WEBVTT 00:00:06.230 --> 00:00:08.990 SRINI DEVADAS: So today, we're going 00:00:08.990 --> 00:00:11.300 to look at that classic puzzle. 00:00:11.300 --> 00:00:14.060 It's called the eight queens puzzle, 00:00:14.060 --> 00:00:16.309 and it's very easy to describe. 00:00:16.309 --> 00:00:19.910 I'll write down the rules, if you happen not to know chess 00:00:19.910 --> 00:00:23.430 and how a queen moves on a chess board, in a moment. 00:00:23.430 --> 00:00:25.550 But before I get into that, I just 00:00:25.550 --> 00:00:29.780 wanted to mention that the programming paradigms 00:00:29.780 --> 00:00:33.400 that we're going to be looking at today are-- 00:00:33.400 --> 00:00:35.600 is, I should say the programming paradigm-- 00:00:35.600 --> 00:00:39.160 is one of exhaustive enumeration. 00:00:39.160 --> 00:00:45.080 And what we want to do algorithmically 00:00:45.080 --> 00:00:48.260 is to look at all possible cases, 00:00:48.260 --> 00:00:51.020 all possible combinations, possibilities, 00:00:51.020 --> 00:00:54.470 because we want to exhaust the space 00:00:54.470 --> 00:00:56.570 to determine if there's a solution 00:00:56.570 --> 00:01:01.320 to this particular instance of the problem or not. 00:01:01.320 --> 00:01:04.019 And one of the problems, as I mentioned, 00:01:04.019 --> 00:01:05.360 is the eight queen's problem. 00:01:05.360 --> 00:01:08.660 But you could say, I'd like to solve a smaller 00:01:08.660 --> 00:01:12.350 problem, the five queens problem or the three queens problem. 00:01:12.350 --> 00:01:16.280 And you'd like, obviously, to be able to write code that 00:01:16.280 --> 00:01:20.600 solves this generalized problem, where 00:01:20.600 --> 00:01:24.210 you can vary the number of rows and columns on a board. 00:01:24.210 --> 00:01:26.870 It's no longer a chessboard if it's not eight rows and eight 00:01:26.870 --> 00:01:30.080 columns, but you can still obviously concoct 00:01:30.080 --> 00:01:33.800 a puzzle associated with an arbitrary-sized board. 00:01:33.800 --> 00:01:35.960 And then the other thing that we'll do 00:01:35.960 --> 00:01:39.260 today is look at data structures, 00:01:39.260 --> 00:01:45.800 and so this obviously is going to be a data structure that's 00:01:45.800 --> 00:01:47.900 immediately clear from what I have 00:01:47.900 --> 00:01:50.300 up there on the board, a matrix data structure 00:01:50.300 --> 00:01:51.440 to solve the problem. 00:01:51.440 --> 00:01:53.840 But can you do things more efficiently 00:01:53.840 --> 00:01:56.730 with a different selection of a data structure? 00:01:56.730 --> 00:01:58.100 All right? 00:01:58.100 --> 00:02:02.540 So the eight queens problem simply 00:02:02.540 --> 00:02:13.610 says, place eight queens on a chessboard-- 00:02:13.610 --> 00:02:18.590 and that implies immediately that it's eight by eight, 00:02:18.590 --> 00:02:21.050 as I've drawn over there-- 00:02:21.050 --> 00:02:33.470 such that no queen attacks any other queen. 00:02:37.560 --> 00:02:41.030 Now, if you had no background in solving this problem, 00:02:41.030 --> 00:02:46.164 perhaps you knew chess, and you might think there's a solution. 00:02:46.164 --> 00:02:47.330 You might think there's not. 00:02:47.330 --> 00:02:48.788 I mean, a queen is pretty powerful. 00:02:52.430 --> 00:02:55.320 This implies, because a queen can move horizontally. 00:02:55.320 --> 00:02:56.690 It can move vertically. 00:02:56.690 --> 00:03:02.060 It can move diagonally, back and forth, in both directions, 00:03:02.060 --> 00:03:07.700 up and down, left and right, and I guess up and down diagonally. 00:03:07.700 --> 00:03:09.590 It's a powerful piece. 00:03:09.590 --> 00:03:11.480 It's the most powerful piece in chess. 00:03:11.480 --> 00:03:16.619 And first time I heard about this problem, 00:03:16.619 --> 00:03:19.160 my guess was that there wasn't a solution to the eight queens 00:03:19.160 --> 00:03:21.080 problem. 00:03:21.080 --> 00:03:22.940 And you know, how many of you think there's 00:03:22.940 --> 00:03:25.342 a solution to this problem? 00:03:25.342 --> 00:03:27.050 Do you know there's a solution, or do you 00:03:27.050 --> 00:03:28.130 think there's a solution? 00:03:28.130 --> 00:03:31.249 AUDIENCE: For sure, because knights, 00:03:31.249 --> 00:03:33.298 because they can't move like in the way 00:03:33.298 --> 00:03:35.510 a knight can, so it's got to be-- 00:03:35.510 --> 00:03:38.160 SRINI DEVADAS: So Ganatra says there's a solution. 00:03:38.160 --> 00:03:41.480 How many of you, before you heard him, 00:03:41.480 --> 00:03:44.430 thought there wasn't a solution? 00:03:44.430 --> 00:03:46.632 Well, you know, years ago, I was convinced 00:03:46.632 --> 00:03:48.840 that there was no solution, because I tried very hard 00:03:48.840 --> 00:03:50.070 and I couldn't find one. 00:03:50.070 --> 00:03:52.560 I tried very hard for a half hour, OK? 00:03:52.560 --> 00:03:56.080 And then I decided there wasn't one. 00:03:56.080 --> 00:04:01.440 But then, you know, now at this point, it got to the point 00:04:01.440 --> 00:04:04.980 where I could write a computer program in about a half hour 00:04:04.980 --> 00:04:06.930 or an hour to solve this problem, 00:04:06.930 --> 00:04:09.120 and I did discover a solution. 00:04:09.120 --> 00:04:10.500 So I'm giving it away. 00:04:10.500 --> 00:04:12.670 It turns out there's many solutions, 00:04:12.670 --> 00:04:18.720 and there's certainly a slightly different code 00:04:18.720 --> 00:04:22.830 for finding one solution versus finding all of them, 00:04:22.830 --> 00:04:25.080 and we'll do both. 00:04:25.080 --> 00:04:29.850 So without further ado, you want to translate 00:04:29.850 --> 00:04:36.650 the movement of the queens into a set of rules 00:04:36.650 --> 00:04:41.730 that tell you if you have found a solution or not. 00:04:41.730 --> 00:04:45.320 And there's essentially three rules, 00:04:45.320 --> 00:04:50.375 because a queen can move in three different ways. 00:04:55.470 --> 00:04:57.790 No two queens can be on the same column. 00:04:57.790 --> 00:05:00.960 No two queens can be in the same row. 00:05:00.960 --> 00:05:03.780 No two queens can be in the same diagonal. 00:05:03.780 --> 00:05:06.446 And remember, there's two diagonals, one of which 00:05:06.446 --> 00:05:08.820 that goes this way, and the other one that goes that way. 00:05:18.050 --> 00:05:23.390 Right, so there's a lot of combinations here. 00:05:23.390 --> 00:05:27.980 If you think about this and you look at this board over here, 00:05:27.980 --> 00:05:34.420 I have eight different rows and eight different columns. 00:05:34.420 --> 00:05:42.620 And the way that someone might try to solve this problem 00:05:42.620 --> 00:05:49.070 is go either left or right or top down. 00:05:49.070 --> 00:05:53.360 And I've written the constraint no two queens can 00:05:53.360 --> 00:05:55.310 be in the same column first. 00:05:55.310 --> 00:05:57.910 So you can imagine, then, this is what we're going to do. 00:05:57.910 --> 00:06:01.380 You could certainly modify the code and modify our strategy, 00:06:01.380 --> 00:06:03.020 but we're going to go left to right. 00:06:03.020 --> 00:06:06.080 So what we're going to do is, we know if we put a queen up here, 00:06:06.080 --> 00:06:11.000 we can't put a queen in any other row on that first column, 00:06:11.000 --> 00:06:13.280 obviously, because of the first constraint. 00:06:13.280 --> 00:06:15.320 So maybe I'll put a queen up here. 00:06:15.320 --> 00:06:18.680 And then at that point, I can move to the second column 00:06:18.680 --> 00:06:21.770 in my manual strategy. 00:06:21.770 --> 00:06:26.060 And I couldn't put a queen up here 00:06:26.060 --> 00:06:27.800 because of the second constraint, 00:06:27.800 --> 00:06:29.910 no two queens can be on the same row. 00:06:29.910 --> 00:06:32.490 And if I had a queen up here-- let me just write that out. 00:06:32.490 --> 00:06:35.030 So I can't put one here, I can't put one here. 00:06:35.030 --> 00:06:36.650 I could put one here, and this goes 00:06:36.650 --> 00:06:38.191 back to what Ganatra said, about this 00:06:38.191 --> 00:06:40.230 is how a knight moves in chess. 00:06:40.230 --> 00:06:44.540 And so that gets me two queens. 00:06:44.540 --> 00:06:47.120 And obviously, not a solution to the problem of the eight 00:06:47.120 --> 00:06:51.570 queens, but maybe I'm a quarter of the way there, right? 00:06:51.570 --> 00:06:55.040 And it turns out that you have to do 00:06:55.040 --> 00:06:57.500 what is called backtracking. 00:06:57.500 --> 00:07:00.260 And backtracking simply means you 00:07:00.260 --> 00:07:03.650 may have to undo a decision you made 00:07:03.650 --> 00:07:07.010 with respect to the placement of a particular queen 00:07:07.010 --> 00:07:10.310 in this exhaustive search strategy, right? 00:07:10.310 --> 00:07:13.400 And the important thing when you do 00:07:13.400 --> 00:07:15.920 a search, as I mentioned early on, 00:07:15.920 --> 00:07:19.520 you have to convince yourself that the code you've written 00:07:19.520 --> 00:07:21.920 is going to exhaust all of the possibilities, 00:07:21.920 --> 00:07:24.650 and that it won't miss solutions. 