# Codejam Solution Archive - Vestigium

## Description

Vestigium means “trace” in Latin. In this problem we work with Latin squares and matrix traces.

The trace of a square matrix is the sum of the values on the main diagonal (which runs from the upper left to the lower right).

An N-by-N square matrix is a Latin square if each cell contains one of N different values, and no value is repeated within a row or a column. In this problem, we will deal only with “natural Latin squares” in which the N values are the integers between 1 and N.

Given a matrix that contains only integers between 1 and N, we want to compute its trace and check whether it is a natural Latin square. To give some additional information, instead of simply telling us whether the matrix is a natural Latin square or not, please compute the number of rows and the number of columns that contain repeated values.

## Input

The first line of the input gives the number of test cases, T. T test cases follow. Each starts with a line containing a single integer N: the size of the matrix to explore. Then, N lines follow. The i-th of these lines contains N integers Mi,1, Mi,2 …, Mi,N. Mi,j is the integer in the i-th row and j-th column of the matrix.

## Output

For each test case, output one line containing Case #x: k r c, where x is the test case number (starting from 1), k is the trace of the matrix, r is the number of rows of the matrix that contain repeated elements, and c is the number of columns of the matrix that contain repeated elements.

## Solution

```
# input() reads a string with a line of input, stripping the ' ' (newline) at the end.
# This is all you need for most Code Jam problems.
t = int(input()) # read a line with a single integer
for i in range(1, t + 1):
N = int(input())
input_matrix = []
for _ in range(N):
raw_input = input().split(" ")
input_matrix.append([int(s) for s in raw_input])
main_diagonal = 0
repeated_row = 0
repeated_column = 0
for j in range(N):
main_diagonal += input_matrix[j][j]
tmp_set = set(input_matrix[j])
if len(tmp_set) != N:
repeated_row += 1
for j in range(N):
tmp_set = set()
for k in range(N):
tmp_set.add(input_matrix[k][j])
if len(tmp_set) != N:
repeated_column += 1
print("Case #{}: {} {} {}".format(i, main_diagonal, repeated_row, repeated_column))
# check out .format's specification for more formatting options
'''
3
4
1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
4
2 2 2 2
2 3 2 3
2 2 2 3
2 2 2 2
3
2 1 3
1 3 2
1 2 3
Case #1: 4 0 0
Case #2: 9 4 4
Case #3: 8 0 2
'''
```

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