WEBVTT 00:00:00.000 --> 00:00:03.727 Hi. In this lecture we’re talking about problem solving. 00:00:03.727 --> 00:00:08.958 And we’re talking about the role that diverse perspectives play in finding solutions to problems. 00:00:08.958 --> 00:00:10.561 So when you think about a problem, 00:00:10.561 --> 00:00:12.688 perspective is how you represent it. 00:00:12.688 --> 00:00:16.636 So remember from the previous lecture, we talked about landscapes. 00:00:16.636 --> 00:00:19.011 We talked about landscape being a way to represent 00:00:19.011 --> 00:00:21.877 the solutions along this axis 00:00:21.877 --> 00:00:26.571 and the value of the solutions as the height. 00:00:26.571 --> 00:00:29.737 And so this is metaphorically a way to represent 00:00:29.737 --> 00:00:33.110 how someone might think about solving a problem: 00:00:33.125 --> 00:00:36.751 Finding high points on their landscape. 00:00:36.751 --> 00:00:39.800 What we want to do is take this metaphor and formalize it 00:00:39.800 --> 00:00:43.066 and part of the reason for this course is to get better logic, 00:00:43.066 --> 00:00:45.248 [in order to] think through things in a clear way. 00:00:45.248 --> 00:00:49.403 So I’m going to take this landscape metaphor and turn it into a formal model. 00:00:49.403 --> 00:00:50.499 So how do we do it? 00:00:50.499 --> 00:00:54.774 The first thing we do is we formally define what a perspective is. 00:00:54.774 --> 00:00:56.724 So we speak math to metaphor. 00:00:56.724 --> 00:00:59.022 So what a perspective is going to be is 00:00:59.022 --> 00:01:01.498 it’s going to be a representation of all possible solutions. 00:01:01.498 --> 00:01:05.389 So it’s some encoding of the set of possible solutions to the problem. 00:01:05.389 --> 00:01:08.870 Once we have that encoding of the set of possible solutions, 00:01:08.870 --> 00:01:13.399 then we can create our landscape by just assigning a value to each one of those solutions. 00:01:13.399 --> 00:01:16.358 And that will give us a landscape picture like you saw before. 00:01:16.358 --> 00:01:19.806 Now most of us are familiar with perspectives, 00:01:19.806 --> 00:01:21.409 even though we don’t know it. 00:01:21.409 --> 00:01:22.567 Let me give some examples. 00:01:22.567 --> 00:01:24.534 Remember when we took seventh grade math? 00:01:24.534 --> 00:01:27.675 We learned about how to represent a point, how to plot points. 00:01:27.675 --> 00:01:29.884 And we typically learned two ways to do it. 00:01:29.884 --> 00:01:32.547 The first way was Cartesian coordinates. 00:01:32.547 --> 00:01:34.774 So given a point, we would represent it 00:01:34.774 --> 00:01:38.755 by and an X and a Y value in space. 00:01:38.755 --> 00:01:40.379 So, it might be five units, 00:01:40.379 --> 00:01:42.371 this would be the point, let’s say (5, 2). 00:01:42.371 --> 00:01:45.894 It’s five units in the X direction, two units in the Y direction. 00:01:45.894 --> 00:01:48.715 But we also learned another way to represent points, 00:01:48.715 --> 00:01:50.709 and that was [polar] coordinates. 00:01:50.709 --> 00:01:52.432 So we can take the same point and say, 00:01:52.432 --> 00:01:54.936 there’s a radius, which is its distance from the origin, 00:01:54.936 --> 00:01:56.652 and then there’s some angle theta, 00:01:56.652 --> 00:01:58.496 which says how far we have to sweep out, 00:01:58.496 --> 00:02:02.714 in order to sweep that radius out in order to get to the point. 00:02:02.714 --> 00:02:05.585 So two totally reasonable ways to represent a point: 00:02:05.585 --> 00:02:07.652 X and Y, R and theta. 00:02:07.652 --> 00:02:09.688 Cartesian, polar. 00:02:09.688 --> 00:02:11.360 Which is better? 00:02:11.360 --> 00:02:12.814 Well, the answer? It depends. 