00:07:24.650 --> 00:07:26.750 And if you don't exhaust all the possibilities, 00:07:26.750 --> 00:07:28.910 your code will terminate, and you 00:07:28.910 --> 00:07:31.910 may throw up your hands, because you had buggy code, 00:07:31.910 --> 00:07:34.820 or because you had a buggy search strategy, 00:07:34.820 --> 00:07:36.680 and say, there's no solution to the problem, 00:07:36.680 --> 00:07:38.600 whereas there is a solution to the problem. 00:07:38.600 --> 00:07:41.080 All right? 00:07:41.080 --> 00:07:44.780 And the way you do this search, this exhaustive search-- 00:07:44.780 --> 00:07:50.300 because it's quite intense, there's a lot of combinations, 00:07:50.300 --> 00:07:55.200 and they grow exponentially with the number of rows and columns 00:07:55.200 --> 00:07:56.810 on our board-- 00:07:56.810 --> 00:07:58.680 is important. 00:07:58.680 --> 00:08:00.590 And so you want to be careful that you 00:08:00.590 --> 00:08:04.190 don't do redundant work, that you don't do things 00:08:04.190 --> 00:08:08.610 that you don't really have to, because the combinations blow 00:08:08.610 --> 00:08:09.110 up. 00:08:09.110 --> 00:08:12.350 So there's hundreds of millions of combinations with respect 00:08:12.350 --> 00:08:16.190 to putting queens up here, so if you really think about it, 00:08:16.190 --> 00:08:19.100 there's like eight different places 00:08:19.100 --> 00:08:23.610 you can put a queen here, eight, eight, eight et cetera. 00:08:23.610 --> 00:08:26.950 So you know, that's 8 raised to 8 right? 00:08:26.950 --> 00:08:28.620 If you think about that. 00:08:28.620 --> 00:08:30.200 And that's a large number, right? 00:08:30.200 --> 00:08:34.610 And if you translate that, I don't even know what that is. 00:08:34.610 --> 00:08:35.830 But you could do-- 00:08:35.830 --> 00:08:39.110 you know, this is 2 raised to 3, raised to 8, 00:08:39.110 --> 00:08:45.260 so that's 2 raised to 24, which is in the millions. 00:08:45.260 --> 00:08:48.610 And if you had larger board-- 00:08:48.610 --> 00:08:52.700 it isn't a chessboard-- then it would be much, much larger 00:08:52.700 --> 00:08:54.860 as you grew the size of the board. 00:08:54.860 --> 00:08:58.280 But that 2 raised to 24 doesn't scare us these days, 00:08:58.280 --> 00:09:00.020 because computers are so fast. 00:09:00.020 --> 00:09:04.190 But back when I was your age, that was a large number, OK? 00:09:04.190 --> 00:09:06.800 And you didn't want to write code 00:09:06.800 --> 00:09:10.430 that took 2 raised to 24 steps to solve problems. 00:09:10.430 --> 00:09:13.580 Now it takes a matter of fractions of seconds 00:09:13.580 --> 00:09:15.380 to run through those kinds of combinations, 00:09:15.380 --> 00:09:18.530 because you're running at a gigahertz on computers. 00:09:18.530 --> 00:09:21.140 But before we dive into the specifics 00:09:21.140 --> 00:09:25.370 of a particular strategy, let's look at smaller problems. 00:09:25.370 --> 00:09:29.120 You always get intuition about problems 00:09:29.120 --> 00:09:31.650 by shrinking them and making them simpler. 00:09:31.650 --> 00:09:35.630 So let's look at the case of a two-by-two board. 00:09:35.630 --> 00:09:38.900 Can I solve the two queens problem on a two-by-two board? 00:09:38.900 --> 00:09:40.130 No. 00:09:40.130 --> 00:09:43.910 So if I put a queen here, well, immediately I cannot put 00:09:43.910 --> 00:09:44.870 a queen-- 00:09:44.870 --> 00:09:47.750 it attacks all of these other positions. 00:09:47.750 --> 00:09:50.210 So the two-by-two does not have a solution. 00:09:50.210 --> 00:09:51.841 Let's look at a three-by-three. 00:09:55.761 --> 00:09:56.260 Right. 00:09:56.260 --> 00:09:59.262 So let's say I put a queen up here. 00:09:59.262 --> 00:10:01.720 So the way I'm doing this, by the way, is column by column, 00:10:01.720 --> 00:10:03.680 and that's kind of going to be our strategy. 00:10:03.680 --> 00:10:05.630 As I said, you could flip it to row by row, 00:10:05.630 --> 00:10:08.510 but no reason to do both. 00:10:08.510 --> 00:10:09.852 Just pick one. 00:10:09.852 --> 00:10:11.060 And so I put a queen up here. 00:10:11.060 --> 00:10:13.010 I can't put one here, I can't put one here, 00:10:13.010 --> 00:10:14.300 I can't put one here, right? 00:10:14.300 --> 00:10:16.250 So I go ahead and put that over here. 00:10:16.250 --> 00:10:17.630 Can I put a queen here? 00:10:17.630 --> 00:10:19.220 No, because that attacks it. 00:10:19.220 --> 00:10:20.360 Can I put a queen here? 00:10:20.360 --> 00:10:22.190 No, because this attacks it. 00:10:22.190 --> 00:10:24.140 And clearly, I can't do it over here. 00:10:24.140 --> 00:10:26.270 So am I done? 00:10:26.270 --> 00:10:27.560 Can I just give up now? 00:10:30.510 --> 00:10:33.000 What should I do? 00:10:33.000 --> 00:10:34.170 I don't want to give up now. 00:10:34.170 --> 00:10:37.260 I don't want to say that a three-by-three-- at this point, 00:10:37.260 --> 00:10:39.360 I don't know what the answer is in terms 00:10:39.360 --> 00:10:42.460 of whether a solution exists for a three-by-three or not. 00:10:42.460 --> 00:10:47.790 But at this point, given that I put a queen up here, 00:10:47.790 --> 00:10:53.760 and I went through this process, I cannot give up, right? 00:10:53.760 --> 00:10:55.180 And why cannot I give up? 00:10:55.180 --> 00:10:58.057 What should I do next, once I've failed here? 00:10:58.057 --> 00:11:00.640 I put a queen here and a queen here, and I said, um, you know, 00:11:00.640 --> 00:11:02.220 I can't put a queen anywhere here. 00:11:02.220 --> 00:11:03.410 What do I do? 00:11:03.410 --> 00:11:05.620 Right? 00:11:05.620 --> 00:11:06.590 Yeah, go ahead, Fadi. 00:11:06.590 --> 00:11:08.965 AUDIENCE: You can change the position of the first queen. 00:11:08.965 --> 00:11:10.714 SRINI DEVADAS: You can change the position 00:11:10.714 --> 00:11:11.510 of the first queen. 00:11:11.510 --> 00:11:13.070 That's exactly right. 00:11:13.070 --> 00:11:17.000 So I need to backtrack, and so I tried a few different things 00:11:17.000 --> 00:11:17.624 here. 00:11:17.624 --> 00:11:19.040 And it was exactly the same thing, 00:11:19.040 --> 00:11:20.780 where I had a queen here. 00:11:20.780 --> 00:11:24.830 And let's say I have a very straightforward strategy that 00:11:24.830 --> 00:11:28.700 is going to put queens down and run a piece of code that is 00:11:28.700 --> 00:11:30.260 going to check for conflicts. 00:11:30.260 --> 00:11:35.930 This is going to be our most important subroutine that we're 00:11:35.930 --> 00:11:38.600 going to write, and it's going to be dependent on the data 00:11:38.600 --> 00:11:42.890 structure that we picked, but that is our fundamental check, 00:11:42.890 --> 00:11:44.840 you know, the conflict checker. 00:11:44.840 --> 00:11:46.167 So I could put a queen here. 00:11:46.167 --> 00:11:47.750 And obviously, with one queen, there's 00:11:47.750 --> 00:11:49.560 no reason to run the conflict checker. 00:11:49.560 --> 00:11:51.860 When I put a queen here, I run the conflict checker, 00:11:51.860 --> 00:11:53.790 and it says, conflict. 00:11:53.790 --> 00:11:54.750 Here, conflict. 00:11:54.750 --> 00:11:55.970 Here, no conflict. 00:11:55.970 --> 00:11:56.570 Right? 00:11:56.570 --> 00:11:59.540 And then I move on to the next, and I get conflict 00:11:59.540 --> 00:12:01.310 for all three of them, right? 00:12:01.310 --> 00:12:03.650 So at this point, I have to have a way in my code-- 00:12:03.650 --> 00:12:07.130 this is exactly the enumeration that I need in my code, 00:12:07.130 --> 00:12:10.790 to go back and say, just like I tried different combinations 00:12:10.790 --> 00:12:13.190 for the second column before I converged 00:12:13.190 --> 00:12:15.860 on putting a queen down here, I need 00:12:15.860 --> 00:12:21.380 to go back and change this to do this. 00:12:21.380 --> 00:12:25.720 So I put a queen here, then can I put a queen here? 00:12:25.720 --> 00:12:26.880 Here? 00:12:26.880 --> 00:12:27.920 Here? 00:12:27.920 --> 00:12:28.900 No, right? 00:12:28.900 --> 00:12:31.430 So that doesn't work either. 00:12:31.430 --> 00:12:35.350 So then I go up and put a queen here, 00:12:35.350 --> 00:12:39.020 and now you know where this is headed, because of symmetry. 00:12:39.020 --> 00:12:41.840 The only place I can put a queen is over here. 00:12:41.840 --> 00:12:43.790 And once I put the queen over there, 00:12:43.790 --> 00:12:46.644 I can't put a queen in any of these spots. 00:12:46.644 --> 00:12:49.060 So it turns out a three-by-three does not have a solution. 