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