1
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

2
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.

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.

4
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

5
00:00:24,500 --> 00:00:30,130
means that this value is forced. This has to be negative 4, and then can we

6
00:00:30,130 --> 00:00:36,460
pick a value here, and have this row and this column still sum to 9? Yeah,

7
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

8
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

9
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

10
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

11
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

12
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

13
00:01:11,950 --> 00:01:18,260
these entries but then these ones would be forced. This number of tiles is n

14
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

15
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

16
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

17
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

18
00:01:44,336 --> 00:01:47,724
n minus 1 squared.