WEBVTT 00:00:03.040 --> 00:00:07.046 You've probably all had some problem you've  worked at. 00:00:07.046 --> 00:00:10.765 And you keep going at it, and you keep going at it... 00:00:10.765 --> 00:00:13.901 Nothing seems to work. You go away. 00:00:13.901 --> 00:00:16.924 Suddenly, the next day you think, 'What problem? It's easy!' 00:00:22.160 --> 00:00:26.616 It's something I said I've  certainly had, and I'm sure you've encountered. 00:00:26.616 --> 00:00:32.336 And psychologists and other people studying creativity have looked at these kind of 00:00:32.336 --> 00:00:37.456 issues – why it is that we get to these impasses and how it is that we break out of them. 00:00:39.600 --> 00:00:44.805 *Fixation* is one of the keywords that  psychologists use here: 00:00:44.805 --> 00:00:49.526 the way in which when you start to deal with a problem, you have some sort of approach, 00:00:49.526 --> 00:00:57.185 some sort of way of dealing with it, and it's incredibly hard to stop just continuing with that same way. 00:00:57.185 --> 00:01:03.263 Once you've got that original idea, that original approach, it's very hard to go to a new one. 00:01:03.263 --> 00:01:08.033 And sometimes – you know – an approach that might work for one problem doesn't work for 00:01:08.033 --> 00:01:13.080 another, but of course if you're stuck in that approach, you can't try other ones. 00:01:13.238 --> 00:01:17.186 This can get worse depending on how you encounter the problem. 00:01:17.186 --> 00:01:21.640 And this is sometimes used in sort of trick questions to get you 00:01:21.640 --> 00:01:25.674 thinking down the wrong track and then it's  very hard to get down the right one. 00:01:25.674 --> 00:01:31.541 And this is sometimes called – if I can say it right! – the *Einstellung effect* 00:01:31.541 --> 00:01:35.504 which is about being fixed in a position. 00:01:35.504 --> 00:01:39.680 These guys called Luchins and Luchins back – I think – in the 1940s 00:01:39.680 --> 00:01:43.325 did experiments with water jars. And you've probably seen this sort of puzzle. 00:01:43.325 --> 00:01:46.409 You've got a number of jars, and the idea is they're different sizes 00:01:46.409 --> 00:01:52.745 and you're trying to get, say... two pints of water and you've got a 17-pint jug 00:01:52.745 --> 00:01:56.458 and a 3-pint jug and you've got to pour water back and forth. 00:01:56.458 --> 00:02:01.725 And there sometimes are really complicated solutions where you have to like fill one up 00:02:01.725 --> 00:02:06.619 from another, and then what's left in  this one is some nice magic amount. 00:02:06.619 --> 00:02:10.616 So, if you've got a 3-liter jug and a 7-liter jug and you fill the 7-liter one 00:02:10.616 --> 00:02:14.626 and you pour it into the 3-liter jug and then throw the three liters away, 00:02:14.626 --> 00:02:19.252 you've got four liters in your 7-liter jug. And you have these complex solutions. 00:02:19.252 --> 00:02:23.696 So, what Luchins and Luchins did was they gave people a number of 00:02:23.840 --> 00:02:29.167 examples which they worked through which required that kind of complex solution. 00:02:29.167 --> 00:02:33.351 And then they gave them four more. But the four more they gave them were ones where you could 00:02:33.351 --> 00:02:39.505 actually solve it very, very simply. You know – possibly just filling up two and pouring them into another. 00:02:39.840 --> 00:02:44.080 So, some subjects were given that; some  were just given the four easy problems. 00:02:44.080 --> 00:02:48.717 The ones who were just given the four easy problems solved it in the easy way. 00:02:48.717 --> 00:02:54.348 The ones who *started off with the complex ones applied the same complex heuristics* 00:02:54.348 --> 00:02:57.999 to the easier problems and took *much longer* to do it. 00:02:57.999 --> 00:03:03.144 So, they got so stuck in their ways,  having been introduced to this set of problems. 