00:12:49.060 --> 00:12:52.600 At this point, because I've gone through all of the combinations 00:12:52.600 --> 00:12:58.510 with respect to the starting points on the first column, 00:12:58.510 --> 00:13:01.090 and then with each of those starting points, 00:13:01.090 --> 00:13:03.700 I've gone through all of the combinations 00:13:03.700 --> 00:13:07.060 in the second column and the third column, 00:13:07.060 --> 00:13:09.790 I've effectively done the 3 raised 00:13:09.790 --> 00:13:13.750 to 3, which was basically what I described to you previously 00:13:13.750 --> 00:13:15.950 with respect to these different combinations. 00:13:15.950 --> 00:13:19.750 Now I don't know if the 8-- 00:13:19.750 --> 00:13:22.210 we kind of think at this point that there 00:13:22.210 --> 00:13:25.540 is a solution in those 8 raised to 8 combinations for the eight 00:13:25.540 --> 00:13:26.800 queens problem. 00:13:26.800 --> 00:13:29.352 But there was no solution in the 2 raised 00:13:29.352 --> 00:13:32.920 to 2 combinations for the two queens problem. 00:13:32.920 --> 00:13:36.565 And there were no solutions in the 3 raised 00:13:36.565 --> 00:13:41.280 to 3 combinations or placements for the three queens problem. 00:13:41.280 --> 00:13:41.920 OK? 00:13:41.920 --> 00:13:42.970 So let's do four queens. 00:13:48.060 --> 00:13:51.450 And the reason I'm doing this is I 00:13:51.450 --> 00:13:56.590 want to point out one thing, which is an efficiency 00:13:56.590 --> 00:14:00.430 trick that is going to be important to make sure 00:14:00.430 --> 00:14:02.560 that our code runs in a reasonable amount of time. 00:14:02.560 --> 00:14:03.060 Right? 00:14:03.060 --> 00:14:05.830 And which is kind of implicit in what we've done here, 00:14:05.830 --> 00:14:07.470 but I want to point it out. 00:14:07.470 --> 00:14:09.970 So in the case of four queens, I'm going to go off 00:14:09.970 --> 00:14:12.010 and I'm going to put a queen up here. 00:14:12.010 --> 00:14:14.620 And I'm going to run through this fairly quickly. 00:14:14.620 --> 00:14:16.750 No, no, yes. 00:14:16.750 --> 00:14:18.700 And so I move on. 00:14:18.700 --> 00:14:24.401 No, no, no, no, no. 00:14:24.401 --> 00:14:24.900 Right? 00:14:24.900 --> 00:14:28.470 So oops-- but now, I don't need to go back 00:14:28.470 --> 00:14:30.750 all the way to the first. 00:14:30.750 --> 00:14:33.920 I can put a queen here, right? 00:14:33.920 --> 00:14:38.880 So now, no, yes, so I get a little further. 00:14:38.880 --> 00:14:41.920 And then I could move to the fourth column. 00:14:41.920 --> 00:14:45.402 No, no, no, no. 00:14:45.402 --> 00:14:47.610 I mean, they attacked from here or attack from there. 00:14:47.610 --> 00:14:48.840 Oops, all right. 00:14:48.840 --> 00:14:52.310 Let's just go-- can I put it over here? 00:14:52.310 --> 00:14:53.720 No, no. 00:14:53.720 --> 00:14:57.510 I went back to the most recently placed queen, OK? 00:14:57.510 --> 00:15:00.860 So one thing that is important, which you kind of get a sense 00:15:00.860 --> 00:15:03.140 for when you increase the size of the board, 00:15:03.140 --> 00:15:07.460 is when you fail, when check conflicts returns false, 00:15:07.460 --> 00:15:10.730 when you fail, you go back to the most recent decision, most 00:15:10.730 --> 00:15:14.150 recent column that you've placed your queen, 00:15:14.150 --> 00:15:17.240 and you change that, and you enumerate those possibilities. 00:15:17.240 --> 00:15:17.960 OK? 00:15:17.960 --> 00:15:19.410 And when you do that-- 00:15:19.410 --> 00:15:21.720 the other thing that we've done, by the way, 00:15:21.720 --> 00:15:27.920 is for the first queen, we put a queen down with impunity. 00:15:27.920 --> 00:15:32.360 For the second queen, we check that lone, existing queen 00:15:32.360 --> 00:15:35.510 on the board for a conflict, right? 00:15:35.510 --> 00:15:40.910 For the third queen, what am I doing here for the third queen? 00:15:40.910 --> 00:15:42.890 What conflicts am I checking when 00:15:42.890 --> 00:15:45.320 I place the third queen on one of the rows 00:15:45.320 --> 00:15:48.370 of the third column, specifically? 00:15:48.370 --> 00:15:50.200 What conflicts am I checking? 00:15:50.200 --> 00:15:54.290 I mean, who am I checking with? 00:15:54.290 --> 00:15:55.380 Yeah, go ahead back there. 00:15:55.380 --> 00:15:57.070 AUDIENCE: The two queens that you-- 00:15:57.070 --> 00:15:57.910 SRINI DEVADAS: Yeah, the two queens. 00:15:57.910 --> 00:15:59.701 And what am I doing for the existing queens 00:15:59.701 --> 00:16:00.970 that are already on the board? 00:16:00.970 --> 00:16:02.550 Do I do anything for them? 00:16:02.550 --> 00:16:03.220 No. 00:16:03.220 --> 00:16:06.470 So I guess it doesn't seem like much, 00:16:06.470 --> 00:16:08.451 but it's an important notion. 00:16:08.451 --> 00:16:08.950 OK? 00:16:08.950 --> 00:16:11.140 The notion here is I'm doing a certain amount 00:16:11.140 --> 00:16:15.880 of incremental checking when I place a new queen, 00:16:15.880 --> 00:16:19.390 and it's only the new relationships that 00:16:19.390 --> 00:16:20.680 the new queen-- 00:16:20.680 --> 00:16:22.930 or the conflicts that the new queen would have that 00:16:22.930 --> 00:16:24.460 need to be checked. 00:16:24.460 --> 00:16:27.640 And in general, you know, if I had seven queens up 00:16:27.640 --> 00:16:30.370 on the board, and I've set it up nicely 00:16:30.370 --> 00:16:33.070 with these seven queens-- they're all happy, right? 00:16:33.070 --> 00:16:35.080 They're non-conflicting. 00:16:35.080 --> 00:16:40.210 No reason to go look at each of those pairs of seven queens 00:16:40.210 --> 00:16:42.610 when I put an eighth queen down, as I change the position 00:16:42.610 --> 00:16:43.870 of the eighth queen. 00:16:43.870 --> 00:16:47.740 All I care about is that the eighth queen doesn't conflict 00:16:47.740 --> 00:16:49.420 with the first seven queens. 00:16:49.420 --> 00:16:53.050 So I have to check, in general, if I'm putting the k-th queen 00:16:53.050 --> 00:16:55.360 down, and I'm effectively checking 00:16:55.360 --> 00:17:03.354 for k minus 1 pairs of conflicts, or k minus 1 queens 00:17:03.354 --> 00:17:04.270 were already put down. 00:17:04.270 --> 00:17:06.460 The k-th queen is being put down, 00:17:06.460 --> 00:17:09.530 and I need to check for the ones that already existed. 00:17:09.530 --> 00:17:11.510 I don't have to do k minus 1 squared, right? 00:17:11.510 --> 00:17:14.799 You know, because the k minus 1 were in there. 00:17:14.799 --> 00:17:17.410 There's different pairs over there. 00:17:17.410 --> 00:17:18.972 I'm all done with that, all right? 00:17:18.972 --> 00:17:19.930 I'm all done with that. 00:17:19.930 --> 00:17:22.819 All right, so let's keep going here. 00:17:22.819 --> 00:17:23.920 So where was I? 00:17:23.920 --> 00:17:26.470 I forget exactly where I was at, but we 00:17:26.470 --> 00:17:28.540 decided that this didn't work, because I 00:17:28.540 --> 00:17:31.060 failed on column four. 00:17:31.060 --> 00:17:34.540 But I can put it over here, and I can put it over here, right? 00:17:34.540 --> 00:17:41.260 So now, at this point, I've gone through and I've said, 00:17:41.260 --> 00:17:45.980 look, what it means now is that for these two positions 00:17:45.980 --> 00:17:48.340 for the first two columns, there's 00:17:48.340 --> 00:17:51.130 no solution to the four queens problem. 00:17:51.130 --> 00:17:52.530 OK? 00:17:52.530 --> 00:18:02.870 And so I exhausted the positions, for this position. 00:18:02.870 --> 00:18:05.830 I've also exhausted-- so I can say something even stronger 00:18:05.830 --> 00:18:08.110 than that, because for this position, 00:18:08.110 --> 00:18:10.780 I looked at the two possibilities, which 00:18:10.780 --> 00:18:14.110 were this and that, and neither of them work as well, right? 00:18:14.110 --> 00:18:18.010 So the stronger statement now is that there's 00:18:18.010 --> 00:18:20.650 no solution to the four queens problem 00:18:20.650 --> 00:18:23.005 if you put a queen on the top-left corner, right? 00:18:23.005 --> 00:18:25.630 But it doesn't mean that there's no solution to the four queens 00:18:25.630 --> 00:18:26.330 problem. 00:18:26.330 --> 00:18:27.688 Ganatra you had a question. 00:18:27.688 --> 00:18:28.689 AUDIENCE: Can't you also say there's 00:18:28.689 --> 00:18:30.897 no solution when you put the queen on the bottom one? 00:18:30.897 --> 00:18:33.220 SRINI DEVADAS: Symmetry, yes, wonderful. 00:18:33.220 --> 00:18:37.150 So it turns out that you can do rotations and reflections, 00:18:37.150 --> 00:18:40.900 and if you are willing to take your code. 00:18:40.900 --> 00:18:43.060 Remember that computers are dumb. 00:18:43.060 --> 00:18:45.670 You have to explain to them what a rotation is 00:18:45.670 --> 00:18:47.050 and what a reflection is. 00:18:47.050 --> 00:18:48.790 But if you do that, then it turns out 00:18:48.790 --> 00:18:52.480 you can get significant reductions in 8 raised to 8, 00:18:52.480 --> 00:18:53.440 or what have you. 00:18:53.440 --> 00:18:54.010 Right? 00:18:54.010 --> 00:18:55.690 And I'll go back and I'll tell you 00:18:55.690 --> 00:18:59.170 about the number of solutions to the eight queens problem, 00:18:59.170 --> 00:19:01.120 if you just looked at it this way, 00:19:01.120 --> 00:19:04.420 without rotating yourself or reflecting, 00:19:04.420 --> 00:19:06.940 and then tell you how many unique solutions there actually 00:19:06.940 --> 00:19:10.450 are, if you took into account the symmetries associated 00:19:10.450 --> 00:19:12.280 with rotation and reflection. 