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The problems that Frost's poetry poses for us as readers are not problems of reference. They can't be solved by footnotes.
他的诗歌给我们的困难,不是参考的问题，它们不能就用脚注解决。
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and how using your knowledge of math can help you solve something in biology.
如何用数学方面的知识解决生物方面的问题。
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So let me turn to a different approach to solving the question, answering the question,which one should we believe?
所以让我用另一个方法来解决这个问题，回答该相信哪一个理论的问题？
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So if I try to apply the same logic, well how can I divide and conquer this problem.
如果我想要用同样的逻辑，那么怎样划分,并解决这个问题呢？
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So when you're trying to think about is this a problem that I can solve with dynamic programming, these are the two questions you ask.
所以当你想知道能否,用动态编程来解决这个问题时,你可以问自己两个问题。
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Mine is to demonstrate over and over how little mathematics you need to get the job done.
我的方法就是,反复演算,用最少的数学来解决问题
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Partly because I think it's a good exercise, but also I don't want this class to be a class where you sit there, with your cup of Willoughby's coffee if needed to keep awake, and you watch me solve models because that's not how you learn.
一方面是由于我认为这是个好的锻炼机会,此外,我也不希望这个课程,变成你们坐在这,用威洛比咖啡来保持清醒,一边看我解决问题,因为你们不应那样学习
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The way we use technology to overcome that is we have a lot of simulations and visualizations set up, built around in the virtual spaces.
我们用技术来解决这个问题的方法,是我们用很多模拟实验,以及直观化实验，放在模拟空间里。
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You can have your own mnemonic device to figure that out, but anyway.
反正你可以用你自己的记忆方法,来解决这个问题。
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so that you can solve it in the best and most efficient manner.
这样便可以用最好，最有效的方式解决问题。
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And basically you solve biological problems using a computer and yeah.
主要是用计算机来解决生物问题。
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So things that are good candidates for divide And conquer are problems where it's easy to figure out how to divide down, and the combination is of little complexity.
因为适合用分治算法解决的问题,最好是能够简单的将问题进行分解,并且合并的过程不是非常的复杂,只要比线性方案要小。
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If we make--If we solve the problem of boredom setting in with progressive memory loss and radical personality changes, maybe boredom won't set in, but it's not anything that I especially want for myself.
如果用持续失忆,和剧烈人格变化来解决无聊的问题，也许不会产生无聊，但那不是我特别想做的事。
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If you'd like to sit down with one of us more intimately and more sort of tranquilly, and work through problems for more than just a couple minutes at a time.
如果你希望和我们其中一个人坐下来，更亲密的,更平静的，每次用几分钟的时间,来解决问题。
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All right. So it's fun, and I got a little bit of applause out of it, which is always nice for me, but I also showed you how to think about it recursively.
好，挺有趣的，我也得到了一点掌声，这让我挺高兴的，但是我想让大家明白,怎么用迭代的思想来解决这个问题。
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Next time down I've got eight versions.
我可以用替代来解决这个问题。
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t In the b even case, again I'm going to let t of b be the number of steps I want to go through.
如果b为偶数，那么我还是要用,来代表解决这个问题需要的步骤数。
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At the end of the day this is all about equipping you with skills and knowledge and understanding with which to take a problem, take a large data set maybe and actually produce something useful-- produce a project that is of interest to you.
下午放学后,就用学到的技能，知识和理解力来,解决问题，或者处理超大数据集,或制造一些有用的东西-,例如发起一个你们感兴趣的项目。
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It's probably among the most useful ideas there is for solving complicated problems.
这大概是解决复杂问题,最有用的思想之一。
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So no harm is done, once we've clarified the question that we're trying to ask, if we summarize that question in a bit of a jargon or slogan.
其实没什么的,一但我们证实了我们,试图想要解决的问题,如果把这个问题用术语或行话简化
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The point is the arrow A, somebody has chosen to write in terms of i prime and j prime as Ax prime and Ay prime.
解决问题的关键在于矢量 A,可以用这样的形式来描述,i' ? Ax' + j' ? Ay'
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And I'm going to do it only using multiplication and addition and some simple tests. All right?
我只能用乘法和加法,以及一些小测试来解决这个问题？
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- And actually, it's worth noting an int -- 4 billion -- we didn't even solve that problem.
事实上，int数据类型还是没什么用--40亿-,我们还是没有解决那个问题。
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Once you hear that description, it's easy to write the code, in fact. This is a place where the recursive version of it is much easier to think about than the iterative one.
实际上一旦你听到了大致描述,就能很轻松的写出代码来,在这一点上应用递归来解决问题,比用迭代要容易理解多了。
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And I would suggest palindrome as a great example of that. That's easy to think about recursively. It's much harder to think about iteratively. And you want to get into the habit of deciding which is the right one for you to use. And with that, we'll see you next time.
其它的问题可能用递归的方式,可以更好的解决,而直接思考可能很困难,这时你就得养成做出,正确选择的好习惯了,下周再见。
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Typically up till now, we've looked at things that can be done in sublinear time. Or, at worst, polynomial time. We'll now look at a problem that does not fall into that. And we'll start with what's called the continuous knapsack problem.
至今为止我们已经处理过,亚线性问题，最多也就是多项式问题,我们现在要看的问题则是不能用这些解决的,我们将要开始讲连续背包问题。
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So that you will come away with a sense of what you can do, what you can't do, and what kinds of things you should use to tackle complex problems.
因此你们才能对,你们能做的事儿，不能做的事儿,和用何方法去,解决棘手问题有一个基本概念。
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Not so easy to see. All right, but this is actually a great one of those educational moments. This is a great example to think recursively. If I wanted to think about this problem recursively- what do I mean by thinking recursively?
看不太出来，好，但实际上是一个有教育意义的时刻,这是一个很好的关于递归的例子,如果我用递归的思想,去考虑这个问题-,我该怎么用递归去解决这个问题呢？
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So let's actually see how we solve this with this function with actual code.
然我们看看怎样用,这个函数和代码解决这个问题。
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System of linear equations. What are the equations here? Well, I could say, you know, the number of pigs plus the number of 20 chickens equals 20, right? Because we've got 20 heads. And then what else do I have?
如何解决这个问题呢？,用线性方程式的办法来解决，等式是什么呢？你应该知道,猪的数量加鸡的数量等于,对吧？因为我们有20个头？
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