possible values of $x \mod 3 = 0,1,2$
$\\ L = \left \{ w|( \ n_a(w) - n_b(w) \ ) \mod 3 > 1 \right \} \\ \\ L = \left \{ w|( \ n_a(w) - n_b(w) \ ) \mod 3 = 2 \right \}$
How the edges are forming ,
Few examples : at first one character strings w = 'a' or w = 'b'
$n_a(w) - n_b(w) = \ ?$
if w = 'a' then
$n_a(w) = 1 \ \ , \ n_b(w) = 0 \\ \\ => 1 - 0 = 1 \\ \\=> (1) \mod 3 = +1$
i.e. we make a transition from state(0) to state(1) for signle character string 'a'
if w = 'b' then
$n_a(w) = 0 \ \ , \ n_b(w) = 1 \\ \\ => 0 - 1 = -1 \\ \\=> (-1) \mod 3 = +2$
i.e. we make a transition from state(0) to state(2) for signle character string 'b'
We finished all one character string , now for two character string we take (for example) w = 'ab'
if w = 'ab' then
$n_a(w) = 1 \ \ , \ n_b(w) = 1 \\ \\ => 1 - 1 = 0 \\ \\=> (0) \mod 3 = 0$
i.e. read 'a' move from state(0) to state(1) [ result 1]
and read 'b' move from state(1) to state(0)
etc.
We go on checking for all two character strings and move to the resultant state in two steps, by utilizing one character string results.
Likewise calculate for three character strings. And by this time all remaining edges are fiiled up and we do not need to check for
$|w| > 3$.