00:03:04.000 --> 00:03:08.480 One that I encountered myself, and this gives you an opportunity to laugh at me if you want to, 00:03:08.480 --> 00:03:12.633 because you'll probably, most of you I'm guessing might find this easy, but 00:03:12.633 --> 00:03:16.447 I was in university and I was once given this puzzle. 00:03:16.447 --> 00:03:20.880 And the puzzle starts off – and there is this stage where you're driven 00:03:20.880 --> 00:03:25.075 down the wrong path – anyway, so I'm giving you a hint by telling you that. 00:03:25.075 --> 00:03:29.603 So, you often get these puzzles where you're supposed to chop things up into identical pieces. 00:03:29.603 --> 00:03:34.305 So, I've got a square up there, and obviously you can chop a square into four identical 00:03:34.305 --> 00:03:37.809 smaller squares. Or you can chop it into four triangles. 00:03:37.809 --> 00:03:41.440 So, the triangles are all the same shape as each other, the same size as each other. 00:03:41.440 --> 00:03:46.156 In mathematical terms, they are *congruent to* each other. 00:03:46.156 --> 00:03:48.800 But they're not necessarily squares. 00:03:48.800 --> 00:03:51.979 I've got some triangles, and we can chop the triangles into four; 00:03:51.979 --> 00:03:54.426 they're different kinds of triangles in different ways. 00:03:54.426 --> 00:03:58.790 And on the right I've got one of the ones you sometimes see as a trick puzzle in books; 00:03:58.830 --> 00:04:04.880 and – or sometimes actually  cut out pieces as a sort of trivia-type puzzle. 00:04:04.880 --> 00:04:11.476 And the idea there is to have four pieces that will make up that 'L' shape. 00:04:11.476 --> 00:04:16.305 And certainly there may be other ways of doing it, but the way I've always seen is with these 00:04:16.305 --> 00:04:20.390 four pieces like that. So, they're all 'L' shapes themselves and they fit together. 00:04:20.390 --> 00:04:23.380 How about cutting things into three? 00:04:23.380 --> 00:04:28.341 Well, obviously, the 'L' shape is an easy one, and things that start off 00:04:28.341 --> 00:04:34.914 with three degrees of similarity like a triangle or hexagon are relatively straightforward. 00:04:34.954 --> 00:04:41.570 So, the crucial question is, can you chop a square into three pieces that are congruent 00:04:41.570 --> 00:04:46.041 with one another? Not all squares, but some shape – they could be triangular; 00:04:46.041 --> 00:04:49.569 they could be more complex – that are each identical. 00:04:49.569 --> 00:04:55.538 Can you do it? Now, my guess is you've probably already thought of how to do it and you'll be right. 00:04:55.760 --> 00:05:00.484 It took me three days to solve this because I was a mathematician 00:05:00.484 --> 00:05:04.757 and the whole thing was framed in geometry, mathematics, 00:05:04.757 --> 00:05:08.200 and I was trying to apply really complex mathematics. 00:05:08.443 --> 00:05:13.724 And despite numerous hints, it took me three days to solve this. 00:05:14.527 --> 00:05:17.923 If you haven't solved it, you're probably a budding mathematician. 00:05:19.680 --> 00:05:24.080 So, *fixation is a problem*. There's also –  fixation is more about the methods you 00:05:24.080 --> 00:05:27.661 use to apply things, but there's also *bias* – when you *evaluate* things. 00:05:27.661 --> 00:05:31.110 When we look at them, we have 00:05:31.280 --> 00:05:35.347 tendencies to see things in one way or another. And that's about all sorts of things. 00:05:35.347 --> 00:05:40.000 That can be about serious personal issues. It can be about trivial things. 00:05:40.402 --> 00:05:46.617 Often, the way in which something is *framed* to us can actually create a *bias* as well. 00:05:46.766 --> 00:05:50.000 A classic example, and there are various 00:05:50.000 --> 00:05:54.320 ways you can do this, is what's called  *anchoring*. So... if we're asked something 00:05:54.320 --> 00:05:58.150 and given something that *suggests a value* 00:05:58.150 --> 00:06:03.715 even if it's told that it's just there for guesswork purposes or something, 00:06:03.715 --> 00:06:08.183 it tends to hold us and move where we see our estimate. 