12球称重问题

问题：
12个外表一样的小球，有11个球重量相同，另1个球为“坏球”，可能比较重，也可能比较轻。 如何利用天平称重3次来找出这个“坏球”？

---

---

---

---

---

---

---

---

分析：
12个球中的某一个球为坏球，该事件的信息熵 H1 即为
∑12112−log(112)=log(12) ,
而该坏球是重球和轻球的信息熵 H2 为
∑2−12log(12)=log(2).

故问题的总信息量 H3=H1+H2=log(24) 。

天平称重，每次给出的结果有三种: 左重，平衡，右重。 此时对应的信息熵 H4=∑3−13log(13)=log(3) ;
天平称重3次给出的总信息熵即为 H5=3∗H4=log(27) .

由于 H5>H4 , 天平3次称重能够提供足够的信息量来找到对应的坏球。不仅如此，其实还可以得出这个坏球是重还是轻。

答案：

代码中A==B情况的处理稍有差异。

思维导图

#include <iostream>
#include <random>
#include <assert.h>
#include <string>

using namespace std;

string str[4] = {"st","nd","rd","th"};
int compare_time = 0;

void initweight (int weight[],int n)
{
    // 数组长度定义
    std::random_device rd;
    std::mt19937 gen(rd());
    std::uniform_int_distribution<> dis(0, n-1);
    for (int i =0 ; i < n; i++)
    {
        weight[i] = 2;
    }
    int idx = dis(gen);
    int type = dis(gen) > ((float)(n-1))/2 ? 1 : -1 ;

//  calculate the probability of type value
//  int count[2] = {0,0};
//  for(int i = 0; i < 1000000; i++)
//  {
//      int idx = dis(gen) > (float)(n-1)/2 ? 1 : 0 ;
//      count[idx]++;
//  }
//  
//  cout<<count[0]<<" "<<count[1]<<endl;
    weight[idx] += type;  // type = 1， 重球； type = 0, 轻球
}

int compare_weight(int left_side[], int right_side[], int n)
{
    //string name[3] = {"1st","2nd","3rd"};
    if(compare_time < 3)
        cout<< " the "<< compare_time<<str[compare_time++] << " compare : ";
    else
        return 0;
    // left minus right , >: left heavy  =: equal  <: light
    int sum_weight = 0;
    for (int i =0 ; i < n ; i++)
    {
        sum_weight += (left_side[i] - right_side[i]);
    }

    if(sum_weight > 0 )
        return 1;

    if(sum_weight == 0)
        return 0;

    if(sum_weight < 0)
        return -1;


}

int AheavythanB(int A[], int B[],int C[])  // A > B
{
    int D[4] = {A[0],C[1],C[2],C[3]};
    int E[4] = {B[0],A[1],A[2],A[3]};
    int bad_ball = -1;
    int info = compare_weight(D,E,4);
    cout<<"     "<< info<< endl;
    switch(info)
    {
    case 1: // D has bad ball, and A[0] is the heavy ball or B[0] is the light ball
        {

            info = compare_weight(A,C,1);
            cout<<"     "<< info<< endl;
            if(info == 1)
            {
                bad_ball = 0;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<A[0]<<endl;
            }
            else
            {
                bad_ball = 4;  // B[0]
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<B[0]<<endl;
            }

            break;
        }
    case 0:
        {
            // B[1],B[2],B[3] has bad ball and is light ball
            info = compare_weight(B+1,B+2,1);
            cout<<"     "<< info<< endl;
            if(info == 0)
            {
                bad_ball = 7; // B[3]
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<B[3]<<endl;
                break;
            }
            else if(info > 0)
            {
                bad_ball = 6; // B[2]
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<B[2]<<endl;
                break;
            }
            else
            {
                bad_ball = 5; // B[1]
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<B[1]<<endl;
                break;
            }
            break;
        }
    case -1:  // D( A[0],C[1],C[2],C[3]) < E(B[0],A[1],A[2],A[3] )
        {
            // A[1],A[2],A[3] is the heavy ball
            info = compare_weight(A+1,A+2,1);
            cout<<"     "<< info<< endl;
            if(info > 0)
            {
                bad_ball = 1;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<A[1]<<endl;
                break;
            }
            else if(info == 0)
            {
                bad_ball = 3;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<A[3]<<endl;
                break;
            }
            else
            {
                bad_ball = 2;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<A[2]<<endl;
                break;
            }

