Description

To check students’ understanding of positional numeral systems, the famous scientist Lu Si gave the following addition table, where letters represent digits. For example:

$$\def\arraystretch{2} \begin{array}{c||c|c|c|c} \rm + & \kern{.5cm} \rm \mathclap{L} \kern{.5cm} & \kern{.5cm} \rm \mathclap{K} \kern{.5cm} & \kern{.5cm} \rm \mathclap{V} \kern{.5cm} & \kern{.5cm} \rm \mathclap{E} \kern{.5cm} \\ \hline\hline \rm L & \rm L & \rm K & \rm V & \rm E \\ \hline \rm K & \rm K & \rm V & \rm E & \rm \mathclap{KL} \\ \hline \rm V & \rm V & \rm E & \rm \mathclap{KL} & \rm \mathclap{KK} \\ \hline \rm E & \rm E & \rm \mathclap{KL} & \rm \mathclap{KK} & \rm \mathclap{KV} \\ \end{array}$$

Its meaning is:

$L+L=L$，$L+K=K$，$L+V=V$，$L+E=E$

$K+L=K$，$K+K=V$，$K+V=E$，$K+E=KL$

$\cdots$

$E+E=KV$

From these rules, we can deduce: $L=0$，$K=1$，$V=2$，$E=3$.

We can also determine that the table represents base $4$ addition.

Input Format

The first line contains an integer $n$ ($3 \le n \le 9$) representing the number of rows.

The following $n$ lines each contain $n$ strings separated by spaces.

Let $s_{i,j}$ denote the string in row $i$ and column $j$. The testdata guarantees $s_{1,1}=\texttt{+}$, $s_{i,1}=s_{1,i}$, $|s_{i,1}|=1$, and $s_{i,1} \ne s_{j,1}$ ($i \ne j$).

It is guaranteed that there is at most one solution.

Output Format

On the first line, output which number each letter represents, in the format like: L=0 K=1 $\cdots$ sorted in the given letter order. Different letters must represent different digits.

On the second line, output the base of the addition.

If it is impossible to form a valid addition table, output ERROR!.

5
+ L K V E
L L K V E
K K V E KL
V V E KL KK
E E KL KK KV

L=0 K=1 V=2 E=3
4

Hint

NOIP 1998 Senior Problem 3.

Translated by ChatGPT 5