00:02:12.814 --> 00:02:14.088 Let me show you why. 00:02:14.088 --> 00:02:16.007 Suppose I wanted to describe this line. 00:02:16.007 --> 00:02:19.542 In order to describe this line I should use Cartesian coordinates, 00:02:19.542 --> 00:02:23.632 ’cause I can just say Y=3 and X moves from two to five. 00:02:23.632 --> 00:02:25.061 It’s really easy. 00:02:25.061 --> 00:02:28.632 But suppose I wanna describe this arc. 00:02:28.632 --> 00:02:29.968 If I wanna describe this arc, 00:02:29.968 --> 00:02:32.728 now Cartesian coordinates are gonna be fairly complicated, 00:02:32.728 --> 00:02:34.864 and I’d be better off using polar coordinates, 00:02:34.864 --> 00:02:35.835 because the radius is fixed 00:02:35.835 --> 00:02:38.713 and I just talked about how the radius is—you know, 00:02:38.713 --> 00:02:39.738 there’s this distance R, 00:02:39.738 --> 00:02:42.585 and theta just moves from, you know, A to B, let’s say. 00:02:42.585 --> 00:02:44.656 So depending on what I want to do. 00:02:44.656 --> 00:02:47.033 If I want to look at straight lines, I should use Cartesian. 00:02:47.033 --> 00:02:50.151 And if I want to look at arcs, I should probably use polar. 00:02:50.151 --> 00:02:52.384 So, perspectives depend on the problem. 00:02:52.384 --> 00:02:54.949 Now let’s think about where we want to go. 00:02:54.949 --> 00:02:58.683 We want to talk about how perspectives help us find solutions to problems 00:02:58.683 --> 00:03:01.732 and how perspectives help us be innovative. 00:03:01.732 --> 00:03:04.472 Well, if you look at the history of science a lot of great breakthroughs— 00:03:04.472 --> 00:03:06.288 you know, we think about Newton, 00:03:06.288 --> 00:03:07.801 you know, his theory of gravity— 00:03:07.801 --> 00:03:11.485 you can think about people actually having new perspectives on old problems. 00:03:11.485 --> 00:03:13.296 Let’s take an example. 00:03:13.296 --> 00:03:16.968 So, Mendeleev came up with the periodic table, 00:03:16.968 --> 00:03:20.409 and in the periodic table he represents the elements by atomic weight. 00:03:20.409 --> 00:03:22.440 He’s got them in these different columns. 00:03:22.440 --> 00:03:26.114 In doing so, by organizing the elements by atomic weight 00:03:26.114 --> 00:03:27.784 he found all sorts of structure. 00:03:27.784 --> 00:03:30.936 So all the metals line one column, stuff like that. 00:03:30.936 --> 00:03:33.002 Remember—from high school chemistry class. 00:03:33.002 --> 00:03:36.936 That’s a perspective: It’s a representation of a set of possible elements. 00:03:36.936 --> 00:03:39.072 He could’ve organized them alphabetically. 00:03:39.072 --> 00:03:41.100 But that wouldn’t have made much sense. 00:03:41.100 --> 00:03:44.679 So alphabetic representation wouldn’t give us any structure. 00:03:44.679 --> 00:03:47.425 Atomic weight representation gives us a lot of structure. 00:03:47.425 --> 00:03:50.859 In fact, when Mendeleev wrote down 00:03:50.859 --> 00:03:53.862 all the elements that were around at the time according to atomic weight, 00:03:53.862 --> 00:03:56.644 there were gaps in his representation. 00:03:56.644 --> 00:03:59.155 There were holes for elements that were missing. 00:03:59.155 --> 00:04:02.230 Those elements became scandium, gallium, and germanium. 00:04:02.230 --> 00:04:04.568 They were eventually found ten to fifteen years later, 00:04:04.568 --> 00:04:06.306 after he’d written down the periodic table: 00:04:06.306 --> 00:04:08.944 People went out and were able to find the missing elements. 00:04:08.944 --> 00:04:11.056 That perspective, atomic weight, 00:04:11.056 --> 00:04:16.056 ended up being a very useful way to organize our thinking about the elements. 00:04:17.148 --> 00:04:19.314 We do it all the time now. 00:04:19.314 --> 00:04:20.912 When you have any sort of task, 00:04:20.912 --> 00:04:23.671 you’ll find that you’re actually using some sort of perspective. 00:04:23.671 --> 00:04:25.502 Suppose that you’re hiring someone. 