00:19:12.280 --> 00:19:16.510 And the last part is important, because it can cut down 00:19:16.510 --> 00:19:19.540 your combinatorial search, if you do take that into account, 00:19:19.540 --> 00:19:20.710 right? 00:19:20.710 --> 00:19:24.400 But as I said, you know, for our purposes, 2 raised to 24 00:19:24.400 --> 00:19:27.370 is not a large number anymore, so we can just 00:19:27.370 --> 00:19:29.050 blast through and not have to do that. 00:19:29.050 --> 00:19:30.370 All right? 00:19:30.370 --> 00:19:33.910 But now what I need to do is I've erased everything. 00:19:33.910 --> 00:19:36.240 You know, I tried that first queen in the corner. 00:19:36.240 --> 00:19:38.140 It's probably not going to work here, right? 00:19:38.140 --> 00:19:40.472 And so because of-- 00:19:40.472 --> 00:19:41.400 well, no probably. 00:19:41.400 --> 00:19:42.740 It will not work here. 00:19:42.740 --> 00:19:44.380 Sorry about that. 00:19:44.380 --> 00:19:46.150 But you know, this is different, right? 00:19:46.150 --> 00:19:48.400 I mean, that clearly we have to check, right? 00:19:48.400 --> 00:19:50.390 So here, no. 00:19:50.390 --> 00:19:51.000 Here, no. 00:19:51.000 --> 00:19:51.700 Here, no. 00:19:51.700 --> 00:19:53.110 Yes. 00:19:53.110 --> 00:19:54.190 Right? 00:19:54.190 --> 00:19:56.980 And then here, yes, right? 00:19:56.980 --> 00:19:59.120 So I could put that over here. 00:19:59.120 --> 00:20:01.270 And then I move on. 00:20:01.270 --> 00:20:02.740 Here, no. 00:20:02.740 --> 00:20:03.670 Here, no. 00:20:03.670 --> 00:20:06.110 Here, yes. 00:20:06.110 --> 00:20:07.350 That's right. 00:20:07.350 --> 00:20:12.250 So we don't know if there's not another solution. 00:20:12.250 --> 00:20:13.490 And in fact, there is. 00:20:13.490 --> 00:20:15.330 There's two solutions. 00:20:15.330 --> 00:20:20.090 We won't go there, but 4 by 4-- 00:20:20.090 --> 00:20:24.695 I'm sorry, four queens has a solution. 00:20:24.695 --> 00:20:25.194 OK? 00:20:28.860 --> 00:20:31.100 So now we're ready to code, OK? 00:20:31.100 --> 00:20:32.730 Now we're ready to code. 00:20:32.730 --> 00:20:36.290 The one thing we need to do is decide on a data structure. 00:20:36.290 --> 00:20:39.320 But before I show you the code, or even talk 00:20:39.320 --> 00:20:41.300 about the data structure, I do want 00:20:41.300 --> 00:20:45.770 you to keep in mind this search strategy that we employed, 00:20:45.770 --> 00:20:46.490 right? 00:20:46.490 --> 00:20:49.520 And the search strategy was a column-by-column search 00:20:49.520 --> 00:20:54.860 strategy, and it requires enumeration and changing 00:20:54.860 --> 00:20:57.050 positions of queens. 00:20:57.050 --> 00:20:59.870 And deciding when to do that is going 00:20:59.870 --> 00:21:03.050 to be based on whether check conflicts gives you back 00:21:03.050 --> 00:21:04.740 true or false. 00:21:04.740 --> 00:21:07.640 And we can set up check conflicts, 00:21:07.640 --> 00:21:09.722 so it's not checking-- 00:21:09.722 --> 00:21:11.180 there's two ways you could do this. 00:21:11.180 --> 00:21:13.010 You could say, here's a board. 00:21:13.010 --> 00:21:17.030 I'm going to check all pairs of conflicts associated 00:21:17.030 --> 00:21:18.650 with all the queens on the board, 00:21:18.650 --> 00:21:22.310 and I'm going to just use that procedure over and over. 00:21:22.310 --> 00:21:25.360 That's not what we did in our search strategy. 00:21:25.360 --> 00:21:29.330 What we did was we have a check conflict strategy that really 00:21:29.330 --> 00:21:40.210 takes a new queen as an argument and the existing board 00:21:40.210 --> 00:21:42.580 as the second argument, right? 00:21:42.580 --> 00:21:46.840 And it's really checking for the new queen's conflicts 00:21:46.840 --> 00:21:49.600 with the existing queens, and that cuts down 00:21:49.600 --> 00:21:51.880 on the amount of computation that you want to do. 00:21:51.880 --> 00:21:52.790 All right? 00:21:52.790 --> 00:21:57.490 So given that, what is the most natural data structure that you 00:21:57.490 --> 00:22:04.630 can think of that would be applicable to the eight queens 00:22:04.630 --> 00:22:07.450 problem or the four queens problem? 00:22:07.450 --> 00:22:09.070 What's the most natural data structure 00:22:09.070 --> 00:22:09.986 that you can think of? 00:22:12.520 --> 00:22:13.530 Someone? 00:22:13.530 --> 00:22:14.460 Yeah, go ahead, Ryan. 00:22:14.460 --> 00:22:15.520 AUDIENCE: Two-dimensional array. 00:22:15.520 --> 00:22:17.061 SRINI DEVADAS: Two-dimensional array. 00:22:17.061 --> 00:22:19.740 Two-dimensional list, and that's exactly right. 00:22:19.740 --> 00:22:28.840 So let's say that I decided to do exactly what Ryan suggested, 00:22:28.840 --> 00:22:30.970 and what I'm going to do is, I'm going 00:22:30.970 --> 00:22:34.260 to have a two-dimensional array, or two-dimensional list. 00:22:34.260 --> 00:22:36.910 In Python, you know, we call them lists, 00:22:36.910 --> 00:22:39.430 which looks like this. 00:22:39.430 --> 00:22:43.570 And this is going to look exactly like the board 00:22:43.570 --> 00:22:44.410 that you have there. 00:22:59.280 --> 00:23:00.310 Two-dimensional. 00:23:00.310 --> 00:23:07.890 This is the list of lists, and you have lists in here. 00:23:07.890 --> 00:23:12.330 And so the important thing, people-- 00:23:12.330 --> 00:23:14.730 all of you, if you haven't already, 00:23:14.730 --> 00:23:20.130 will spend some debugging time in your programming careers, 00:23:20.130 --> 00:23:23.290 figuring out why things don't work, 00:23:23.290 --> 00:23:25.230 to try and figure out why things don't work. 00:23:25.230 --> 00:23:27.240 And it'll be because you flipped the I and J 00:23:27.240 --> 00:23:30.510 indices of a two-dimensional array, or a list, OK? 00:23:30.510 --> 00:23:34.080 Because you'll think that some things are rows and some things 00:23:34.080 --> 00:23:35.896 are columns, and you'll flip them around. 00:23:35.896 --> 00:23:37.020 So I don't want to do that. 00:23:37.020 --> 00:23:39.690 I already did that enough times in my life, 00:23:39.690 --> 00:23:40.890 so I don't want to do that. 00:23:40.890 --> 00:23:46.990 And so right here, I think of B0 as the first row. 00:23:46.990 --> 00:23:47.490 Right? 00:23:47.490 --> 00:23:51.430 So this thing here is B0. 00:23:51.430 --> 00:23:58.760 So when I say something like B 2, 3, 00:23:58.760 --> 00:24:04.990 what is the value of B 2, 3 in my list B? 00:24:04.990 --> 00:24:07.220 1. 00:24:07.220 --> 00:24:08.020 All right? 00:24:08.020 --> 00:24:15.290 And B 3, 2 is 0, 00:24:15.290 --> 00:24:15.790 OK? 00:24:15.790 --> 00:24:18.760 So that's-- I mean, I could obviously have confused myself. 00:24:18.760 --> 00:24:21.870 There'd be a bug in the code if I'd 00:24:21.870 --> 00:24:25.390 switched the identities of rows and columns, right? 00:24:25.390 --> 00:24:27.110 That's exactly the point. 00:24:27.110 --> 00:24:31.780 So I can now imagine that check conflicts, given 00:24:31.780 --> 00:24:34.360 what we said about incremental checking-- 00:24:34.360 --> 00:24:36.520 but it still has to do, obviously, 00:24:36.520 --> 00:24:40.690 computation on the B array, or look at the B array, 00:24:40.690 --> 00:24:42.610 and decide what's going on. 00:24:42.610 --> 00:24:46.570 I'm going to be putting in 1s and 0s inside of the B array. 00:24:46.570 --> 00:24:49.240 And the first two-- 00:24:49.240 --> 00:24:50.810 I mean, they're all straightforward. 00:24:50.810 --> 00:24:53.330 I mean, it's not complicated code here. 00:24:53.330 --> 00:24:57.030 But the first two checks are slightly easier 00:24:57.030 --> 00:24:58.600 than the third check. 00:24:58.600 --> 00:25:02.860 No two Queens on the same column implies 00:25:02.860 --> 00:25:09.150 that I need to actually look at the first index changing, 00:25:09.150 --> 00:25:11.080 you know, from 0 to 3. 00:25:11.080 --> 00:25:14.890 And for some particular-- 00:25:14.890 --> 00:25:16.900 I could just write this out here. 00:25:16.900 --> 00:25:18.820 So no two Queens on the same coin 00:25:18.820 --> 00:25:23.650 them would mean that I want to go 0 through-- 00:25:23.650 --> 00:25:25.060 let's just call it-- 00:25:25.060 --> 00:25:29.140 7, because I have the 8 Queens problem. 00:25:29.140 --> 00:25:33.940 And then I need to look at i where i varies-- 00:25:33.940 --> 00:25:36.640 actually, let me just call that j. 00:25:36.640 --> 00:25:38.980 I like i for rows and j for columns. 00:25:38.980 --> 00:25:41.530 And j varies-- j would be 0. 00:25:41.530 --> 00:25:51.890 And then you check that B 0, 7, 0 only has one 1, all right? 00:25:51.890 --> 00:25:54.020 That's essentially what I need to check for. 00:25:54.020 --> 00:25:56.474 And then I need to check it for j equals 0. 