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Hi. In this lecture we’re talking about problem solving.

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And we’re talking about the role that diverse perspectives play in finding solutions to problems.

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So when you think about a problem,

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perspective is how you represent it.

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So remember from the previous lecture, we talked about landscapes.

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We talked about landscape being a way to represent

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the solutions along this axis

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and the value of the solutions as the height.

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And so this is metaphorically a way to represent

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how someone might think about solving a problem:

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Finding high points on their landscape.

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What we want to do is take this metaphor and formalize it

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and part of the reason for this course is to get better logic,

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[in order to] think through things in a clear way.

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So I’m going to take this landscape metaphor and turn it into a formal model.

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So how do we do it?

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The first thing we do is we formally define what a perspective is.

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So we speak math to metaphor.

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So what a perspective is going to be is

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it’s going to be a representation of all possible solutions.

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So it’s some encoding of the set of possible solutions to the problem.

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Once we have that encoding of the set of possible solutions,

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then we can create our landscape by just assigning a value to each one of those solutions.

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And that will give us a landscape picture like you saw before.

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Now most of us are familiar with perspectives,

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even though we don’t know it.

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Let me give some examples.

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Remember when we took seventh grade math?

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We learned about how to represent a point, how to plot points.

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And we typically learned two ways to do it.

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The first way was Cartesian coordinates.

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So given a point, we would represent it

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by and an X and a Y value in space.

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So, it might be five units,

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this would be the point, let’s say (5, 2).

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It’s five units in the X direction, two units in the Y direction.

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But we also learned another way to represent points,

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and that was [polar] coordinates.

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So we can take the same point and say,

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there’s a radius, which is its distance from the origin,

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and then there’s some angle theta,

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which says how far we have to sweep out,

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in order to sweep that radius out in order to get to the point.

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So two totally reasonable ways to represent a point:

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X and Y, R and theta.

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Cartesian, polar.

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Which is better?

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Well, the answer? It depends.

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Let me show you why.

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Suppose I wanted to describe this line.

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In order to describe this line I should use Cartesian coordinates,

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’cause I can just say Y=3 and X moves from two to five.

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It’s really easy.

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But suppose I wanna describe this arc.

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If I wanna describe this arc,

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now Cartesian coordinates are gonna be fairly complicated,

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and I’d be better off using polar coordinates,

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because the radius is fixed

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and I just talked about how the radius is—you know,

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there’s this distance R,

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and theta just moves from, you know, A to B, let’s say.

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So depending on what I want to do.

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If I want to look at straight lines, I should use Cartesian.

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And if I want to look at arcs, I should probably use polar.

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So, perspectives depend on the problem.

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Now let’s think about where we want to go.

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We want to talk about how perspectives help us find solutions to problems

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and how perspectives help us be innovative.

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Well, if you look at the history of science a lot of great breakthroughs—

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you know, we think about Newton,

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you know, his theory of gravity—

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you can think about people actually having new perspectives on old problems.

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Let’s take an example.

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So, Mendeleev came up with the periodic table,

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and in the periodic table he represents the elements by atomic weight.

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He’s got them in these different columns.

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In doing so, by organizing the elements by atomic weight

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he found all sorts of structure.

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So all the metals line one column, stuff like that.

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Remember—from high school chemistry class.

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That’s a perspective: It’s a representation of a set of possible elements.

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He could’ve organized them alphabetically.

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But that wouldn’t have made much sense.

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So alphabetic representation wouldn’t give us any structure.

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Atomic weight representation gives us a lot of structure.

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In fact, when Mendeleev wrote down

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all the elements that were around at the time according to atomic weight,

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there were gaps in his representation.

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There were holes for elements that were missing.

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Those elements became scandium, gallium, and germanium.

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They were eventually found ten to fifteen years later,

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after he’d written down the periodic table:

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People went out and were able to find the missing elements.

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That perspective, atomic weight,

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ended up being a very useful way to organize our thinking about the elements.

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We do it all the time now.

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When you have any sort of task,

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you’ll find that you’re actually using some sort of perspective.

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Suppose that you’re hiring someone.

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And you’ve got a bunch of recent college graduates who apply for a job.

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And you’ve gotta think,

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“Okay, how do I organize all these applicants?”

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Let’s say 500 applicants.

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One thing you could do is you could organize them by GPA:

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Take the highest GPA down to the lowest GPA.

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That’s be one representation.

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And you might do that if you valued competence or achievement.

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But you might also value work ethic.

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And if that were the case you might instead organize

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those same CV’s or application files by how thick they are.

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[Those who’re going to do the] really thick ones are people who work really, really hard.