00:06:08.183 --> 00:06:12.456 So, you ask somebody, 'How high is the Eiffel Tower?' 00:06:12.456 --> 00:06:17.760 You might have a vague idea that it's big, but you probably don't know exactly how high. 00:06:17.800 --> 00:06:21.595 You might ask one set of people and give them a scale and say, 'Put it on this scale. Just draw 00:06:21.595 --> 00:06:27.230 a cross where you think on this scale, from 250 meters high to 2,500 meters high, 00:06:27.230 --> 00:06:30.468 how high on that scale? 00:06:30.468 --> 00:06:34.440 And people put crosses on the scale – you can see where they were, or put a number. 00:06:34.840 --> 00:06:39.027 But alternatively, you might give people, instead of having a scale of 00:06:39.200 --> 00:06:46.520 250 meters to 2,500 meters, you might give them a scale between 50 meters and 500 meters. 00:06:46.720 --> 00:06:52.240 Now, actually the Eiffel Tower falls on both of those scales: the actual height's around 300 meters. 00:06:52.240 --> 00:06:55.542 But what you find is people don't know the answer; 00:06:55.542 --> 00:07:02.158 given the larger, higher scale, they will tend to put something that is larger and higher 00:07:02.158 --> 00:07:05.349 even though they're told it's just a scale. 00:07:05.349 --> 00:07:08.224 And, actually, on the larger scale it should be right at the bottom. 00:07:08.224 --> 00:07:11.531 Here it should be about two thirds of the way up the scale. 00:07:11.531 --> 00:07:17.181 But what happens is – by framing it with big numbers, people tend to guess a bigger number. 00:07:17.221 --> 00:07:21.422 If you frame it with smaller numbers,  people guess a smaller number. 00:07:21.422 --> 00:07:25.587 *They're anchored by the nature of the way the question is posed.* 00:07:25.627 --> 00:07:28.689 So, how might you get away from some of this fixation? 00:07:28.689 --> 00:07:32.760 We'll talk about some other things later, other ways later. 00:07:33.051 --> 00:07:36.924 But one of the ways to actually break some of these biases and fixations is to 00:07:36.924 --> 00:07:40.240 *deliberately mix things up*. 00:07:40.240 --> 00:07:45.366 So, what you might do is, say you're given the problem of building the Eiffel Tower. 00:07:45.366 --> 00:07:50.171 And the Eiffel Tower I said is about 300 meters tall, so about a thousand feet tall. 00:07:50.211 --> 00:07:53.617 So, you might think, 'Oh crumbs! How are we going to build this?' 00:07:53.617 --> 00:07:58.949 So, one thing you might do is say, 'Imagine, instead of being 300 meters tall, 00:07:58.949 --> 00:08:02.953 it was just *three* meters tall. How would I go about building it, then?' 00:08:02.953 --> 00:08:06.430 And you might think, 'Well, I'd build a big, perhaps a scaffolding or 30 meters tall. 00:08:06.430 --> 00:08:10.278 I might build a scaffolding and just hoist things up to the top.' 00:08:10.575 --> 00:08:15.212 So, then you say, 'Well, OK, can I build  a scaffolding at 300 meters? Does that make sense?' 00:08:15.440 --> 00:08:20.935 Alternatively, you might, say, perhaps it's *300,000 miles tall* 00:08:20.935 --> 00:08:23.520 – basically reaching as high as the Moon. 00:08:24.559 --> 00:08:28.196 How might you build that now? Well, there's no way you're going to hoist things up a scaffold 00:08:28.196 --> 00:08:32.240 – all the workers at the top would have no oxygen because they'd be above the atmosphere. 00:08:32.657 --> 00:08:36.400 So, you might then think about  hoisting it up from the bottom, building it, 00:08:36.400 --> 00:08:40.240 building the top first, hoisting the whole thing up,  building the next layer, 00:08:40.240 --> 00:08:43.301 hoisting the whole thing up, building the next layer. You know... 00:08:43.301 --> 00:08:46.160 like jacking a car and then sticking bits underneath. 