Truth-teller and Liar Detection: Determining the Number of Liars in a Group

This problem involves analyzing statements made by a group of people, some of whom are truth-tellers and others are liars. Each person claims a certain number of liars are present in the group. The task is to determine if the statements are contradictory, and if possible, find the number of liars.

Problem Statement:

There is a group of 'n' people. Some of them might be liars, who always tell lies. Other people always tell the truth. The 'i'-th person says 'There are at least 'li' liars amongst us'. Determine if what people are saying is contradictory, or if it is possible. If it is possible, output the number of liars in the group. If there are multiple possible answers, output any one of them.

Input:

The first line contains a single integer 't' (1≤t≤1000) — the number of test cases.

The first line of each test case contains a single integer 'n' (1≤n≤100).

The second line of each test case contains 'n' integers 'li' (0≤li≤n) — the number said by the 'i'-th person.

It's guaranteed that the sum of all 'n' does not exceed 104.

Output:

For each test case output a single integer. If what people are saying is contradictory, output '-1'. Otherwise, output the number of liars in the group. If there are multiple possible answers, output any one of them.

Example:

inputCopy 7 2 1 2 2 2 2 2 0 0 1 1 1 0 5 5 5 3 3 5 6 5 3 6 6 3 5

outputCopy 1 -1 0 -1 0 3 4

Explanation:

• In the first test case, the first person says there is at least one liar. If the first person is a truth-teller, the second person must be a liar. If the first person is a liar, the second person must be a truth-teller. In both cases, there is exactly one liar.
• In the second test case, both people say there are at least two liars. This is impossible, as there are only two people in the group. So the answer is -1.
• In the third test case, both people say there are no liars. This is possible if both people are truth-tellers.
• In the fourth test case, the only person says there is at least one liar. This is impossible, as there is only one person in the group. So the answer is -1.
• In the fifth test case, the first and second people say there are at least five liars, the third and fourth people say there are at least three liars, and the fifth person says there are at least five liars. This is possible if the first, second, and fifth people are liars, and the third and fourth people are truth-tellers.
• In the sixth test case, the first and fifth people say there are at least five liars, the second and fourth people say there are at least three liars, and the third and sixth people say there are at least six liars. This is possible if the first, third, and sixth people are liars, and the second, fourth, and fifth people are truth-tellers.

Solution:

The key observation is that the maximum number of liars is limited by the minimum number of liars claimed by any person, and the minimum number of liars is limited by the maximum number of people who could be liars based on the maximum number of liars claimed by any person. To determine the number of liars, we need to find the maximum and minimum number of liars based on the statements.

#include<bits/stdc++.h>
using namespace std;

int main()
{
    int t;
    cin>>t;
    while(t--)
    {
        int n;
        cin>>n;
        int minn=0,maxx=n,ok=1;
        for(int i=0;i<n;i++)
        {
            int x;
            cin>>x;
            if(x>minn)
            {
                minn=x;
            }
            if(n-x<maxx)
            {
                maxx=n-x;
            }
            if(x>n)
            {
                ok=0;
            }
        }
        if(n-minn>maxx)
        {
            if(ok)
            {
                cout<<n-minn<<endl;
            }
            else
            {
                cout<<'-1'<<endl;
            }
        }
        else
        {
            if(ok)
            {
                cout<<maxx<<endl;
            }
            else
            {
                cout<<'-1'<<endl;
            }
        }
    }
    return 0;
}

Explanation:

1. Initialization: We initialize 'minn' to 0, 'maxx' to 'n', and 'ok' to 1. 'minn' represents the minimum number of liars claimed, 'maxx' represents the maximum number of liars possible, and 'ok' indicates if the statements are consistent.
2. Looping Through Statements: The loop iterates through the statements made by each person.
3. Finding Minimum and Maximum: For each statement, we update 'minn' if the current claim ('x') is greater than the current 'minn'. We update 'maxx' if 'n-x' is less than the current 'maxx'. 'n-x' represents the maximum number of people who could be truth-tellers if the current statement is true.
4. Checking for Contradictions: If 'x' is greater than 'n' (meaning the person claims there are more liars than people in the group), we set 'ok' to 0, indicating a contradiction.
5. Outputting the Result:
   • If 'n-minn' is greater than 'maxx' (meaning the minimum number of liars based on 'minn' is greater than the maximum number of liars possible based on 'maxx'), we output '-1' if 'ok' is 1 (indicating no contradictions) or '-1' if 'ok' is 0 (indicating a contradiction).
   • Otherwise, we output 'maxx' if 'ok' is 1 (indicating no contradictions) or '-1' if 'ok' is 0 (indicating a contradiction). This is because the maximum number of liars possible is limited by 'maxx'.

This solution efficiently determines if the statements are contradictory and, if possible, finds the number of liars in the group. It utilizes the logic of minimizing the number of liars and maximizing the number of truth-tellers based on the claims made.