00:25:56.474 --> 00:25:57.890 I need to check it for j equals 1. 00:25:57.890 --> 00:26:00.110 I need to check it for j equals 2, et cetera, right? 00:26:00.110 --> 00:26:01.070 That make sense? 00:26:01.070 --> 00:26:03.830 And then, no two Queens on the same row, 00:26:03.830 --> 00:26:09.940 I need to check that B i and i can go from 0 to 7. 00:26:17.080 --> 00:26:19.540 I'm sorry, no two Queens can be on the same row. 00:26:19.540 --> 00:26:21.120 Yeah, that's right. 00:26:21.120 --> 00:26:23.260 So this goes from 0 to 7. 00:26:23.260 --> 00:26:25.270 And then I fix the value of the column. 00:26:33.540 --> 00:26:35.930 This is fixed. 00:26:35.930 --> 00:26:38.620 So right now, let's just say that I look at the first row. 00:26:38.620 --> 00:26:39.830 I'd go 0 here. 00:26:39.830 --> 00:26:42.380 And then I go 0 through 7 here. 00:26:42.380 --> 00:26:47.210 And then I make sure that, on B 0, there's exactly one 1, 00:26:47.210 --> 00:26:48.030 right? 00:26:48.030 --> 00:26:48.850 All right. 00:26:48.850 --> 00:26:52.240 So we're good here? 00:26:52.240 --> 00:26:52.980 Yeah? 00:26:52.980 --> 00:26:53.970 OK. 00:26:53.970 --> 00:26:57.420 The diagonal is the i and j both change, all right? 00:26:57.420 --> 00:26:59.021 And the thing with the diagonal is-- 00:26:59.021 --> 00:27:00.520 and I'll show you the code for this. 00:27:00.520 --> 00:27:04.620 I think it's much easier to explain the code as opposed 00:27:04.620 --> 00:27:08.070 to drawing things now. 00:27:08.070 --> 00:27:12.060 Because it's about the specifics of things. 00:27:12.060 --> 00:27:14.220 And so here it is. 00:27:14.220 --> 00:27:16.410 So what I have here is-- 00:27:16.410 --> 00:27:18.420 I call it noConflicts(). 00:27:18.420 --> 00:27:23.810 noConflicts() has the board, which is our B array here, 00:27:23.810 --> 00:27:25.560 or B list. 00:27:25.560 --> 00:27:28.560 In general, we're going to be working incrementally 00:27:28.560 --> 00:27:30.252 on the current column. 00:27:30.252 --> 00:27:31.210 So that's what that is. 00:27:31.210 --> 00:27:35.160 So you move from 0 to all the way down. 00:27:35.160 --> 00:27:38.670 And then you have something that we're 00:27:38.670 --> 00:27:44.250 going to call qindex, which is where you are. 00:27:44.250 --> 00:27:46.860 So you have two things that you need to worry about. 00:27:46.860 --> 00:27:49.620 You're working on a column that's current. 00:27:49.620 --> 00:27:52.200 And then you also have the position 00:27:52.200 --> 00:27:55.000 that corresponds to the row where 00:27:55.000 --> 00:27:56.250 you're putting the Queen down. 00:27:56.250 --> 00:28:01.590 So you obviously need two of these values for-- 00:28:01.590 --> 00:28:03.540 to point into the 2-dimensional list. 00:28:03.540 --> 00:28:05.370 So that's your qindex over here. 00:28:05.370 --> 00:28:08.420 And this simply is setting n to be 4. 00:28:08.420 --> 00:28:09.240 I mean, I could-- 00:28:09.240 --> 00:28:11.400 this is a general procedure. 00:28:11.400 --> 00:28:14.870 But because I'm calling this in a 4-Queens procedure, 00:28:14.870 --> 00:28:16.530 we're going to do iteratively now. 00:28:16.530 --> 00:28:20.460 So I need to know exactly how many rows and columns they 00:28:20.460 --> 00:28:21.640 are on my board. 00:28:21.640 --> 00:28:23.940 So I've just set that up to be n equals 4. 00:28:23.940 --> 00:28:26.260 But that's essentially that dimension. 00:28:26.260 --> 00:28:28.440 So as you can, I'm going to check 00:28:28.440 --> 00:28:31.020 that there's a single 1 in the current column. 00:28:31.020 --> 00:28:33.910 This is what I was trying to go over over there. 00:28:33.910 --> 00:28:36.990 But I think this code is probably more evocative. 00:28:36.990 --> 00:28:40.440 I'm simply checking to see that the number of 1s 00:28:40.440 --> 00:28:44.350 in the current column is not greater than 1. 00:28:44.350 --> 00:28:47.025 If it is greater than 1, then, immediately, I'd return False. 00:28:50.160 --> 00:28:52.860 And then here, this is a little bit easier. 00:28:52.860 --> 00:28:57.480 I check that the current Queen-- 00:28:57.480 --> 00:29:00.420 this is-- the row on which the current Queen is on only 00:29:00.420 --> 00:29:02.460 has one Queen on it. 00:29:02.460 --> 00:29:10.860 And for the-- remember, as I'm going, 00:29:10.860 --> 00:29:14.980 there's a single 1 in the current column. 00:29:14.980 --> 00:29:19.720 And this thing here says I need to check 00:29:19.720 --> 00:29:24.285 that, when I put a Queen down here, that this is the-- 00:29:24.285 --> 00:29:27.830 I need to check this entire row to make sure that this Queen is 00:29:27.830 --> 00:29:29.410 the only one on it, all right? 00:29:29.410 --> 00:29:36.080 And then I also need to check this and that, OK? 00:29:36.080 --> 00:29:42.460 And so those two checks are over here, this and that. 00:29:42.460 --> 00:29:44.290 So the diagonal, the left diagonal, 00:29:44.290 --> 00:29:47.740 is the first one, the first check. 00:29:47.740 --> 00:29:50.080 And then the one that goes over to the right 00:29:50.080 --> 00:29:51.910 is the second check, all right? 00:29:51.910 --> 00:29:53.807 And again-- yeah, go ahead, Fadi. 00:29:53.807 --> 00:29:55.182 AUDIENCE: In the first procedure, 00:29:55.182 --> 00:29:57.542 like, when we were checking for that only one Queen 00:29:57.542 --> 00:30:02.387 was in one column-- so I see in my mind that 00:30:02.387 --> 00:30:04.912 is qindex equal to i. 00:30:04.912 --> 00:30:06.290 Why is that line added? 00:30:06.290 --> 00:30:07.965 Like here, we're checking that if we 00:30:07.965 --> 00:30:10.905 add the Queen in that position, that there's no conflict. 00:30:10.905 --> 00:30:12.362 SRINI DEVADAS: Are you talking about this line here? 00:30:12.362 --> 00:30:13.320 AUDIENCE: Yes, exactly. 00:30:13.320 --> 00:30:18.580 And also, because our algorithm makes sure that as we progress, 00:30:18.580 --> 00:30:21.960 we add, at each stack, one Queen per column, 00:30:21.960 --> 00:30:23.418 why do we have, in the first place, 00:30:23.418 --> 00:30:24.626 to check for column conflict? 00:30:24.626 --> 00:30:25.980 SRINI DEVADAS: Beautiful. 00:30:25.980 --> 00:30:27.220 That's a great question. 00:30:27.220 --> 00:30:33.550 So I'm going to answer that in a minute or so, all right? 00:30:33.550 --> 00:30:37.060 So Fadi was asking about whether the checks are redundant 00:30:37.060 --> 00:30:38.485 or not. 00:30:38.485 --> 00:30:42.160 And he was pointing to a particular piece of code. 00:30:42.160 --> 00:30:43.810 Let me explain this code. 00:30:43.810 --> 00:30:46.900 And then I'll answer Fadi's question, OK? 00:30:46.900 --> 00:30:49.960 So what I have here is our-- 00:30:49.960 --> 00:30:53.009 this is actually the search algorithm, all right? 00:30:53.009 --> 00:30:54.550 And this is exactly what we've talked 00:30:54.550 --> 00:30:57.160 about up until this point. 00:30:57.160 --> 00:30:59.920 You're going to go over-- 00:30:59.920 --> 00:31:02.620 you're going to go column by column, right? 00:31:02.620 --> 00:31:05.950 And then you're going to go row by row. 00:31:05.950 --> 00:31:08.585 And so that's essentially what the two loops are. 00:31:08.585 --> 00:31:10.460 There's lots of loops here, obviously, right? 00:31:10.460 --> 00:31:12.490 You know, this is horrible code, right? 00:31:12.490 --> 00:31:15.520 I mean, I'm not saying this is pretty code, right? 00:31:15.520 --> 00:31:18.370 But this is horrible code. 00:31:18.370 --> 00:31:21.820 And what happens here is, over here, I'm 00:31:21.820 --> 00:31:26.130 going to go ahead and say, this is-- 00:31:26.130 --> 00:31:27.820 I'm starting with-- 00:31:27.820 --> 00:31:29.410 I'm going to vary-- 00:31:29.410 --> 00:31:31.820 I'm starting with the first column. 00:31:31.820 --> 00:31:34.014 So the column is fixed here, OK? 00:31:34.014 --> 00:31:35.680 Remember, I'm going to column by column. 00:31:35.680 --> 00:31:37.324 So that's why this is a 0. 00:31:37.324 --> 00:31:39.490 And then I'm going to vary the position of the Queen 00:31:39.490 --> 00:31:40.840 on the row, OK? 00:31:40.840 --> 00:31:42.485 And that i varies, all right? 00:31:42.485 --> 00:31:44.110 And then the next thing I'm going to do 00:31:44.110 --> 00:31:46.870 is I'm going to go to the second column. 00:31:46.870 --> 00:31:49.200 And I'm going to-- 00:31:49.200 --> 00:31:50.560 so that's why this is a 1. 00:31:50.560 --> 00:31:52.840 And then I'm going to vary the position of the Queen 00:31:52.840 --> 00:31:55.050 on each of the rows corresponding 00:31:55.050 --> 00:31:58.270 to the second column, and so on and so forth, all right? 00:31:58.270 --> 00:31:59.230 That's it, all right? 00:31:59.230 --> 00:32:02.950 It's brutal code, OK? 00:32:02.950 --> 00:32:06.070 And I'll show you, I wrote code for eight Queens. 00:32:06.070 --> 00:32:10.060 It goes all the way to there, right? 00:32:10.060 --> 00:32:10.682 So that's it. 00:32:10.682 --> 00:32:12.640 I mean, you know, you can see it's simple code. 00:32:12.640 --> 00:32:16.180 I'm not-- you know, sometimes brutish means simple-- 00:32:16.180 --> 00:32:18.130 not always. 