00:08:46.780 --> 00:08:51.605 So, *by just thinking of a completely different scale*, you start to think of *different kinds* 00:08:51.605 --> 00:08:55.530 of *solutions*. It forces you out of that fixation. 00:08:55.530 --> 00:09:00.885 You might *just swap things around*. I mean, this works quite well if you're worried about 00:09:00.885 --> 00:09:04.023 – that you're using some sort of racial or gender bias; 00:09:04.023 --> 00:09:07.260 you just swap the genders of the people involved in a story 00:09:07.260 --> 00:09:13.473 or swap their ethnic background, and often the way you look at the story differently 00:09:13.473 --> 00:09:17.240 might tell you something about some of the biases you bring to it. 00:09:17.680 --> 00:09:22.960 In politics, if you hear a statement from a  politician and you either react positively or 00:09:22.960 --> 00:09:27.680 negatively to it, it might be worth just thinking  what you'd imagine if that statement came to 00:09:27.680 --> 00:09:32.720 the mouth of another politician that was of a  different persuasion; how would you read it then? 00:09:33.107 --> 00:09:39.600 And it's not that you change your views  drastically by doing this, but it helps you to 00:09:39.600 --> 00:09:44.800 perhaps expose why you view these things  differently, and some of that might be valid 00:09:44.800 --> 00:09:50.000 reasons; sometimes you might think, 'Actually, I  need to rethink some of the ways I'm working.' 00:09:51.520 --> 00:09:54.520 So, back to the theory now. 00:09:54.913 --> 00:10:02.200 Psychologists, I said, and people studying the cognitive aspects of creativity talk about *insight problems*. 00:10:02.390 --> 00:10:07.000 And fixation is an aspect of insight problems. 00:10:07.085 --> 00:10:10.651 Some problems you can just start at the beginning and work your way through, 00:10:10.651 --> 00:10:14.796 and so long as you work enough, you get an answer out. Sometimes you can do it faster or slower... 00:10:14.796 --> 00:10:20.647 and you might be lucky in that you approach it and perhaps the first thing you try works. 00:10:20.687 --> 00:10:25.896 You know – and you might go to a maze and you might just say, 'Well, first of all I'll try the right-hand side. 00:10:25.896 --> 00:10:30.160 And if that doesn't work, I'll come back and then I'll turn to the left, try the left-hand side.' 00:10:30.280 --> 00:10:32.720 You can work your way through. 00:10:32.720 --> 00:10:37.264 But some problems aren't like that. No matter how hard you keep working at them, 00:10:37.264 --> 00:10:42.743 you can't solve them. You need some little spark, this bright idea, this different approach 00:10:42.743 --> 00:10:47.200 to come from a different direction in order to solve it. 00:10:47.750 --> 00:10:53.072 Now, that happens at a big scale with big problems, but also there are quite small-scale problems. 00:10:53.072 --> 00:10:57.677 And for experimentation purposes, psychologists often use these, 00:10:57.677 --> 00:11:02.521 give them to people and then perhaps  try and manipulate things, perhaps use suggestive words, 00:11:02.521 --> 00:11:06.720 do different things to help understand  what might change that. 00:11:07.332 --> 00:11:10.776 Here's a couple of examples; so, the first one is a more geometric one. 00:11:10.776 --> 00:11:13.512 And this is again often used in experimentation. 00:11:13.512 --> 00:11:17.063 So, what you have is – you might have seen it and you might know the solution, 00:11:17.063 --> 00:11:21.276 but you have three lines or three dots equally spaced. 00:11:21.276 --> 00:11:25.280 And your job is to try and have four straight lines, 00:11:25.280 --> 00:11:29.739 but where as you draw them you're not allowed  to lift your hand from the paper. 