
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.