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.
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.
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.
Test set 1 (Visible Verdict) Time limit: 20 seconds per test set. Memory limit: 1GB. 1 ≤ T ≤ 100. 2 ≤ N ≤ 100. 1 ≤ Mi,j ≤ N, for all i, j.
Cameron and Jamie's kid is almost 3 years old! However, even though the child is more independent now, scheduling kid activities and domestic necessities is still a challenge for the couple.
Cameron and Jamie have a list of N activities to take care of during the day. Each activity happens during a specified interval during the day. They need to assign each activity to one of them, so that neither of them is responsible for two activities that overlap. An activity that ends at time t is not considered to overlap with another activity that starts at time t.
For example, suppose that Jamie and Cameron need to cover 3 activities: one running from 18:00 to 20:00, another from 19:00 to 21:00 and another from 22:00 to 23:00. One possibility would be for Jamie to cover the activity running from 19:00 to 21:00, with Cameron covering the other two. Another valid schedule would be for Cameron to cover the activity from 18:00 to 20:00 and Jamie to cover the other two. Notice that the first two activities overlap in the time between 19:00 and 20:00, so it is impossible to assign both of those activities to the same partner.
Given the starting and ending times of each activity, find any schedule that does not require the same person to cover overlapping activities, or say that it is impossible.
The first line of the input gives the number of test cases, T. T test cases follow. Each test case starts with a line containing a single integer N, the number of activities to assign. Then, N more lines follow. The i-th of these lines (counting starting from 1) contains two integers Si and Ei. The i-th activity starts exactly Si minutes after midnight and ends exactly Ei minutes after midnight.
For each test case, output one line containing Case #x: y, where x is the test case number (starting from 1) and y is IMPOSSIBLE if there is no valid schedule according to the above rules, or a string of exactly N characters otherwise. The i-th character in y must be C if the i-th activity is assigned to Cameron in your proposed schedule, and J if it is assigned to Jamie.
If there are multiple solutions, you may output any one of them. (See "What if a test case has multiple correct solutions?" in the Competing section of the FAQ. This information about multiple solutions will not be explicitly stated in the remainder of the 2020 contest.)
Time limit: 20 seconds per test set. Memory limit: 1GB. 1 ≤ T ≤ 100. 0 ≤ Si < Ei ≤ 24 × 60.
Test set 1 (Visible Verdict) 2 ≤ N ≤ 10.
Test set 2 (Visible Verdict) 2 ≤ N ≤ 1000.
Indicium means "trace" in Latin. In this problem we work with Latin squares and matrix traces.
A Latin square is an N-by-N square matrix in which each cell contains one of N different values, such that 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.
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).
Given values N and K, produce any N-by-N "natural Latin square" with trace K, or say it is impossible. For example, here are two possible answers for N = 3, K = 6. In each case, the values that contribute to the trace are underlined.
2 1 3 3 1 2 3 2 1 1 2 3 1 3 2 2 3 1
The first line of the input gives the number of test cases, T. T test cases follow. Each consists of one line containing two integers N and K: the desired size of the matrix and the desired trace.
For each test case, output one line containing Case #x: y, where x is the test case number (starting from 1) and y is IMPOSSIBLE if there is no answer for the given parameters or POSSIBLE otherwise. In the latter case, output N more lines of N integers each, representing a valid "natural Latin square" with a trace of K, as described above.
Time limit: 20 seconds per test set. Memory limit: 1GB. N ≤ K ≤ N2.
Test set 1 (Visible Verdict) T = 44. 2 ≤ N ≤ 5.
Test set 2 (Hidden Verdict) 1 ≤ T ≤ 100. 2 ≤ N ≤ 50.