00:11:29.739 --> 00:11:35.075 So, and the straight lines have to go through – every dot has to have a line going through it. 00:11:35.075 --> 00:11:41.807 Now, you're allowed to have two lines going through the same dot, but every dot has to have at least one line. 00:11:41.807 --> 00:11:46.704 So, you can see that if you didn't have to lift your hand off the paper, you could easily do it with three lines 00:11:46.704 --> 00:11:50.117 – just go one, two, three, or one, two three. 00:11:50.117 --> 00:11:54.080 Or, if you keep on the paper, you could go up, down, like do a sort 00:11:54.080 --> 00:11:59.147 of diagonal through, zigzag through, and you could  do it very easily with five lines. 00:11:59.147 --> 00:12:03.120 Or, you could perhaps go around the edge and into the middle – try it. 00:12:03.224 --> 00:12:06.501 Five lines is easy without leaving the paper. 00:12:06.720 --> 00:12:10.168 Three lines is easy, so long as you're allowed to leave the paper. 00:12:10.168 --> 00:12:15.280 The problem is to do four straight lines and you're not allowed to leave the paper. 00:12:16.314 --> 00:12:20.557 I'll come back and give a solution to that later; I'll warn you before I do. 00:12:20.557 --> 00:12:25.425 But you might want to have a think about that as a problem if you've not come across it before. 00:12:25.425 --> 00:12:29.763 So, I'm going to give you another problem now, which is *the candle in the cellar problem*. 00:12:29.763 --> 00:12:34.618 So, you're going into a dark cellar and you are given a candle, 00:12:34.960 --> 00:12:40.220 some drawing pins, or sometimes just one drawing pin – if I can get one out of the box here. 00:12:40.220 --> 00:12:45.330 So, it's just a single drawing pin, but sometimes you're given several. 00:12:45.330 --> 00:12:51.952 And a box of matches – probably smaller than this, usually, is the example one, but a box of matches. 00:12:52.368 --> 00:12:58.320 And you're then given a bit of a story that you're going to have to use both your hands to lift something out of the cellar. 00:12:58.320 --> 00:13:03.843 But it's really dark, so you need the candle to (have) light, but you can't just hold the candle. 00:13:04.480 --> 00:13:10.560 And I'll give you *the solution*  to this now, and there's a standard solution. 00:13:10.560 --> 00:13:14.762 And this is interesting as an insight problem. But we'll come back to that in a bit. 00:13:14.762 --> 00:13:18.872 One of the standard solutions, or the standard solution perhaps, is to 00:13:18.872 --> 00:13:22.640 – and I won't empty them all out, because there's a lot of them, but – oh – I've got a smaller match box, 00:13:22.668 --> 00:13:26.521 I've just realized, inside my bigger match box – is that you 00:13:26.521 --> 00:13:31.183 basically take out the tray and use the tray as a support for your candle 00:13:31.183 --> 00:13:33.895 and then pin the tray to something. 00:13:33.895 --> 00:13:37.426 Obviously that depends on there being something to pin the tray to, 00:13:37.426 --> 00:13:40.717 the cardboard tray to hold the candle without it falling off. 00:13:40.717 --> 00:13:45.760 I'd probably want to try and find an angle  to put it into. But, anyway, there are solutions. 00:13:46.206 --> 00:13:51.120 What you find, though, is in order to have that solution, you have to stop thinking about this 00:13:51.120 --> 00:13:55.489 as a box that contains a match to light the candle. 00:13:55.489 --> 00:13:59.013 So, you actually have to think of the box as being 00:13:59.013 --> 00:14:01.887 something you can use to mechanically solve the problem; 00:14:01.887 --> 00:14:10.040 because what people typically do is they start off thinking of lots of ways of trying to use pins to support the candle 00:14:10.040 --> 00:14:13.972 because pins are for joining things; pins are for connecting things; 00:14:13.972 --> 00:14:17.871 you want to connect the candle some way to a wall or someplace to support it. 00:14:17.871 --> 00:14:21.695 And you have to stop thinking of this  as the way to light the candle 00:14:21.695 --> 00:14:24.435 and think of this as the way to *support* the candle. 