00:04:25.502 --> 00:04:28.348 And you’ve got a bunch of recent college graduates who apply for a job. 00:04:28.348 --> 00:04:29.520 And you’ve gotta think, 00:04:29.520 --> 00:04:32.004 “Okay, how do I organize all these applicants?” 00:04:32.004 --> 00:04:33.752 Let’s say 500 applicants. 00:04:33.752 --> 00:04:36.847 One thing you could do is you could organize them by GPA: 00:04:36.847 --> 00:04:39.604 Take the highest GPA down to the lowest GPA. 00:04:39.604 --> 00:04:40.781 That’s be one representation. 00:04:40.781 --> 00:04:44.679 And you might do that if you valued competence or achievement. 00:04:44.679 --> 00:04:47.520 But you might also value work ethic. 00:04:47.520 --> 00:04:49.296 And if that were the case you might instead organize 00:04:49.296 --> 00:04:53.248 those same CV’s or application files by how thick they are. 00:04:53.248 --> 00:04:56.361 [Those who’re going to do the] really thick ones are people who work really, really hard. 00:04:56.361 --> 00:04:57.560 They’ve accomplished a lot. 00:04:57.560 --> 00:05:00.607 Well, the third thing you might do is you might value creativity. 00:05:00.607 --> 00:05:01.601 And you might say, 00:05:01.601 --> 00:05:05.211 “Well, let’s put the ones that are sort of most colorful, most interesting over here. 00:05:05.211 --> 00:05:08.120 And the ones that are least colorful and least interesting over here.” 00:05:08.120 --> 00:05:09.760 That’s the third way to do it. 00:05:09.760 --> 00:05:11.609 Now depending on what you’re hiring for, 00:05:11.609 --> 00:05:12.900 depending on who the applicants are, 00:05:12.900 --> 00:05:14.720 any one of these might be fine. 00:05:14.720 --> 00:05:20.033 The only point I’m trying to make here is that there’s different ways to organize these applicants. 00:05:20.033 --> 00:05:21.968 In each one of those ways you organize— 00:05:21.968 --> 00:05:22.993 whether it’s in your head, 00:05:22.993 --> 00:05:25.512 or whether it’s formally laying them out in some way— 00:05:25.512 --> 00:05:27.176 is a perspective. 00:05:27.176 --> 00:05:31.944 And those perspectives will determine how hard the problem will be for you. 00:05:31.944 --> 00:05:33.420 Let me explain why. 00:05:33.420 --> 00:05:36.432 Now I want to go back to the landscape metaphor. 00:05:36.432 --> 00:05:38.424 And when I think of that landscape as being rugged, 00:05:38.424 --> 00:05:42.984 and by rugged I mean that it doesn’t look like a single peak, 00:05:42.984 --> 00:05:45.009 that there’s lots of peaks on it. 00:05:45.009 --> 00:05:48.448 And I want to formalize this notion of peaks. 00:05:48.448 --> 00:05:50.315 And I do so as follows: 00:05:50.315 --> 00:05:52.568 I’m going to define what I call a local optima. 00:05:52.568 --> 00:05:55.808 A local optima is a point such that 00:05:55.808 --> 00:05:57.784 if you look at the points on either side of it, 00:05:57.784 --> 00:05:59.125 they’re lower in value. 00:05:59.125 --> 00:06:02.406 So it’s sort of a point that locally is the highest possible value. 00:06:02.406 --> 00:06:04.807 So if I look at this particular rugged landscape again, 00:06:04.807 --> 00:06:07.369 there’s three local optima: 1, 2, 3. 00:06:07.369 --> 00:06:10.351 At any one of these three points, I’d be stuck: 00:06:10.351 --> 00:06:12.683 If I looked to the left or to the right, 00:06:12.683 --> 00:06:14.843 I wouldn’t find a solution that’s better. 00:06:14.843 --> 00:06:18.934 So we think about what makes a good perspective: 00:06:18.934 --> 00:06:23.702 A good perspective is going to be a perspective that doesn’t have many local optima. 00:06:23.702 --> 00:06:27.583 A bad perspective is going to be one that has a lot of local optima. 