0:00:00.200,0:00:05.501
Let's pick any number in here. Let's say 5. Can we choose this number and still

0:00:05.501,0:00:12.690
have this sum to 9, and still have these columns sum to 9? Yeah. Let's say 3.

0:00:12.690,0:00:17.602
So together those make 8, and that means this one is forced it has to be 1.

0:00:17.602,0:00:24.500
Okay, can we pick a value here. Yeah, let's say 8. So those sum to 13, which

0:00:24.500,0:00:30.130
means that this value is forced. This has to be negative 4, and then can we

0:00:30.130,0:00:36.460
pick a value here, and have this row and this column still sum to 9? Yeah,

0:00:36.460,0:00:41.467
let's say 7. Now if this column has to sum to 9, then this entry's forced, it's

0:00:41.467,0:00:47.651
negative 1. And as you can see, this entry's forced too, this adds to 15, so to

0:00:47.651,0:00:54.740
add to 9, this should be negative 6. And this entry's forced as well, this has

0:00:54.740,0:01:00.711
to be 14. Then both this column and this row sum to 9. So in this case there

0:01:00.711,0:01:05.829
are 4 degrees of freedom. But if we have an n by n table, in this case this is

0:01:05.829,0:01:11.950
a 3 by 3 table. This is a 4 by 4 table. Then we would be able to chose all of

0:01:11.950,0:01:18.260
these entries but then these ones would be forced. This number of tiles is n

0:01:18.260,0:01:24.760
minus 1. And this number of tiles is also n minus 1. So the total number that

0:01:24.760,0:01:31.710
we can choose is n minus 1 squared. So here in this 3 by 3 table, we were able

0:01:31.710,0:01:39.262
to choose 2 times 2. In this 4 by 4 table, we were able to choose 3 times 3. So

0:01:39.262,0:01:44.336
when we have an n by n table, we can choose n minus 1 times n minus 1, or just

0:01:44.336,0:01:47.724
n minus 1 squared.