00:14:24.475 --> 00:14:28.188 So, a different way of thinking – so that's one standard way. 00:14:30.000 --> 00:14:34.291 The thing with these insight problems is you have to *change your way of thinking*; 00:14:34.291 --> 00:14:37.280 you can't just  follow the way you were doing it before. 00:14:38.880 --> 00:14:44.000 The psychologists talk about *incubation*, so what  happens – you're trying to solve a problem like this, 00:14:44.000 --> 00:14:48.880 but often – and you'll have found it yourself – you go away, sometimes go to sleep and wake up the next day; 00:14:48.880 --> 00:14:53.680 if you're doing an experiment, they usually just get somebody to do some other tasks for a 00:14:53.680 --> 00:14:59.200 while that eats up their mind, and then return to the problem again. 00:14:59.280 --> 00:15:05.014 And very often when you return to the problem, you just think, 'Ah! Of course!' 00:15:05.014 --> 00:15:08.000 (Sound effect) And it comes out. 00:15:08.664 --> 00:15:12.680 And this is *incubation followed by insight* actually. 00:15:12.680 --> 00:15:17.398 So, you have the problem and then after the incubation often this insight just happens 00:15:17.398 --> 00:15:21.145 in a way that doesn't if you keep  on working at the problem. 00:15:21.799 --> 00:15:25.354 I said you've have probably encountered this yourself, both for big-scale problems 00:15:25.354 --> 00:15:28.360 and also this more puzzly kind of problem. 00:15:28.640 --> 00:15:33.217 So, how does this work? And here the jury is a little bit more out. 00:15:33.217 --> 00:15:36.479 It's well understood that this happens. 00:15:36.479 --> 00:15:42.557 It's less well understood, but... there are lots of works... there is a lot of understanding. 00:15:42.557 --> 00:15:46.193 But it's still less well understood *exactly* how it happens. 00:15:46.400 --> 00:15:50.880 Some of it is clearly just about the fact that you  let the problem out of your mind, but you know what 00:15:50.880 --> 00:15:56.373 the problem is, so you don't have to get it explained again. And of course, the *very explaining of the problem* 00:15:56.752 --> 00:16:00.000 *may create the fixation*. 00:16:00.000 --> 00:16:03.221 So, for instance... as soon as you see the candle and the matches, 00:16:03.221 --> 00:16:06.648 you instantly think 'candles to light, match'; 00:16:06.648 --> 00:16:10.430 whereas when you come back to it, you don't have to think that, because you just know it's the problem. 00:16:10.430 --> 00:16:15.000 So, sometimes it's about getting out of your mind and then your mind is fresh. 00:16:15.896 --> 00:16:20.134 The solution strategies you were following of course are so much in your mind when you're doing it the first 00:16:20.134 --> 00:16:24.696 time, you can't think of anything else; whereas when you come back to it, 00:16:24.696 --> 00:16:28.440 suddenly there's a possibility of  thinking a new way – you're breaking the fixation 00:16:28.440 --> 00:16:32.829 purely by not being there  and then coming back to it. 00:16:33.200 --> 00:16:39.940 There's also quite a lot of discussion about how much happens *unconsciously* during this incubation period. 00:16:39.940 --> 00:16:46.720 And the debates on this – they're clearly things  do happen, that you're in a better position to 00:16:46.720 --> 00:16:52.474 solve the problem often when  you come to it later than before. 00:16:52.920 --> 00:16:57.235 Why that is is interesting. There are some famous things about this; 00:16:57.235 --> 00:17:00.377 like the guy who discovered benzene rings. 00:17:00.377 --> 00:17:05.857 He was trying to work out how on earth their  carbon and hydrogen atoms fitted together. 