00:06:27.583 --> 00:06:29.397 Let me give you an example, okay? 00:06:29.397 --> 00:06:31.090 So, suppose I’m coming up with a candy bar. 00:06:31.090 --> 00:06:33.495 Suppose I’m tasked with coming up with a new candy bar. 00:06:33.495 --> 00:06:39.376 So I have my team of chefs make a whole bunch of different confections for me to try, 00:06:39.376 --> 00:06:41.121 and I want to find the very best one. 00:06:41.121 --> 00:06:43.712 But there’re so many of them, there’s so many possibilities, 00:06:43.712 --> 00:06:45.322 that I’m not even sure how to think about it. 00:06:45.322 --> 00:06:49.145 But one way to represent those candy bars might be by the number of calories that they had. 00:06:49.145 --> 00:06:53.096 So I can organize all the different things they make by number of calories. 00:06:53.096 --> 00:06:55.890 And if I did that, maybe I’d have three local optima. 00:06:55.890 --> 00:06:59.607 So that’s a reasonable way to represent these possible candy bars. 00:07:00.645 --> 00:07:02.991 Alternatively, I might represent those candy bars 00:07:02.991 --> 00:07:05.559 by masticity, which is chew time— 00:07:05.559 --> 00:07:07.174 how long it takes to chew ’em. 00:07:07.174 --> 00:07:10.760 So these would be the ones that maybe only take two minutes to chew. 00:07:10.760 --> 00:07:13.247 And these may take twenty minutes to chew. 00:07:13.247 --> 00:07:17.016 Well, chew time is probably not the best way to look at a candy bar. 00:07:17.016 --> 00:07:20.824 And so, as a result, I’m going to have a landscape with many, many more peaks. 00:07:21.547 --> 00:07:25.409 And so, because it’s got many more peaks, that’s more places I could get stuck. 00:07:25.409 --> 00:07:28.976 So it’s not as good as a way to represent the possible solutions. 00:07:28.976 --> 00:07:30.804 It’s not as good a perspective. 00:07:30.804 --> 00:07:36.001 The best perspective would be what we call a Mount Fuji landscape, 00:07:36.001 --> 00:07:38.047 the ideal landscape that just has one peak. 00:07:38.047 --> 00:07:39.807 And these are called Mount Fuji landscapes 00:07:39.807 --> 00:07:41.152 because if you’ve ever been to Japan, 00:07:41.152 --> 00:07:42.629 and you look at Mount Fuji, it looks pretty much like this. 00:07:42.629 --> 00:07:44.944 Actually not quite like this, there’s like snow on the top. 00:07:44.944 --> 00:07:48.013 But for the most part, it looks just like one giant cone. 00:07:48.013 --> 00:07:49.616 If you’re on a Mount Fuji landscape, 00:07:49.616 --> 00:07:51.128 if you’re sitting at some point, 00:07:51.128 --> 00:07:54.100 you can always just climb your way right up to the top. 00:07:54.100 --> 00:07:55.936 So these single-peak landscapes are really good 00:07:55.936 --> 00:07:57.700 because you’ve basically taken a problem 00:07:57.700 --> 00:07:59.929 and made it very, very simple. 00:08:01.160 --> 00:08:03.913 What would be an example of a Mount Fuji landscape? 00:08:03.913 --> 00:08:06.007 I’m going to take a famous example. 00:08:06.007 --> 00:08:08.536 So, a famous example comes from scientific management, 00:08:08.536 --> 00:08:09.650 and due to Frederick Taylor. 00:08:09.650 --> 00:08:12.487 Taylor famously solved for the optimal size of a shovel. 00:08:12.487 --> 00:08:15.450 So let’s think about the shovel size landscape. 00:08:15.450 --> 00:08:18.252 So, on this axis, I’ve got the size of the shovel. 00:08:18.883 --> 00:08:21.809 And on this axis, I’ve got the value. 00:08:21.809 --> 00:08:23.384 And what do I mean by the value? 00:08:23.384 --> 00:08:24.984 I don’t mean how much I can sell the shovel for, 00:08:24.984 --> 00:08:27.496 I mean it’s like how useful the shovel is at the task. 00:08:27.496 --> 00:08:29.420 So let’s suppose we’re shoveling coal 00:08:29.420 --> 00:08:30.474 and I want to think about 00:08:30.474 --> 00:08:33.396 how many pounds of coal can some[one] shovel in a day 00:08:33.396 --> 00:08:35.441 as a function of the size. 00:08:35.441 --> 00:08:37.896 So let’s start out here where the size is zero. 00:08:37.896 --> 00:08:39.690 So this is the size of the pan. 00:08:39.690 --> 00:08:41.631 If I have a shovel has a pan of size zero, 00:08:41.631 --> 00:08:43.700 that’s commonly known as a stick 00:08:43.700 --> 00:08:45.876 and we can’t get anything. 