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They’ve accomplished a lot.

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Well, the third thing you might do is you might value creativity.

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And you might say,

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“Well, let’s put the ones that are sort of most colorful, most interesting over here.

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And the ones that are least colorful and least interesting over here.”

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That’s the third way to do it.

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Now depending on what you’re hiring for,

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depending on who the applicants are,

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any one of these might be fine.

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The only point I’m trying to make here is that there’s different ways to organize these applicants.

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In each one of those ways you organize—

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whether it’s in your head,

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or whether it’s formally laying them out in some way—

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is a perspective.

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And those perspectives will determine how hard the problem will be for you.

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Let me explain why.

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Now I want to go back to the landscape metaphor.

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And when I think of that landscape as being rugged,

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and by rugged I mean that it doesn’t look like a single peak,

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that there’s lots of peaks on it.

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And I want to formalize this notion of peaks.

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And I do so as follows:

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I’m going to define what I call a local optima.

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A local optima is a point such that

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if you look at the points on either side of it,

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they’re lower in value.

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So it’s sort of a point that locally is the highest possible value.

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So if I look at this particular rugged landscape again,

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there’s three local optima: 1, 2, 3.

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At any one of these three points, I’d be stuck:

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If I looked to the left or to the right,

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I wouldn’t find a solution that’s better.

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So we think about what makes a good perspective:

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A good perspective is going to be a perspective that doesn’t have many local optima.

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A bad perspective is going to be one that has a lot of local optima.

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Let me give you an example, okay?

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So, suppose I’m coming up with a candy bar.

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Suppose I’m tasked with coming up with a new candy bar.

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So I have my team of chefs make a whole bunch of different confections for me to try,

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and I want to find the very best one.

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But there’re so many of them, there’s so many possibilities,

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that I’m not even sure how to think about it.

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But one way to represent those candy bars might be by the number of calories that they had.

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So I can organize all the different things they make by number of calories.

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And if I did that, maybe I’d have three local optima.

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So that’s a reasonable way to represent these possible candy bars.

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Alternatively, I might represent those candy bars

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by masticity, which is chew time—

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how long it takes to chew ’em.

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So these would be the ones that maybe only take two minutes to chew.

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And these may take twenty minutes to chew.

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Well, chew time is probably not the best way to look at a candy bar.

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And so, as a result, I’m going to have a landscape with many, many more peaks.

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And so, because it’s got many more peaks, that’s more places I could get stuck.

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So it’s not as good as a way to represent the possible solutions.

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It’s not as good a perspective.

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The best perspective would be what we call a Mount Fuji landscape,

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the ideal landscape that just has one peak.

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And these are called Mount Fuji landscapes

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because if you’ve ever been to Japan,

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and you look at Mount Fuji, it looks pretty much like this.

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Actually not quite like this, there’s like snow on the top.

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But for the most part, it looks just like one giant cone.

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If you’re on a Mount Fuji landscape,

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if you’re sitting at some point,

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you can always just climb your way right up to the top.

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So these single-peak landscapes are really good

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because you’ve basically taken a problem

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and made it very, very simple.

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What would be an example of a Mount Fuji landscape?

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I’m going to take a famous example.

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So, a famous example comes from scientific management,

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and due to Frederick Taylor.

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Taylor famously solved for the optimal size of a shovel.

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So let’s think about the shovel size landscape.

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So, on this axis, I’ve got the size of the shovel.

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And on this axis, I’ve got the value.

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And what do I mean by the value?

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I don’t mean how much I can sell the shovel for,

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I mean it’s like how useful the shovel is at the task.

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So let’s suppose we’re shoveling coal

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and I want to think about

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how many pounds of coal can some[one] shovel in a day

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as a function of the size.

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So let’s start out here where the size is zero.

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So this is the size of the pan.

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If I have a shovel has a pan of size zero,

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that’s commonly known as a stick

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and we can’t get anything.

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We’re not going to shovel anything with a stick.

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Well, if I make it bigger,

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you know, make it the size of maybe like a little spoon or something,

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then we can shovel a little bit.

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And as I make the shovel bigger and bigger and bigger,

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we, whoever, my workers, can shovel more and more coal.

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But at some point, the shovel’s going to get a little bit too big.

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And it’s going to be too heavy to lift.

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And the worker’s going to get tired,

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and I’ll shovel less,

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he’ll shovel less and less and less and less.

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And then eventually get to some point where the shovel’s so big

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that he can’t even lift it,

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and it’s as useless as the stick.

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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.

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I’m going to get a single-peaked landscape.