00:17:05.857 --> 00:17:11.699 And one of the various stories that are  related is that he imagined these 00:17:11.699 --> 00:17:16.246 – sometimes it's snakes swallowing each other or people dancing with each other hand to hand – 00:17:16.246 --> 00:17:22.173 and then actually does this sort of half asleep and then wakes up and thinks, 'Aha!' 00:17:22.173 --> 00:17:28.091 And one way of looking at that is to say that his unconscious mind was going through all the problem 00:17:28.091 --> 00:17:33.000 and actually had found the solution, was telling him through his dreams. 00:17:33.680 --> 00:17:37.633 Another way to think about this is that actually, of course, especially if 00:17:37.633 --> 00:17:40.888 your mind is wandering, various thoughts come along; 00:17:40.888 --> 00:17:46.000 you have a problem sitting there. If one of those thoughts happens to sort of, say, match the problem, 00:17:46.000 --> 00:17:49.131 you think, 'Aha! I've got it!' 00:17:49.131 --> 00:17:53.002 So, again, that  guy probably woke up a dozen times or 00:17:53.002 --> 00:17:57.211 dozens of times with all sorts of half dreams in his head, 00:17:57.211 --> 00:18:00.200 most of which were completely irrelevant. 00:18:00.240 --> 00:18:05.775 When he wakes up with the one that *is* relevant, of course it suddenly matches the problem and he sees the solution. 00:18:05.775 --> 00:18:11.421 So, it's not clear whether it's random, whether it's thinking in your unconscious. 00:18:11.760 --> 00:18:15.787 But what is well understood is that this makes  a difference. 00:18:15.787 --> 00:18:19.221 So, again from a practical point of view, doing this is something you do, 00:18:19.221 --> 00:18:23.000 making sure you have these breaks when you get stuck. 00:18:24.811 --> 00:18:27.574 So, I'll just give you the dots one. 00:18:27.574 --> 00:18:33.690 One of the nice things about this dots example, it is *literally about thinking out of the box*, 00:18:33.690 --> 00:18:37.520 because the dots create a box and what happens is 00:18:37.520 --> 00:18:44.130 certainly a lot of people tend to try and draw  the lines often starting at the dots. 00:18:44.130 --> 00:18:47.061 The other thing is, how do you draw a line when you've got dots? 00:18:47.061 --> 00:18:50.632 You naturally tend to think of starting on a dot and ending on a dot. 00:18:50.632 --> 00:18:55.000 As soon as you allow your  lines to escape the box, and 00:18:55.477 --> 00:19:00.394 – I was supposed to warn you before, so hopefully you've noticed this if you wanted to think about it; 00:19:00.394 --> 00:19:03.842 if not, stop me now and go off and think more. 00:19:03.842 --> 00:19:07.840 But if you start to think *outside the box*, suddenly solutions come. 00:19:07.840 --> 00:19:10.879 Here is the canonical solution. 00:19:10.879 --> 00:19:14.360 So, you start at the bottom corner; you go *outside* the box, 00:19:14.360 --> 00:19:19.211 and then cut diagonally across, so it's almost like there's an extra dot at the outside. 00:19:19.211 --> 00:19:22.354 And this is weird. If you added dots, this gets easier; 00:19:22.354 --> 00:19:25.543 it's one where you add dots, suddenly the problem gets easier. 00:19:25.543 --> 00:19:29.463 Then you come back down to that bottom left-hand corner, 00:19:29.463 --> 00:19:32.949 and then you take yourself back out again, 00:19:32.949 --> 00:19:37.840 and you've gone through all the dots –  four straight lines, pencil never left the paper. 00:19:40.720 --> 00:19:46.320 So, it's interesting. Now I do know that one well. But there's quite a few times where I saw that 00:19:46.320 --> 00:19:50.573 and thought 'Oh, that's clever!' and then came to it later and found I'd actually forgotten that solution 00:19:50.573 --> 00:19:54.000 and got stuck in exactly the  same way as I had before. 00:19:54.000 --> 00:19:58.441 But you notice here, there's a lot of assumptions that you bring to a problem, 00:19:58.441 --> 00:20:03.280 like the fact that lines perhaps start and end on dots  might be one of the ones that was 00:20:03.280 --> 00:20:08.000 going through your head and stopping you  thinking about some of these solutions.