00:08:46.384 --> 00:08:47.895 We’re not going to shovel anything with a stick. 00:08:47.895 --> 00:08:50.004 Well, if I make it bigger, 00:08:50.004 --> 00:08:52.241 you know, make it the size of maybe like a little spoon or something, 00:08:52.241 --> 00:08:53.693 then we can shovel a little bit. 00:08:53.693 --> 00:08:55.984 And as I make the shovel bigger and bigger and bigger, 00:08:55.984 --> 00:08:58.672 we, whoever, my workers, can shovel more and more coal. 00:08:58.672 --> 00:09:02.616 But at some point, the shovel’s going to get a little bit too big. 00:09:02.616 --> 00:09:04.953 And it’s going to be too heavy to lift. 00:09:04.953 --> 00:09:06.056 And the worker’s going to get tired, 00:09:06.056 --> 00:09:07.216 and I’ll shovel less, 00:09:07.216 --> 00:09:08.460 he’ll shovel less and less and less and less. 00:09:08.460 --> 00:09:11.898 And then eventually get to some point where the shovel’s so big 00:09:11.898 --> 00:09:14.015 that he can’t even lift it, 00:09:14.015 --> 00:09:14.905 and it’s as useless as the stick. 00:09:14.905 --> 00:09:20.828 So if I look at value in terms of how much coal the person can shovel in a day is a function of the size of the shovel. 00:09:20.828 --> 00:09:23.441 I’m going to get a single-peaked landscape. 00:09:23.441 --> 00:09:24.604 That’s going to be an easy problem to solve. 00:09:24.604 --> 00:09:29.542 And this idea, that we could represent scientific problems in this way— 00:09:29.542 --> 00:09:33.943 or we could put engineering problems in this way—and then climb our way to peaks, 00:09:33.943 --> 00:09:36.568 is the basis is something called scientific management 00:09:36.568 --> 00:09:38.040 And the idea was that you could then 00:09:38.040 --> 00:09:40.720 by finding these high points on these landscapes, 00:09:40.720 --> 00:09:42.794 find optimal solutions. 00:09:42.794 --> 00:09:45.733 We’re only going to find out the optimal solution for sure 00:09:45.733 --> 00:09:48.458 if your hill climbed like this—if it’s single peaked. 00:09:48.613 --> 00:09:51.013 If it’s rugged and looks like this mess, 00:09:51.013 --> 00:09:52.411 looks like Mount Fuji landscape you’re fine, 00:09:52.411 --> 00:09:53.417 but if it looks like this mess, this masticity landscape, 00:09:53.417 --> 00:09:55.739 if you have a bad perspective, 00:09:55.739 --> 00:09:57.775 well then if you climbed hills 00:09:57.775 --> 00:10:00.565 you could get stuck just about anywhere. 00:10:00.596 --> 00:10:03.711 So what you’d like is you’d like a Mount Fuji landscape, 00:10:03.711 --> 00:10:07.674 And in the case of simple things like this shovel, that’s easy to get. 00:10:07.674 --> 00:10:09.480 Let me give you another example. 00:10:09.480 --> 00:10:10.517 This one’s a lot of fun. 00:10:10.517 --> 00:10:12.824 This is a favorite game of mine called Sum to fifteen 00:10:12.824 --> 00:10:14.736 and was developed by Herb Simon 00:10:14.736 --> 00:10:17.561 who’s a Nobel Prize winner in economics. 00:10:17.561 --> 00:10:19.827 And Sum to fifteen was developed to show people 00:10:19.827 --> 00:10:22.501 why diverse perspectives are so useful, 00:10:22.501 --> 00:10:25.157 why different ways of representing a problem can make them easy, 00:10:25.157 --> 00:10:26.695 can make them like Mount Fuji, 00:10:26.695 --> 00:10:29.048 or can make them really difficult. 00:10:29.048 --> 00:10:31.313 So here’s how Sum to fifteen works. 00:10:31.313 --> 00:10:34.863 There’s cards numbered from one to nine face up on a table. 00:10:34.863 --> 00:10:36.769 There’s nine cards in front of you. 00:10:36.769 --> 00:10:37.946 There’s two players. 