00:32:18.130 --> 00:32:22.480 And so this no conflicts thing is essentially something that 00:32:22.480 --> 00:32:28.880 is calling this noConflicts() procedure that is doing all 00:32:28.880 --> 00:32:31.630 of the things that I just described to you, right? 00:32:31.630 --> 00:32:34.360 And so if I go ahead and run this, 00:32:34.360 --> 00:32:35.710 and run this with four Queens-- 00:32:39.200 --> 00:32:42.300 oh yikes, this always happens. 00:32:42.300 --> 00:32:46.150 After a while, it's going to come back. 00:32:46.150 --> 00:32:49.390 But when I run this with four Queens-- 00:32:49.390 --> 00:32:54.247 let me-- let's see if this is what we want. 00:32:54.247 --> 00:32:55.233 One more time. 00:33:02.628 --> 00:33:06.090 Ah, sorry. 00:33:06.090 --> 00:33:09.930 I've been fighting this system bug ever since I installed 00:33:09.930 --> 00:33:11.980 macOS Sierra on my laptop. 00:33:14.780 --> 00:33:16.900 There you go. 00:33:16.900 --> 00:33:20.300 So this gave me my two solutions, one of which 00:33:20.300 --> 00:33:21.491 is up there on the board. 00:33:21.491 --> 00:33:23.990 And this is the second solution for the four Queens problem, 00:33:23.990 --> 00:33:25.190 all right? 00:33:25.190 --> 00:33:29.090 So let me go back to answer Fadi's question, right? 00:33:29.090 --> 00:33:30.920 So the idea here-- 00:33:30.920 --> 00:33:34.900 and the way I described this was, look, 00:33:34.900 --> 00:33:41.180 as I go column by column, I only put one Queen down 00:33:41.180 --> 00:33:43.460 on this particular column, the current column 00:33:43.460 --> 00:33:44.870 that I'm working on. 00:33:44.870 --> 00:33:47.060 And the way I have this code set up, 00:33:47.060 --> 00:33:51.830 notice that I'm setting this to be 1, 00:33:51.830 --> 00:33:54.830 and then I'm resetting it down below here, the j, i, 0, 00:33:54.830 --> 00:33:55.920 whether it's i-- 00:33:55.920 --> 00:33:58.700 whether it's this line of code and this line of code, 00:33:58.700 --> 00:34:01.620 they're paired together-- or this line of code 00:34:01.620 --> 00:34:02.870 and this line of code. 00:34:02.870 --> 00:34:06.110 These two things are ensuring that there's exactly 00:34:06.110 --> 00:34:09.230 one Queen on any given column. 00:34:09.230 --> 00:34:11.154 Because I'm only putting that Queen down. 00:34:11.154 --> 00:34:13.570 And then I'm taking it out and putting it in another spot, 00:34:13.570 --> 00:34:14.600 right? 00:34:14.600 --> 00:34:16.969 And that's how I described this entire procedure 00:34:16.969 --> 00:34:18.650 to you manually, right? 00:34:18.650 --> 00:34:20.780 That's how we worked through it. 00:34:20.780 --> 00:34:25.921 So your contention, Fadi, is that this check is redundant, 00:34:25.921 --> 00:34:26.420 right? 00:34:26.420 --> 00:34:28.170 And you're right, right? 00:34:28.170 --> 00:34:30.170 You're absolutely right. 00:34:30.170 --> 00:34:32.449 And so if I comment out that check, 00:34:32.449 --> 00:34:33.710 everything works exactly-- 00:34:33.710 --> 00:34:35.850 it's a little bit faster. 00:34:35.850 --> 00:34:38.150 But everything works exactly the same, all right? 00:34:38.150 --> 00:34:41.179 So there was a reason I put the check in there. 00:34:41.179 --> 00:34:42.140 You asked the question. 00:34:42.140 --> 00:34:43.639 I was going to ask you the question. 00:34:46.900 --> 00:34:48.989 But that is a good question, all right? 00:34:48.989 --> 00:34:52.699 So that's kind of the summary of four Queens. 00:34:52.699 --> 00:34:56.750 You can clearly imagine generalizing this 00:34:56.750 --> 00:34:58.860 to eight Queens. 00:34:58.860 --> 00:35:02.110 It looks even more ugly. 00:35:02.110 --> 00:35:05.340 This is what that looks like, OK? 00:35:05.340 --> 00:35:07.260 And in fact, it would look worse, 00:35:07.260 --> 00:35:13.266 except I decided that I'd employ the continue statement, right? 00:35:13.266 --> 00:35:15.890 So I don't know how many of you know of the continue statement. 00:35:15.890 --> 00:35:18.720 But the continue statement essentially 00:35:18.720 --> 00:35:23.460 says, if there's an F predicate followed by continue, 00:35:23.460 --> 00:35:25.830 and the predicate is true, then you immediately 00:35:25.830 --> 00:35:28.590 start the next iteration of the loop, OK? 00:35:28.590 --> 00:35:31.320 So there's two ways that you could 00:35:31.320 --> 00:35:35.081 say, I don't want to do anything for the rest of the loop, 00:35:35.081 --> 00:35:35.580 right? 00:35:35.580 --> 00:35:39.120 There's two ways you could do this. 00:35:39.120 --> 00:35:43.390 One is you could say-- 00:35:43.390 --> 00:35:46.950 so I have a for loop here, dot, dot, dot. 00:35:46.950 --> 00:36:00.430 And if I say if condition, then you do stuff, right? 00:36:00.430 --> 00:36:03.580 And so this do is indented inside of the if. 00:36:03.580 --> 00:36:16.210 Compare that with, for if not condition, continue. 00:36:16.210 --> 00:36:18.590 And here you would have do stuff, 00:36:18.590 --> 00:36:21.770 the same do stuff, all right? 00:36:21.770 --> 00:36:24.560 Now, if this do stuff has indentations in it, 00:36:24.560 --> 00:36:26.150 then you're better off-- 00:36:26.150 --> 00:36:28.910 I mean, just because I have so many indentations-- you're 00:36:28.910 --> 00:36:31.310 better off doing this strategy, right? 00:36:31.310 --> 00:36:34.580 Because if do stuff goes on and has a bunch of ifs in it, 00:36:34.580 --> 00:36:37.640 then you at least started at this level as opposed 00:36:37.640 --> 00:36:40.490 to starting at this level, which is 2D, all right? 00:36:40.490 --> 00:36:42.080 So that's absolutely the only reason 00:36:42.080 --> 00:36:44.930 why I used continue over here. 00:36:44.930 --> 00:36:47.480 Because it would look even more ugly if I hadn't use 00:36:47.480 --> 00:36:49.010 continue, all right? 00:36:49.010 --> 00:36:51.210 So if we run this-- 00:36:51.210 --> 00:36:53.810 and the only other thing-- 00:36:53.810 --> 00:36:56.000 I'm going to get back to no conflicts in a second. 00:36:56.000 --> 00:36:59.720 But we'll go ahead and run this. 00:36:59.720 --> 00:37:02.930 And you see that there's 92 different solutions 00:37:02.930 --> 00:37:07.400 to the eight Queens problem if you don't take into account 00:37:07.400 --> 00:37:09.680 rotations and reflections, OK? 00:37:09.680 --> 00:37:11.930 And all of those solutions are being printed out. 00:37:11.930 --> 00:37:15.530 And so then, the question is, what does that mean, right? 00:37:15.530 --> 00:37:16.920 What does that mean? 00:37:16.920 --> 00:37:21.770 And that comes to our selection of the data structure, 00:37:21.770 --> 00:37:22.520 all right? 00:37:22.520 --> 00:37:26.045 So we did this. 00:37:26.045 --> 00:37:30.410 This was the data structure that you picked, 00:37:30.410 --> 00:37:33.020 which is a natural data structure. 00:37:33.020 --> 00:37:35.150 You can imagine that you could do 00:37:35.150 --> 00:37:42.250 something that is more compact, much more compact. 00:37:42.250 --> 00:37:47.890 And I want to show you how you could optimize this code, 00:37:47.890 --> 00:37:55.270 both for succinctness as well as memory efficiency, 00:37:55.270 --> 00:37:59.090 by picking a different data structure, all right? 00:37:59.090 --> 00:38:21.170 So let's exploit the fact that each column has only one Queen 00:38:21.170 --> 00:38:24.804 allowed, all right? 00:38:24.804 --> 00:38:26.970 So we want to exploit the fact that each column only 00:38:26.970 --> 00:38:28.650 has one Queen allowed, right? 00:38:28.650 --> 00:38:31.270 So if you think about it, and you say, 00:38:31.270 --> 00:38:38.270 well, I have this big thing here, but each of the columns 00:38:38.270 --> 00:38:42.860 has exactly one Queen, then sure, I could use a bit, 00:38:42.860 --> 00:38:46.410 like I did before, of 0 and 1 to specify it. 00:38:46.410 --> 00:38:50.990 But I do know that this thing is going to be all 0s and a 1, 00:38:50.990 --> 00:38:53.910 followed by all 0s, or 1 followed by all 0s, et cetera, 00:38:53.910 --> 00:38:54.880 et cetera, all right? 00:38:54.880 --> 00:38:58.340 So it's very constrained, if I look at that first column, 00:38:58.340 --> 00:39:01.370 as to what the value of that column is, right? 00:39:01.370 --> 00:39:06.180 And so I kind of gave it away by running this. 00:39:06.180 --> 00:39:10.150 But what could I do now with respect 00:39:10.150 --> 00:39:14.080 to representing the configuration of the board 00:39:14.080 --> 00:39:17.263 by doing some amount of compaction, all right? 00:39:20.090 --> 00:39:22.920 What can I do? 00:39:22.920 --> 00:39:26.920 Those of you, look at the screen and-- 00:39:26.920 --> 00:39:27.960 yeah, go ahead. 00:39:27.960 --> 00:39:29.880 AUDIENCE: variable for a row 00:39:30.840 --> 00:39:33.780 SRINI DEVADAS: I could just have a single variable that 00:39:33.780 --> 00:39:39.750 corresponds to the row number that this particular Queen is 00:39:39.750 --> 00:39:42.000 on, right, in that column, right? 