We will label all the vertices different from \( s \) and \( t \) with letters \( A \), \( B \), \( C \), \( D \), \( E \), \( F \), \( G \), \( H \), \( I \), \( J \), \( K \), \( L \), \( M \), \( N \). We label them column by column starting from left. The first column contains \( A \) and \( B \), the second contains \( C \), \( D \), and \( E \), etc.
Consider the sets \( S=\left\{s, A,B,C,D,E, G\right\} \) and \( T=\left\{F, H, I,J,K,L,M,N,t\right\} \). Observe that \( [S,T] \) is a source/sink cut and that its capacity is 35.
Observe that the flow \( f \) that satisfies:
\begin{eqnarray*}
&& f(s,A)=10, f(A,C)=10, f(C,F)=10, f(F,J)=10, f(J,M)=10,\newline
&& f(s,B)=25, f(B,E)=25, f(E,I)=15, f(I,L)=15, f(L,N)=5,\newline
&& f(N,t)=5, f(E,D)=10, f(D,G)=10, f(G,H)=10, f(H,L)=10,\newline
&& f(L,K)=20, f(K,M)=20, f(M,t)=30 ,\newline
&& f(e)=0,\mbox{ for every other edge }e
\end{eqnarray*}
has value equal to 35. Since the capacity of every cut is greater than or equal to the value of every flow, we conclude that the maximal value of a flow is \( 35 \).