Do you have trouble about Case 1 or Case 2?

Before showing you some example, we will talking about some properties.

Property 1: After executing operators (our table becomes expected table), suppose that we have row X[] and row Y[] (consider X and Y as arrays), there are only two cases: (1): X==Y or (2): X==^Y (^Y means change Y[i] into 1-Y[i]).

For example:

010001
010001
101110
010001

Notes that Property 1 is still right with columns, not only rows.

Property 2: If we "invert" a row on table (choose a row X and let X=^X), our table is still meet required conditions.

For example, invert row 2 in above example:

010001
101110
101110
010001

Note that Property 2 is right not only with rows, but also columns.

Now, let's talking about Case 1 and Case 2.

Example for Case 1:

Case 1 means there is some row is unmodified through operators.

010101
010001
101110
010001

Iterate i from 1 to row count, assume that i=2 (Row[i] = 010001). Choose Row[2] as a unmodified row.

Row[1] has two choices, (1): ==Row[2] and (2): ==^Row[2]. First choice is better (Cost=1). And second choice is worse, Cost=5. So if Row[2] is not modified, Row[1] must be 010001.

It is the same idea with Row[3], and Row[4].

Example for Case 2:

1100
0110
0011

Try all possibility for Column[1], use Property 1, we have Column[2]==Column[1] || Column[2]==^Column[1], choose the better choice, then do the same with Column[3], Column[4].