\section*{Problem 1 - 071021}

In

\begin{tabular}{cccccc} 
& \(\mathrm{F}\) & \(\mathrm{U}\) & \(\mathrm{E}\) & \(\mathrm{N}\) & \(\mathrm{F}\) \\
+ & & \(\mathrm{Z}\) & \(\mathrm{W}\) & \(\mathrm{E}\) & \(\mathrm{I}\) \\
\hline \(\mathrm{S}\) & \(\mathrm{I}\) & \(\mathrm{E}\) & \(\mathrm{B}\) & \(\mathrm{E}\) & \(\mathrm{N}\)
\end{tabular}

the letters should be replaced by digits such that the addition results in a correct sum. The same letters should represent the same digits, and different letters should represent different digits.

Determine how many solutions the problem has!
}

Assuming the conditions of the problem are satisfiable, then F, U, E, N, Z, W, I, S, B are all less than or equal to 9, and it holds that:

(1) \(S \neq 0\) and \(S=1\), since \(U+Z<20\) and \(F+1 \leq 10\) must be true.

(2) \(F=9\), since \(F+1 \geq 10\) must be true.

From this follows (3) \(F+1=10\) and from this in turn (4) \(I=0\).

From the last column of the problem, it follows that \(F+I=N\), so due to (4) \(F=N\).

Since by assumption \(F \neq N\) must be true, the conditions of the problem cannot all be satisfied simultaneously, i.e., the problem has no solution.