个选项：
She’d have a single option:
从1楼开始扔蛋
To start by dropping it from the first floor
一层一层楼往上 直到蛋摔碎
and go up one by one until it breaks.
那么 她就会知道 这层之下
Then she’d know that the floor below that
就是她最终要偷蛋的楼层
is the one she needs to target for the real heist.
但这样可能要试100次
But this could require as many as 100 tries.
拥有另一个复制品蛋让大盗有了更好的选择
Having an additional replica egg gives the thief a better option.
她可以以更大的楼层跨度试着丢下第1个蛋
She can drop the first egg from different floors at larger intervals
以缩小找到临界楼层的范围
in order to narrow down the range where the critical floor can be found.
一旦第1个蛋碎了
And once the first breaks,
她可以用第2个蛋 在第1个蛋试过的两层楼之间再进行尝试
she can use the second egg to explore that interval floor by floor.
楼层跨度大并不特别有效
Large floor intervals don’t work great.
最差的情况是第2个蛋也要试很多次
In the worst case scenario, they require many tests with the second egg.
楼层跨度小点会更奏效
Smaller intervals work much better.
比如 如果她以10楼为跨度试着扔下第1个蛋
For example, if she starts by dropping the first egg from every 10th floor,
一旦蛋碎了 她只要试下面的9层
once it breaks, she’ll only have to test the nine floors below.
这意味着要找到正确的楼层最多需要试19次
That means it’ll take at most 19 tries to find the right floor.
有更好的办法吗？
But can she do even better?
毕竟每次的楼层跨度并不需要一致
After all, there’s no reason every interval has to be the same size.
假设一共只有10层楼
Let’s say there were only ten floors.
如果大盗把第1个蛋
The thief could test this whole building with just four total throws
从4层 7层 9层 扔下来
by dropping the first egg at floors four, seven,
一共只要尝试4次
and nine.
如果彩蛋从4楼扔下就碎了 那第2个蛋只需试3次
If it broke at floor four, it would take up to three throws of the second egg
就能找到要偷的楼层
to find the exact floor.
如果蛋在7楼扔下碎了
If it broke at seven,
第2个蛋只要再扔2次
it would take up to two throws with the second egg.
如果第9楼扔下碎了
And if it broke at floor nine,
第2个蛋只要再试1次
it would take just one more throw of the second egg.
直观来说 在这里我们试的是把整栋楼分成不同部分
Intuitively, what we’re trying to do here is divide the building into sections
这样无论哪一层是正确的
where no matter which floor is correct,
都只需要扔相同的次数以找到对的楼层
it takes up to the same number of throws to find it.
我们想让每个间距比上一个少1层楼
We want each interval to be one floor smaller than the last.
这个等式可以帮我们找到我们要从100层大楼的
This equation can help us solve for the first floor we need to start with
哪一层开始
in the 100 floor building.
解这个等式有很多方法
There are several ways to solve this equation,
包括试错法
including trial and error.
如果我们让n等于2 等式会变成这样
If we plug in two for n, that equation would look like this.
如果n等于3 会得到这个式子
If we plug in three, we get this.
因此我们可以通过增加求和项
So we can find the first n to pass 100
找到第1个让等式大于100的n
by adding more terms until we get to our answer,
这个n等于14
which is 14.
所以大盗应该从第14层开始试
And so our thief starts on the 14th floor,
接下来到27层 之后39层 按这个规律继续向上
moving up to the 27th, the 39th, and so on,
一共需要尝试丢14次蛋
for a maximum of 14 drops.
就像俗语说的 舍不得孩子套不着狼啊
Like the old saying goes, you can’t pull a heist without breaking a few eggs.