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That’s going to be an easy problem to solve.

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And this idea, that we could represent scientific problems in this way—

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or we could put engineering problems in this way—and then climb our way to peaks,

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is the basis is something called scientific management

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And the idea was that you could then

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by finding these high points on these landscapes,

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find optimal solutions.

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We’re only going to find out the optimal solution for sure

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if your hill climbed like this—if it’s single peaked.

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If it’s rugged and looks like this mess,

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looks like Mount Fuji landscape you’re fine,

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but if it looks like this mess, this masticity landscape,

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if you have a bad perspective,

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well then if you climbed hills

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you could get stuck just about anywhere.

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So what you’d like is you’d like a Mount Fuji landscape,

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And in the case of simple things like this shovel, that’s easy to get.

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Let me give you another example.

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This one’s a lot of fun.

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This is a favorite game of mine called Sum to fifteen

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and was developed by Herb Simon

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who’s a Nobel Prize winner in economics.

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And Sum to fifteen was developed to show people

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why diverse perspectives are so useful,

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why different ways of representing a problem can make them easy,

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can make them like Mount Fuji,

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or can make them really difficult.

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So here’s how Sum to fifteen works.

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There’s cards numbered from one to nine face up on a table.

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There’s nine cards in front of you.

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There’s two players.

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Each person.takes turns, taking a card.

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until all the cards are gone, possibly—it could end sooner.

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If anybody ever holds three cards that add up to exactly 15, they win.

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That’s the game. So, really simple.

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Nine cards. Alternate taking cards.

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If you ever get exactly three that sum to fifteen you win.

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So let me show you a game.

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Here’s a game between two people,

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[let’s] call them Paul and David.

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Paul goes first. Now you’d think when you play this game

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the thing to do would be to choose the five.

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Paul chooses the four, which is sort of an odd choice.

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David goes next so he takes the five.

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Paul then takes the six.

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Now the six is a strange choice

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because four plus six plus five equals fifteen.

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So it looks like there is no way that he can win.

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Well this will be confusing to Doug.

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So Doug’s going to take the eight.

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Now notice eight plus five equals thirteen.

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So that means Paul has to take the two.

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So he takes the two.

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Well think about what happens next:

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Four plus two is six.

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So if Doug doesn’t take the nine, he’s going to lose.

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But six plus two is eight.

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So if Doug doesn’t take the seven he’s going to lose.

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So what you’ve got here is that Paul has won.

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No matter what Doug does, Paul’s going to win the game.

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Now this is a pretty tricky game, right?

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It was developed by a Nobel Prize winner.

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You could imagine there’s lots of strategy involved.

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I want to show you this game in a different perspective.

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Remember the magic square from seventh grade math?

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Every row adds up to fifteen—

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8+3+4, 1+5+9, 6+7+2 —

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so does every column—

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8+1+6 sums up to fifteen;

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3+5+7 sums up to fifteen—

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and even the diagonals—

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eight, five, two is fifteen;

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six, five, four is fifteen.

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Every row, every column, every diagonal sum up to fifteen.

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Let me show you this game again on the Magic Square.

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So, it’s just a different perspective on “Sum to Fifteen”.

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Paul goes first, and takes the four.

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Doug goes next and takes the five.

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Paul takes the six, which is an odd choice, because now he can’t win.

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Doug then takes the eight, Paul blocks him with the two.

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But now it turns out, either the nine or seven will let Paul win.

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What game is this?

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Well, you’re right, it’s tic-tac-toe.

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Sum to fifteen is just tic-tac-toe,

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but on a different perspective, using a different perspective.

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So if you turn Sum to Fifteen—

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if you moved the cards 1 to 9 and put them in the magic square—

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what you do is you create a Mount Fuji landscape In a sense:

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You make the problem really simple.

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So a lot of great breakthroughs,

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like the periodic table,

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Newton’s Theory of Gravity,

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those are perspectives on problems

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that turned something that was really difficult to figure out

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into something that suddenly makes a lot of sense,

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very easy to see the solution.

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At least it’s something I call in my book, one of my books,
the difference,

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I call this the Savant Existence Theorem.

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For any problem that’s out there,

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there exists some way to represent it,

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so that you turn it into a Mt. Fuji problem.

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Now, why is that?

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Well, it’s actually fairly straightforward.

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All you have to do is,

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if you’ve got all the solutions here represented on this thing,

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you put the very best one in the middle.

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And then put the worst ones at the end.

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And then just sort of line up the solutions in such a way

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so that you turn it into a Mount Fuji.

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So it’s very straightforward.