00:10:37.946 --> 00:10:41.823 Each person.takes turns, taking a card. 00:10:41.823 --> 00:10:44.897 until all the cards are gone, possibly—it could end sooner. 00:10:45.072 --> 00:10:50.409 If anybody ever holds three cards that add up to exactly 15, they win. 00:10:50.670 --> 00:10:51.923 That’s the game. So, really simple. 00:10:51.923 --> 00:10:54.448 Nine cards. Alternate taking cards. 00:10:54.448 --> 00:10:58.275 If you ever get exactly three that sum to fifteen you win. 00:10:58.275 --> 00:10:59.824 So let me show you a game. 00:10:59.824 --> 00:11:01.528 Here’s a game between two people, 00:11:01.528 --> 00:11:03.892 [let’s] call them Paul and David. 00:11:03.907 --> 00:11:05.251 Paul goes first. Now you’d think when you play this game 00:11:05.251 --> 00:11:07.913 the thing to do would be to choose the five. 00:11:07.913 --> 00:11:11.604 Paul chooses the four, which is sort of an odd choice. 00:11:11.604 --> 00:11:14.402 David goes next so he takes the five. 00:11:14.402 --> 00:11:16.844 Paul then takes the six. 00:11:16.844 --> 00:11:18.920 Now the six is a strange choice 00:11:18.920 --> 00:11:22.872 because four plus six plus five equals fifteen. 00:11:22.872 --> 00:11:25.832 So it looks like there is no way that he can win. 00:11:25.832 --> 00:11:28.234 Well this will be confusing to Doug. 00:11:28.234 --> 00:11:30.255 So Doug’s going to take the eight. 00:11:30.255 --> 00:11:34.505 Now notice eight plus five equals thirteen. 00:11:34.520 --> 00:11:37.712 So that means Paul has to take the two. 00:11:37.712 --> 00:11:39.363 So he takes the two. 00:11:39.363 --> 00:11:41.530 Well think about what happens next: 00:11:41.530 --> 00:11:43.222 Four plus two is six. 00:11:43.222 --> 00:11:45.068 So if Doug doesn’t take the nine, he’s going to lose. 00:11:45.791 --> 00:11:47.563 But six plus two is eight. 00:11:47.563 --> 00:11:49.606 So if Doug doesn’t take the seven he’s going to lose. 00:11:49.606 --> 00:11:52.148 So what you’ve got here is that Paul has won. 00:11:52.148 --> 00:11:55.421 No matter what Doug does, Paul’s going to win the game. 00:11:55.544 --> 00:11:57.002 Now this is a pretty tricky game, right? 00:11:57.002 --> 00:11:58.568 It was developed by a Nobel Prize winner. 00:11:58.568 --> 00:12:00.878 You could imagine there’s lots of strategy involved. 00:12:00.878 --> 00:12:05.502 I want to show you this game in a different perspective. 00:12:05.502 --> 00:12:08.134 Remember the magic square from seventh grade math? 00:12:08.134 --> 00:12:11.388 Every row adds up to fifteen— 00:12:11.388 --> 00:12:15.507 8+3+4, 1+5+9, 6+7+2 — 00:12:15.507 --> 00:12:16.885 so does every column— 00:12:16.885 --> 00:12:20.273 8+1+6 sums up to fifteen; 00:12:20.273 --> 00:12:22.942 3+5+7 sums up to fifteen— 00:12:22.942 --> 00:12:24.734 and even the diagonals— 00:12:24.734 --> 00:12:26.657 eight, five, two is fifteen; 00:12:26.657 --> 00:12:28.469 six, five, four is fifteen. 00:12:28.469 --> 00:12:30.638 Every row, every column, every diagonal sum up to fifteen. 00:12:30.638 --> 00:12:34.108 Let me show you this game again on the Magic Square. 00:12:34.108 --> 00:12:37.397 So, it’s just a different perspective on “Sum to Fifteen”. 00:12:37.397 --> 00:12:39.638 Paul goes first, and takes the four. 00:12:40.099 --> 00:12:42.278 Doug goes next and takes the five. 00:12:42.278 --> 00:12:45.787 Paul takes the six, which is an odd choice, because now he can’t win. 00:12:45.787 --> 00:12:50.202 Doug then takes the eight, Paul blocks him with the two. 00:12:50.202 --> 00:12:55.121 But now it turns out, either the nine or seven will let Paul win. 00:12:55.413 --> 00:12:57.744 What game is this? 00:12:58.021 --> 00:13:00.605 Well, you’re right, it’s tic-tac-toe. 00:13:00.958 --> 00:13:04.061 Sum to fifteen is just tic-tac-toe, 00:13:04.061 --> 00:13:07.316 but on a different perspective, using a different perspective. 