00:39:42.000 --> 00:39:44.750 And that's exactly what's going on out there, all right? 00:39:44.750 --> 00:39:49.050 So if I just take that, and I take-- 00:39:49.050 --> 00:39:51.430 I'll do the bottom one, because that's 00:39:51.430 --> 00:39:52.920 the easiest for me to see. 00:39:52.920 --> 00:39:56.200 But the rows go 0 through 7. 00:40:00.070 --> 00:40:03.340 So what that says is that, on the first column, 00:40:03.340 --> 00:40:06.990 I have a Queen up here, OK? 00:40:06.990 --> 00:40:09.380 And then the third one, I have a Queen. 00:40:09.380 --> 00:40:11.550 So I have a Queen here. 00:40:11.550 --> 00:40:15.810 And 0-- uh-oh, I messed up. 00:40:15.810 --> 00:40:16.630 Thank you. 00:40:16.630 --> 00:40:19.060 Whew, that would have been bad. 00:40:19.060 --> 00:40:22.459 So 3, and then over here-- 00:40:22.459 --> 00:40:23.500 and then, what do I have? 00:40:23.500 --> 00:40:24.370 2, right? 00:40:24.370 --> 00:40:27.280 So 2 is over here. 00:40:27.280 --> 00:40:31.690 I'm going to have a panic attack if I end up getting a conflict. 00:40:31.690 --> 00:40:32.850 Because I drew these-- 00:40:32.850 --> 00:40:34.660 I put these Queens up here. 00:40:34.660 --> 00:40:37.870 My name will be Mud, OK? 00:40:37.870 --> 00:40:42.840 So 5 is over here. 00:40:42.840 --> 00:40:49.980 And then 1 is this one over here, all right? 00:40:49.980 --> 00:40:54.340 And 6 is out here. 00:40:54.340 --> 00:40:59.250 And then finally, 4 is out here, OK? 00:40:59.250 --> 00:41:01.950 All right, don't tell me there's a conflict. 00:41:01.950 --> 00:41:03.780 Even if there's one, I refuse to-- 00:41:03.780 --> 00:41:07.110 I will refuse to admit that, all right? 00:41:07.110 --> 00:41:10.530 So that's one of the solutions to the eight Queens problem. 00:41:10.530 --> 00:41:13.980 And as I mentioned, there's really 12 distinct ones. 00:41:13.980 --> 00:41:17.670 And so it turns out you don't need 00:41:17.670 --> 00:41:22.170 to muck with a 2-dimensional list, which is confusing. 00:41:22.170 --> 00:41:25.470 And you start flipping I's and J's, and your life 00:41:25.470 --> 00:41:26.970 is miserable, right? 00:41:26.970 --> 00:41:30.150 So let's just go with a single-dimensional list, right? 00:41:30.150 --> 00:41:33.900 Let's just say that if I had something 00:41:33.900 --> 00:41:36.720 like this-- and in fact, I'm not putting a Queen down 00:41:36.720 --> 00:41:38.010 on this second column. 00:41:38.010 --> 00:41:40.180 This is not something we would do in this algorithm, 00:41:40.180 --> 00:41:42.520 but I just want to show you a generality, right? 00:41:42.520 --> 00:41:45.780 I'm going to take that, and I'm going to represent this, 00:41:45.780 --> 00:41:51.300 as you can imagine, as a single-dimensional Python list. 00:41:51.300 --> 00:41:52.770 And I'm just going to put-- 00:41:52.770 --> 00:41:56.540 what am I going to put down here according to what I just 00:41:56.540 --> 00:41:58.140 described to you? 00:41:58.140 --> 00:41:59.430 I'm going to put 1. 00:41:59.430 --> 00:42:01.730 And what do you think I should put down here? 00:42:01.730 --> 00:42:03.120 Should I put 0? 00:42:03.120 --> 00:42:05.730 No, no, no, a 0 would be bad, right? 00:42:05.730 --> 00:42:08.070 Because that would imply that there's a Queen here. 00:42:08.070 --> 00:42:10.140 I could put minus 1, right? 00:42:10.140 --> 00:42:15.400 Minus 1 could imply that there's no Queen on that column. 00:42:15.400 --> 00:42:17.280 And I could start with a board that is empty, 00:42:17.280 --> 00:42:20.340 which is all minus 1s, right? 00:42:20.340 --> 00:42:24.300 And then this one would be-- 00:42:24.300 --> 00:42:25.440 this is minus 1, yep. 00:42:25.440 --> 00:42:27.150 Sorry, I wrote 1s. 00:42:27.150 --> 00:42:31.290 Let me write this like that, minus 1. 00:42:31.290 --> 00:42:32.110 That's 1. 00:42:32.110 --> 00:42:37.300 And then I would have 2 here, and then 0 there, OK? 00:42:37.300 --> 00:42:45.130 So what's cool about this is the overall procedure in terms 00:42:45.130 --> 00:42:49.570 of the column by column incremental checking, 00:42:49.570 --> 00:42:53.780 that control flow of the algorithm can stay the same. 00:42:53.780 --> 00:42:55.730 I mean, if it were four Queens, the code 00:42:55.730 --> 00:42:57.169 would look slightly different. 00:42:57.169 --> 00:42:58.960 Because obviously, our board data structure 00:42:58.960 --> 00:43:00.680 is a little bit different. 00:43:00.680 --> 00:43:03.160 I'm going to show you the eight Queens code in a minute. 00:43:03.160 --> 00:43:05.999 But it's effectively the same algorithm, right? 00:43:05.999 --> 00:43:07.540 There's eight loops for eight Queens. 00:43:07.540 --> 00:43:09.290 There's four loops for four Queens. 00:43:09.290 --> 00:43:11.560 And a little bit of bookkeeping with respect 00:43:11.560 --> 00:43:14.770 to putting the Queen on a column would 00:43:14.770 --> 00:43:20.434 imply that I'm changing minus 1 to 0, or I'm changing 2 to 3, 00:43:20.434 --> 00:43:21.850 or something like that, as opposed 00:43:21.850 --> 00:43:25.300 to doing what I did previously, which was putting in a 1 00:43:25.300 --> 00:43:29.890 for a particular element in the 2-dimensional list, 00:43:29.890 --> 00:43:32.230 and then making it a 0, and then putting a 1 00:43:32.230 --> 00:43:33.802 in a different spot, right? 00:43:33.802 --> 00:43:35.260 So even that gets a little simpler, 00:43:35.260 --> 00:43:37.000 because I'm just overwriting. 00:43:37.000 --> 00:43:41.320 When I go 2 the 3, it means that I'm moving this Queen from here 00:43:41.320 --> 00:43:42.980 to there, OK? 00:43:42.980 --> 00:43:45.430 So there's that, which is fairly minor. 00:43:45.430 --> 00:43:49.100 What is much more interesting is our checkConflicts() procedure 00:43:49.100 --> 00:43:52.090 or our noConflicts() procedure and what that would look like, 00:43:52.090 --> 00:43:52.600 right? 00:43:52.600 --> 00:43:54.040 And we had a bunch of code-- 00:43:54.040 --> 00:43:56.590 some of it was redundant-- 00:43:56.590 --> 00:43:58.090 for noConflicts(). 00:43:58.090 --> 00:44:02.650 But let's look at the code for noConflicts() in this case. 00:44:02.650 --> 00:44:03.430 And that's it. 00:44:03.430 --> 00:44:06.340 It's four lines of code, OK? 00:44:06.340 --> 00:44:09.310 And this is not redundant, right? 00:44:09.310 --> 00:44:11.690 So if you guys can make this code smaller, 00:44:11.690 --> 00:44:14.810 A plus right off the bat, OK? 00:44:14.810 --> 00:44:18.820 Well, it is a pass-fail class, but I'll write you a letter, 00:44:18.820 --> 00:44:21.320 OK? 00:44:21.320 --> 00:44:23.950 So this is essentially the code that 00:44:23.950 --> 00:44:27.850 would take this new data structure 00:44:27.850 --> 00:44:30.670 and check conflicts in an incremental way, 00:44:30.670 --> 00:44:31.490 OK-- same thing. 00:44:31.490 --> 00:44:34.201 You know, I'm going to add a new Queen to the mix. 00:44:34.201 --> 00:44:35.950 And I'm going to be checking the conflicts 00:44:35.950 --> 00:44:39.602 of this new Queen versus the existing Queens, all right? 00:44:39.602 --> 00:44:40.810 And so let's go through this. 00:44:40.810 --> 00:44:44.200 Because I have some time, and that's the last thing 00:44:44.200 --> 00:44:46.420 I want to do. 00:44:46.420 --> 00:44:54.490 So what I have here is, I'm not going to be checking anything 00:44:54.490 --> 00:44:56.500 about the columns, all right? 00:44:56.500 --> 00:44:59.710 Because I know that the number is going to-- 00:44:59.710 --> 00:45:02.560 basically, the invariant here is there's 00:45:02.560 --> 00:45:03.940 going to be some number here. 00:45:03.940 --> 00:45:06.940 It it's a negative number, there's nothing on that column. 00:45:06.940 --> 00:45:08.460 And if there's a positive number, 00:45:08.460 --> 00:45:11.950 it better be between 0 and the number of rows minus 1. 00:45:11.950 --> 00:45:13.450 And that is going to tell me where 00:45:13.450 --> 00:45:14.800 the one Queen is, all right? 00:45:14.800 --> 00:45:16.850 So I'm done with that. 00:45:16.850 --> 00:45:24.490 But I do have to check that there's a row-- 00:45:24.490 --> 00:45:25.600 if there's a row problem. 00:45:25.600 --> 00:45:27.266 There could be a row problem, all right? 00:45:27.266 --> 00:45:33.740 So I could put a Queen up here. 00:45:33.740 --> 00:45:36.800 And obviously, I have to detect that, correct? 00:45:36.800 --> 00:45:41.364 So the way I detect that is by the highlighted code, where 00:45:41.364 --> 00:45:43.280 I'm just going to check to see whether there's 00:45:43.280 --> 00:45:48.332 two numbers that are repeated corresponding to-- 00:45:48.332 --> 00:45:51.695 well, the current-- so I know what the value of current is. 00:45:51.695 --> 00:45:53.570 It's not going to be minus 1, because current 00:45:53.570 --> 00:45:56.564 is going to be a real Queen put into a particular position. 