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Now the thing is, in order to make the Mount Fuji,

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you’d have to know the solution already.

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This isn’t a good way to solve problems

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but the point is, it exists.

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So it’s always the possibility

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that someone could look at particular problem and said,

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“Hey, what if think of it this way?”

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And doing so turn something that was really rugged

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into something that looks like Mount Fuji.

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Here is the flip side though.

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There is a ton of bad perspectives.

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So just like there’s these Mount Fuji perspectives,

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there’s also lots and lots of horrible ways to look at problems.

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Think about this: Suppose I have just ten alternatives

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and I want to think about what are all the different ways I can just put them in a line.

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Well there’s ten things I could put first,

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nine things I could put second,

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eight things I could put third and so on.

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So there’s 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 perspectives.

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Most of those are going to not be very good.

347
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They’re not going to organize this set of solutions in any useful way.

348
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Particularly, only a few of them are going to create Mount Fujis.

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So we think about the value of perspectives, what we get is this:

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There’s really good ones out there,

351
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that insightful, smart people can come up

352
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with really good representations of problem[s]

353
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to make the landscapes less rugged.

354
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If we just think about things in random ways,

355
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we’re likely to get a landscape that’s so rugged

356
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that we’re going to get stuck just about everywhere.

357
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We’re not going to be able to find good solutions to the problem.

358
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And we’re going to hit things that look like the masticity landscape,

359
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and we’re going to get things with lots and lots of peaks.

360
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Let’s move on now and talk about how we move on these landscapes.

361
00:15:32,511 --> 00:15:35,935
So once I got our landscape, how do I find better solutions?

362
00:15:35,935 --> 00:15:38,616
Are there other alternatives to just sort of climbing a hill?

363
00:15:38,616 --> 00:15:42,207
Because that hill climbing idea really only works in one dimension.

364
00:15:42,207 --> 00:15:43,966
What if I’ve got all sorts of dimensions?

365
00:15:43,966 --> 00:15:45,036
How do I think about…

366
00:15:46,374 --> 00:15:47,012
(Just a sec…)

367
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So what have we learned?

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First thing we’ve learned is that when we go about trying to solve a problem,

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when we encode it in some way,

370
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that’s a perspective.

371
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And a perspective creates peaks; it creates these local optima.

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So a better perspectives have fewer local optima.

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Worse perspectives have lots of local optima.

374
00:16:13,258 --> 00:16:15,962
And if you think about how many perspectives are out there,

375
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we just saw there’s billions of them.

376
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Because there’s billions of perspectives,

377
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most of those probably aren’t very useful.

378
00:16:21,425 --> 00:16:25,256
Some of them, though, turn problems into Mount Fujis.

379
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And sometimes it takes a genius—

380
00:16:27,118 --> 00:16:28,582
it takes a Newton, it takes a Mendeleev—

381
00:16:28,582 --> 00:16:30,784
to come up with a way of representing reality

382
00:16:30,784 --> 00:16:32,958
so that something that was incredibly rugged

383
00:16:32,958 --> 00:16:34,561
becomes Mount Fuji–like.

384
00:16:34,561 --> 00:16:36,914
Other times, if you think about the size of a shovel,

385
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,

386
00:16:42,352 --> 00:16:44,416
so that it becomes a Mount Fuji.

387
00:16:44,416 --> 00:16:45,366
The big point is this:

388
00:16:45,366 --> 00:16:48,969
When we go about solving problems, the first thing we do is we encode them.

389
00:16:48,969 --> 00:16:51,262
We have some representation of the problem.

390
00:16:51,262 --> 00:16:55,519
That representation determines how hard the problem will be.

391
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If we represent it in such a way that it’s a Mount Fuji, it’s easy.

392
00:16:58,384 --> 00:17:01,912
If we represent it in such a way that it looks like that masticity landscape,

393
00:17:01,912 --> 00:17:04,150
it’s probably going to be fairly hard.

394
00:17:04,150 --> 00:17:05,794
Where we want to go next,

395
00:17:05,794 --> 00:17:09,791
is we want to talk about once we’ve got this representation of the possible solutions,

396
00:17:09,791 --> 00:17:11,831
once we have that landscape, so to speak,

397
00:17:11,831 --> 00:17:13,412
how do we search on that landscape?

398
00:17:13,412 --> 00:17:14,509
So one thing we’ve talked about was climbing hills.

399
00:17:14,509 --> 00:17:17,200
But there’s lots of different ways you can climb hills.

400
00:17:17,200 --> 00:17:20,919
That’s what we’ll talk about next: the heuristics we use on a landscape.

401
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Thanks.