00:13:07.446 --> 00:13:09.308 So if you turn Sum to Fifteen— 00:13:09.308 --> 00:13:12.237 if you moved the cards 1 to 9 and put them in the magic square— 00:13:12.237 --> 00:13:16.166 what you do is you create a Mount Fuji landscape In a sense: 00:13:16.166 --> 00:13:18.549 You make the problem really simple. 00:13:18.549 --> 00:13:20.499 So a lot of great breakthroughs, 00:13:20.499 --> 00:13:21.831 like the periodic table, 00:13:21.831 --> 00:13:23.254 Newton’s Theory of Gravity, 00:13:23.254 --> 00:13:25.716 those are perspectives on problems 00:13:25.716 --> 00:13:27.981 that turned something that was really difficult to figure out 00:13:27.981 --> 00:13:31.005 into something that suddenly makes a lot of sense, 00:13:31.005 --> 00:13:32.519 very easy to see the solution. 00:13:32.519 --> 00:13:34.836 At least it’s something I call in my book, one of my books, the difference, 00:13:34.836 --> 00:13:37.309 I call this the Savant Existence Theorem. 00:13:37.309 --> 00:13:39.504 For any problem that’s out there, 00:13:39.504 --> 00:13:41.725 there exists some way to represent it, 00:13:41.725 --> 00:13:44.518 so that you turn it into a Mt. Fuji problem. 00:13:44.518 --> 00:13:45.750 Now, why is that? 00:13:45.750 --> 00:13:47.262 Well, it’s actually fairly straightforward. 00:13:47.262 --> 00:13:49.609 All you have to do is, 00:13:49.609 --> 00:13:53.023 if you’ve got all the solutions here represented on this thing, 00:13:53.023 --> 00:13:54.670 you put the very best one in the middle. 00:13:54.670 --> 00:13:57.354 And then put the worst ones at the end. 00:13:57.354 --> 00:13:58.899 And then just sort of line up the solutions in such a way 00:13:58.899 --> 00:14:01.282 so that you turn it into a Mount Fuji. 00:14:01.282 --> 00:14:02.650 So it’s very straightforward. 00:14:02.650 --> 00:14:04.386 Now the thing is, in order to make the Mount Fuji, 00:14:04.386 --> 00:14:07.134 you’d have to know the solution already. 00:14:07.134 --> 00:14:09.069 This isn’t a good way to solve problems 00:14:09.069 --> 00:14:11.881 but the point is, it exists. 00:14:11.881 --> 00:14:13.480 So it’s always the possibility 00:14:13.480 --> 00:14:15.224 that someone could look at particular problem and said, 00:14:15.224 --> 00:14:17.399 “Hey, what if think of it this way?” 00:14:17.399 --> 00:14:20.095 And doing so turn something that was really rugged 00:14:20.095 --> 00:14:22.653 into something that looks like Mount Fuji. 00:14:24.145 --> 00:14:26.002 Here is the flip side though. 00:14:26.002 --> 00:14:28.398 There is a ton of bad perspectives. 00:14:28.398 --> 00:14:30.622 So just like there’s these Mount Fuji perspectives, 00:14:30.622 --> 00:14:34.065 there’s also lots and lots of horrible ways to look at problems. 00:14:34.065 --> 00:14:37.205 Think about this: Suppose I have just ten alternatives 00:14:37.205 --> 00:14:40.286 and I want to think about what are all the different ways I can just put them in a line. 00:14:40.286 --> 00:14:42.424 Well there’s ten things I could put first, 00:14:42.424 --> 00:14:44.064 nine things I could put second, 00:14:44.064 --> 00:14:45.924 eight things I could put third and so on. 00:14:45.924 --> 00:14:51.348 So there’s 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 perspectives. 00:14:51.348 --> 00:14:54.169 Most of those are going to not be very good. 00:14:54.169 --> 00:14:58.381 They’re not going to organize this set of solutions in any useful way. 00:14:58.381 --> 00:15:01.187 Particularly, only a few of them are going to create Mount Fujis. 00:15:01.187 --> 00:15:03.794 So we think about the value of perspectives, what we get is this: 00:15:03.794 --> 00:15:06.583 There’s really good ones out there, 00:15:06.583 --> 00:15:09.726 that insightful, smart people can come up 00:15:09.726 --> 00:15:11.825 with really good representations of problem[s] 00:15:11.825 --> 00:15:14.415 to make the landscapes less rugged. 