[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.20,0:00:05.50,Default,,0000,0000,0000,,Let's pick any number in here. Let's say 5. Can we choose this number and still
Dialogue: 0,0:00:05.50,0:00:12.69,Default,,0000,0000,0000,,have this sum to 9, and still have these columns sum to 9? Yeah. Let's say 3.
Dialogue: 0,0:00:12.69,0:00:17.60,Default,,0000,0000,0000,,So together those make 8, and that means this one is forced it has to be 1.
Dialogue: 0,0:00:17.60,0:00:24.50,Default,,0000,0000,0000,,Okay, can we pick a value here. Yeah, let's say 8. So those sum to 13, which
Dialogue: 0,0:00:24.50,0:00:30.13,Default,,0000,0000,0000,,means that this value is forced. This has to be negative 4, and then can we
Dialogue: 0,0:00:30.13,0:00:36.46,Default,,0000,0000,0000,,pick a value here, and have this row and this column still sum to 9? Yeah,
Dialogue: 0,0:00:36.46,0:00:41.47,Default,,0000,0000,0000,,let's say 7. Now if this column has to sum to 9, then this entry's forced, it's
Dialogue: 0,0:00:41.47,0:00:47.65,Default,,0000,0000,0000,,negative 1. And as you can see, this entry's forced too, this adds to 15, so to
Dialogue: 0,0:00:47.65,0:00:54.74,Default,,0000,0000,0000,,add to 9, this should be negative 6. And this entry's forced as well, this has
Dialogue: 0,0:00:54.74,0:01:00.71,Default,,0000,0000,0000,,to be 14. Then both this column and this row sum to 9. So in this case there
Dialogue: 0,0:01:00.71,0:01:05.83,Default,,0000,0000,0000,,are 4 degrees of freedom. But if we have an n by n table, in this case this is
Dialogue: 0,0:01:05.83,0:01:11.95,Default,,0000,0000,0000,,a 3 by 3 table. This is a 4 by 4 table. Then we would be able to chose all of
Dialogue: 0,0:01:11.95,0:01:18.26,Default,,0000,0000,0000,,these entries but then these ones would be forced. This number of tiles is n
Dialogue: 0,0:01:18.26,0:01:24.76,Default,,0000,0000,0000,,minus 1. And this number of tiles is also n minus 1. So the total number that
Dialogue: 0,0:01:24.76,0:01:31.71,Default,,0000,0000,0000,,we can choose is n minus 1 squared. So here in this 3 by 3 table, we were able
Dialogue: 0,0:01:31.71,0:01:39.26,Default,,0000,0000,0000,,to choose 2 times 2. In this 4 by 4 table, we were able to choose 3 times 3. So
Dialogue: 0,0:01:39.26,0:01:44.34,Default,,0000,0000,0000,,when we have an n by n table, we can choose n minus 1 times n minus 1, or just
Dialogue: 0,0:01:44.34,0:01:47.72,Default,,0000,0000,0000,,n minus 1 squared.