        }
    }

    return bad_ball;
}

int AlightthanB(int A[], int B[], int C[]) // A < B
{
    int D[4] = { B[0],C[1],C[2],C[3]};
    int E[4] = { A[0],B[1],B[2],B[3]};

    int bad_ball = -1;
    int info = compare_weight(D,E,4);
    cout<<"     "<< info<< endl;
    switch(info)
    {
    case 1:  // B[0] heavy ball or A[0] light ball
        {

            info = compare_weight(A,C,1);
            cout<<"     "<< info<< endl;
            if(info < 0)
            {
                bad_ball = 0;
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<A[0]<<endl;               
            }
            else
            {
                bad_ball = 4;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<B[0]<<endl;   
            }
            break;
        }

    case 0:  // A[1],A[2],A[3] has light ball
        {
            info = compare_weight(A+1,A+2,1);
            cout<<"     "<< info<< endl;
            if( info > 0 )
            {
                bad_ball = 2;
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<A[2]<<endl;   
            }
            else if(info == 0)
            {
                bad_ball = 3;
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<A[3]<<endl;   
            }
            else
            {
                bad_ball = 1;
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<A[1]<<endl;   
            }
            break;
        }
    case -1: // B[1],B[2],B[3] has heavy ball
        {
            info = compare_weight(B+1,B+2,1);
            cout<<"     "<< info<< endl;
            if(info > 1)
            {
                bad_ball = 5;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<B[1]<<endl;   
            }
            else if(info == 0)
            {
                bad_ball = 7;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<B[3]<<endl;   
            }
            else
            {
                bad_ball = 6;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<B[2]<<endl;   
            }

            break;
        }

    }
    return bad_ball;
}

int Aequal2B(int C[],int A[])  // C has the bad ball
{
    int bad_ball = -1;
    int info = compare_weight(C,A,2); // 2rd compare
    cout<<"     "<< info<< endl;
    switch(info)
    {
    case 1: // C(0,1) > A(0,1) ,  C[0],C[1] has the bad ball and is heavy ball
        {
            info = compare_weight(C,A,1);
            cout<<"     "<< info<< endl;
            if(info > 0)
            {
                bad_ball = 8;                       
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<C[0]<<endl;
            }
            else // info == 0 , because info < 0 not exist
            {
                bad_ball = 9;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<C[1]<<endl;
            }
            break;
        }
    case 0:
        {
            // (C(2,3)) has the bad all and 
            info = compare_weight(C+2,A,1);
            cout<<"     "<< info<< endl;
            if(info > 0)
            {
                bad_ball = 10;
                cout <<"the heavy ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<C[2]<<endl;
            }
            else if(info < 0)
            {
                bad_ball = 10;
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<C[2]<<endl;
            }
            else
            {
                bad_ball = 11;
                cout <<"the heavy/light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<C[3]<<endl;
            }

            break;
        }
    case -1:
        {
            // C(0,1) < A(0,1) ,  C has the bad ball and is light ball
            info = compare_weight(C,A,1);
            cout<<"     "<< info<< endl;
            if(info < 0)
            {
                bad_ball = 8;
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<C[0]<<endl;
            }
            else
            {
                bad_ball = 9;
                cout <<"the light ball is "<< bad_ball+1 << str[min(bad_ball,3)]<< " : "<<C[1]<<endl;
            }
            break;
        }

    } //2nd

    return bad_ball;
}

int find_ball(int weight[], int n)
{
    assert(n == 12); // this the 12 balls' method

    int bad_ball = -1;

    int A[4],B[4],C[4];
    for(int i = 0; i < 4; i++)
    {
        A[i] = weight[i];
        B[i] = weight[i+4];
        C[i] = weight[i+8];
    }

    // first compare
    int info = compare_weight(A,B,4);
    cout<<"     "<< info<< endl;
    switch(info)
    {
    case 1:  // A > B 
        {
            bad_ball = AheavythanB(A,B,C);          
            break;
        }

    case 0:  // A = B , C has the bad all 
        {

            bad_ball = Aequal2B(C,A);
            break;
        }

    case -1:  // A < B
        {
            bad_ball = AlightthanB(A,B,C);
            break;
        }
    }

    return bad_ball;
}

int _tmain(int argc, _TCHAR* argv[])
{

    int w[12];

    int n = 12;
    int err = 0;
    for(int i = 0; i < n; i++)
    {
        compare_time = 0;

        initweight(w,12);
        // the input data
        for (int i = 0; i < 12; i++)
        {
            cout<<i+1<<":"<<w[i]<<", ";
        }
        cout<<endl;

        int bad = find_ball(w,12);

        if(w[bad] == 2)
            err++;
    }

    cout<<"find error count: "<< err<<endl;


    return 0;
}