WEBVTT 00:00:00.200 --> 00:00:05.501 Let's pick any number in here. Let's say 5. Can we choose this number and still 00:00:05.501 --> 00:00:12.690 have this sum to 9, and still have these columns sum to 9? Yeah. Let's say 3. 00:00:12.690 --> 00:00:17.602 So together those make 8, and that means this one is forced it has to be 1. 00:00:17.602 --> 00:00:24.500 Okay, can we pick a value here. Yeah, let's say 8. So those sum to 13, which 00:00:24.500 --> 00:00:30.130 means that this value is forced. This has to be negative 4, and then can we 00:00:30.130 --> 00:00:36.460 pick a value here, and have this row and this column still sum to 9? Yeah, 00:00:36.460 --> 00:00:41.467 let's say 7. Now if this column has to sum to 9, then this entry's forced, it's 00:00:41.467 --> 00:00:47.651 negative 1. And as you can see, this entry's forced too, this adds to 15, so to 00:00:47.651 --> 00:00:54.740 add to 9, this should be negative 6. And this entry's forced as well, this has 00:00:54.740 --> 00:01:00.711 to be 14. Then both this column and this row sum to 9. So in this case there 00:01:00.711 --> 00:01:05.829 are 4 degrees of freedom. But if we have an n by n table, in this case this is 00:01:05.829 --> 00:01:11.950 a 3 by 3 table. This is a 4 by 4 table. Then we would be able to chose all of 00:01:11.950 --> 00:01:18.260 these entries but then these ones would be forced. This number of tiles is n 00:01:18.260 --> 00:01:24.760 minus 1. And this number of tiles is also n minus 1. So the total number that 00:01:24.760 --> 00:01:31.710 we can choose is n minus 1 squared. So here in this 3 by 3 table, we were able 00:01:31.710 --> 00:01:39.262 to choose 2 times 2. In this 4 by 4 table, we were able to choose 3 times 3. So 00:01:39.262 --> 00:01:44.336 when we have an n by n table, we can choose n minus 1 times n minus 1, or just 00:01:44.336 --> 00:01:47.724 n minus 1 squared.