00:45:56.564 --> 00:45:57.980 I mean, the last thing you want is 00:45:57.980 --> 00:46:01.040 to check two minus 1s for equality 00:46:01.040 --> 00:46:02.750 and throw up a conflict. 00:46:02.750 --> 00:46:04.710 Because I mean, that's just empty versus empty. 00:46:04.710 --> 00:46:06.100 You know, there's no conflict. 00:46:06.100 --> 00:46:10.130 But board current is guaranteed, by the invocation procedure, 00:46:10.130 --> 00:46:11.780 to be a positive number-- 00:46:11.780 --> 00:46:13.340 I should say a non-negative number, 00:46:13.340 --> 00:46:16.730 either is 0 up until 7 in our case of our eight Queens 00:46:16.730 --> 00:46:17.570 problem. 00:46:17.570 --> 00:46:19.730 And I just need to check to see that that value-- 00:46:19.730 --> 00:46:20.900 let's call it 4-- 00:46:20.900 --> 00:46:24.830 doesn't exist in any of the other columns, correct? 00:46:24.830 --> 00:46:25.910 So that's it. 00:46:25.910 --> 00:46:30.050 It's a little loop, but it's three lines of code. 00:46:30.050 --> 00:46:31.790 And in that same loop, we're going 00:46:31.790 --> 00:46:36.020 to be actually checking for, it turns out, 00:46:36.020 --> 00:46:39.930 both classes of diagonal conflicts, OK? 00:46:42.570 --> 00:46:48.020 In this line of code here, can someone explain me-- 00:46:48.020 --> 00:46:50.620 explain to me what this line of code is doing, all right? 00:46:53.360 --> 00:46:54.540 What am I doing here? 00:46:54.540 --> 00:46:57.510 I mean, even pictorially would be fine. 00:46:57.510 --> 00:47:01.090 Or you know, use your arms and wave, right? 00:47:01.090 --> 00:47:03.500 What is that line of code doing? 00:47:03.500 --> 00:47:05.810 I mean, roughly, it's checking for diagonal conflicts, 00:47:05.810 --> 00:47:08.260 but I want more, all right? 00:47:08.260 --> 00:47:10.457 How is it checking for diagonal conflicts? 00:47:14.280 --> 00:47:15.005 Yeah, go ahead. 00:47:15.005 --> 00:47:17.330 AUDIENCE: I think the difference between the index 00:47:17.330 --> 00:47:21.959 numbers of that one we had is that the values have 00:47:21.959 --> 00:47:22.792 the same difference. 00:47:22.792 --> 00:47:24.351 SRINI DEVADAS: That's exactly right. 00:47:24.351 --> 00:47:26.350 So if you think about what's going on, remember, 00:47:26.350 --> 00:47:27.830 we're talking squares here. 00:47:27.830 --> 00:47:29.470 You know, no rectangles, right? 00:47:29.470 --> 00:47:32.050 Maybe we could have a homework assignment where 00:47:32.050 --> 00:47:34.420 we have rectangular chessboards, and do things, 00:47:34.420 --> 00:47:36.110 and confuse everybody, right? 00:47:36.110 --> 00:47:38.830 But the good news here is that it's a square, right? 00:47:38.830 --> 00:47:41.200 I like squares, OK? 00:47:41.200 --> 00:47:43.210 And squares are easy to deal with, 00:47:43.210 --> 00:47:46.720 because you end up in a situation 00:47:46.720 --> 00:47:52.780 where you know that how much you move in this direction 00:47:52.780 --> 00:47:55.930 should be exactly the same that you 00:47:55.930 --> 00:47:58.000 move in the other direction, all right? 00:47:58.000 --> 00:47:59.680 That's what a diagonal in a chessboard 00:47:59.680 --> 00:48:02.080 is, right-- you know, on a square board is. 00:48:02.080 --> 00:48:06.340 You need to move in the same amount of movement, all right? 00:48:06.340 --> 00:48:08.380 So you're not talking about diagonals 00:48:08.380 --> 00:48:10.240 that look like that, right? 00:48:10.240 --> 00:48:11.870 That's not what you're talking about. 00:48:11.870 --> 00:48:15.490 So what this means is-- if you go look at this line of code, 00:48:15.490 --> 00:48:17.680 what it says is-- 00:48:17.680 --> 00:48:19.510 my current is telling me-- 00:48:23.660 --> 00:48:29.615 for this particular column, it's telling me what row I'm on, OK? 00:48:29.615 --> 00:48:30.865 That's really what the value-- 00:48:33.600 --> 00:48:39.040 the value-- I'm sorry, this is the column number. 00:48:39.040 --> 00:48:41.150 Excuse me, I misspoke. 00:48:41.150 --> 00:48:45.130 Current is telling me what the column number is, OK? 00:48:45.130 --> 00:48:54.970 And this board current is telling me 00:48:54.970 --> 00:48:57.670 what row number there is. 00:48:57.670 --> 00:49:08.210 So board current gives me the row number 00:49:08.210 --> 00:49:12.410 for the current column, all right? 00:49:12.410 --> 00:49:20.210 And i is whatever column that I'm comparing against. 00:49:20.210 --> 00:49:23.360 I need to enumerate the different columns that I'm 00:49:23.360 --> 00:49:25.280 comparing against, all right? 00:49:25.280 --> 00:49:28.850 So if I look at this and that, you'll 00:49:28.850 --> 00:49:34.540 find that the difference between-- 00:49:34.540 --> 00:49:37.130 this is 1, 1-- 00:49:37.130 --> 00:49:38.520 2, 1, right? 00:49:38.520 --> 00:49:44.270 So this column is 2, and the row is 1, right? 00:49:44.270 --> 00:49:46.700 So I'll just say 2 equals column, 00:49:46.700 --> 00:49:50.870 and row equals 1 for this thing over here, all right? 00:49:50.870 --> 00:49:54.080 And then I'll use a different color for this one. 00:49:54.080 --> 00:49:58.060 Here I have, the column 3-- 00:49:58.060 --> 00:50:00.140 the column is equal to 3. 00:50:00.140 --> 00:50:08.230 And the row equals 0, OK? 00:50:08.230 --> 00:50:09.310 Right? 00:50:09.310 --> 00:50:11.510 Clearly, that's a conflict, correct? 00:50:11.510 --> 00:50:14.350 And the reason there's a conflict, mathematically 00:50:14.350 --> 00:50:19.360 speaking, is because the difference in the columns 00:50:19.360 --> 00:50:24.060 is the same as the difference in the rows, OK? 00:50:24.060 --> 00:50:26.250 3 minus 2-- well 2 minus 3-- 00:50:26.250 --> 00:50:27.905 2 happens to be greater than-- 00:50:27.905 --> 00:50:29.640 sorry, 2 happens to be less than 3. 00:50:29.640 --> 00:50:31.200 3 happens to be greater than 2. 00:50:31.200 --> 00:50:33.630 But the absolute value of the difference 00:50:33.630 --> 00:50:36.690 is something that you need to take into account. 00:50:36.690 --> 00:50:41.940 Because you could have diagonals in both directions, right? 00:50:41.940 --> 00:50:46.230 So that ABS-- the absolute value over there 00:50:46.230 --> 00:50:48.030 is key, because it tells you that it's 00:50:48.030 --> 00:50:50.640 checking for both this diagonal as well as this diagonal, 00:50:50.640 --> 00:50:51.790 right? 00:50:51.790 --> 00:50:52.920 And so that's it. 00:50:52.920 --> 00:50:55.950 So if you think about this being a square, 00:50:55.950 --> 00:50:59.580 and you go look at the comparisons between the values 00:50:59.580 --> 00:51:02.070 on the rows, and if that value, the difference, 00:51:02.070 --> 00:51:04.950 is the same, in absolute terms, as the difference 00:51:04.950 --> 00:51:07.680 in the columns, you have a conflict. 00:51:07.680 --> 00:51:09.660 Otherwise, you don't have a conflict, right? 00:51:09.660 --> 00:51:10.740 Because then you're going off. 00:51:10.740 --> 00:51:12.660 The diagonal is going away, through the board, 00:51:12.660 --> 00:51:14.970 but a Queen is not on it, all right? 00:51:14.970 --> 00:51:19.020 So I guess I did run it, so I won't run it again. 00:51:19.020 --> 00:51:22.560 But I'll leave you with the thought-- 00:51:22.560 --> 00:51:24.380 and we'll talk about this next time-- 00:51:24.380 --> 00:51:28.800 is how in heck can we make this code look prettier, all right? 00:51:28.800 --> 00:51:31.890 And we will go and at a-- 00:51:31.890 --> 00:51:34.800 from here on out, we'll be looking at a programming 00:51:34.800 --> 00:51:36.895 paradigm that's very powerful. 00:51:36.895 --> 00:51:37.770 I won't give it away. 00:51:37.770 --> 00:51:39.180 You probably know what it is. 00:51:39.180 --> 00:51:43.416 But we'll talk about it first thing tomorrow. 00:51:43.416 --> 00:51:44.790 And one of the things we will do, 00:51:44.790 --> 00:51:47.130 we'll do a different puzzle tomorrow as well. 00:51:47.130 --> 00:51:50.730 But we'll take a look at the eight Queens code, 00:51:50.730 --> 00:51:53.160 make it prettier, and also get it 00:51:53.160 --> 00:51:57.750 to the point where it solves the n Queen's problem where n is 00:51:57.750 --> 00:52:01.260 an input to do the procedure. 00:52:01.260 --> 00:52:04.650 The problem with this code is it only solves eight Queens. 00:52:04.650 --> 00:52:06.090 I showed you code for four Queens 00:52:06.090 --> 00:52:07.950 that only solved four Queens. 00:52:07.950 --> 00:52:12.180 If I told you that I needed you to write code that 00:52:12.180 --> 00:52:18.720 solves n Queens where n is an argument to the procedure, 00:52:18.720 --> 00:52:22.740 you wouldn't be able to do that with this iterative strategy 00:52:22.740 --> 00:52:26.880 unless you bounded and wrote n different pieces of code, 00:52:26.880 --> 00:52:29.436 right, one of which has four loops, 00:52:29.436 --> 00:52:30.810 and the other one has five loops, 00:52:30.810 --> 00:52:32.143 and the other one has six loops. 00:52:32.143 --> 00:52:34.620 And you would still need to know what n is, OK? 00:52:34.620 --> 00:52:37.670 All right, so see you next time.