00:15:14.415 --> 00:15:16.978 If we just think about things in random ways, 00:15:16.978 --> 00:15:18.915 we’re likely to get a landscape that’s so rugged 00:15:18.915 --> 00:15:21.284 that we’re going to get stuck just about everywhere. 00:15:21.284 --> 00:15:23.408 We’re not going to be able to find good solutions to the problem. 00:15:23.408 --> 00:15:26.561 And we’re going to hit things that look like the masticity landscape, 00:15:26.561 --> 00:15:29.210 and we’re going to get things with lots and lots of peaks. 00:15:29.210 --> 00:15:32.511 Let’s move on now and talk about how we move on these landscapes. 00:15:32.511 --> 00:15:35.935 So once I got our landscape, how do I find better solutions? 00:15:35.935 --> 00:15:38.616 Are there other alternatives to just sort of climbing a hill? 00:15:38.616 --> 00:15:42.207 Because that hill climbing idea really only works in one dimension. 00:15:42.207 --> 00:15:43.966 What if I’ve got all sorts of dimensions? 00:15:43.966 --> 00:15:45.036 How do I think about… 00:15:46.374 --> 00:15:47.012 (Just a sec…) 00:15:53.601 --> 00:15:55.238 So what have we learned? 00:15:55.238 --> 00:15:57.953 First thing we’ve learned is that when we go about trying to solve a problem, 00:15:57.953 --> 00:15:59.709 when we encode it in some way, 00:15:59.709 --> 00:16:01.770 that’s a perspective. 00:16:01.770 --> 00:16:06.755 And a perspective creates peaks; it creates these local optima. 00:16:06.755 --> 00:16:09.748 So a better perspectives have fewer local optima. 00:16:09.748 --> 00:16:13.258 Worse perspectives have lots of local optima. 00:16:13.258 --> 00:16:15.962 And if you think about how many perspectives are out there, 00:16:15.962 --> 00:16:18.078 we just saw there’s billions of them. 00:16:18.078 --> 00:16:19.390 Because there’s billions of perspectives, 00:16:19.390 --> 00:16:21.425 most of those probably aren’t very useful. 00:16:21.425 --> 00:16:25.256 Some of them, though, turn problems into Mount Fujis. 00:16:25.256 --> 00:16:27.118 And sometimes it takes a genius— 00:16:27.118 --> 00:16:28.582 it takes a Newton, it takes a Mendeleev— 00:16:28.582 --> 00:16:30.784 to come up with a way of representing reality 00:16:30.784 --> 00:16:32.958 so that something that was incredibly rugged 00:16:32.958 --> 00:16:34.561 becomes Mount Fuji–like. 00:16:34.561 --> 00:16:36.914 Other times, if you think about the size of a shovel, 00:16:36.914 --> 00:16:42.352 that problem most of us could probably figure out a way that problem just by shovel size, 00:16:42.352 --> 00:16:44.416 so that it becomes a Mount Fuji. 00:16:44.416 --> 00:16:45.366 The big point is this: 00:16:45.366 --> 00:16:48.969 When we go about solving problems, the first thing we do is we encode them. 00:16:48.969 --> 00:16:51.262 We have some representation of the problem. 00:16:51.262 --> 00:16:55.519 That representation determines how hard the problem will be. 00:16:55.519 --> 00:16:58.384 If we represent it in such a way that it’s a Mount Fuji, it’s easy. 00:16:58.384 --> 00:17:01.912 If we represent it in such a way that it looks like that masticity landscape, 00:17:01.912 --> 00:17:04.150 it’s probably going to be fairly hard. 00:17:04.150 --> 00:17:05.794 Where we want to go next, 00:17:05.794 --> 00:17:09.791 is we want to talk about once we’ve got this representation of the possible solutions, 00:17:09.791 --> 00:17:11.831 once we have that landscape, so to speak, 00:17:11.831 --> 00:17:13.412 how do we search on that landscape? 00:17:13.412 --> 00:17:14.509 So one thing we’ve talked about was climbing hills. 00:17:14.509 --> 00:17:17.200 But there’s lots of different ways you can climb hills. 00:17:17.200 --> 00:17:20.919 That’s what we’ll talk about next: the heuristics we use on a landscape. 00:17:20.919 --> 99:59:59.999 Thanks.