how to use pipe to connect two child process in C

Question

I am self-studying an OS textbook and doing its homework:

Write a program that creates two children, and connects the standard output of one to the standard input of the other, using the pipe() system call.

then I try to write my code like this:

#include <stdio.h>
#include <unistd.h>
#include <stdlib.h>
#include <fcntl.h>
int main()
{
    printf("parent in pid: (%d)\n", (int)getpid());
    fflush(stdout);
    int p1[2], p2[2];
    pipe(p1);
     pipe(p2);
    
    char buff1[100];
    char buff2[100];
    int rc = fork();
    if (rc < 0)
    {
        fprintf(stderr, "fork failed");
        exit(1);
    }
    else if (rc == 0)
    {
        int fd = open("test1.txt", O_CREAT|O_WRONLY|O_TRUNC, S_IRWXU);
        printf("%d\n", fd);
        printf("entering child1 process at pid : (%d)\n", (int)getpid());
        dup2(p1[0], STDIN_FILENO);
        dup2(p2[1], STDOUT_FILENO);
        int n = read(STDIN_FILENO, buff1, sizeof buff1);
        
        puts("this is a message from child 1");
        write(fd, buff1, n);
    }
    else
    {
        int rc2 = fork();
        if (rc2 == 0)
        {
            int fd2 = open("test2.txt", O_CREAT|O_WRONLY|O_TRUNC, S_IRWXU);
            printf("%d\n", fd2);
            printf("entering child2 process at pid : (%d)\n", (int)getpid());
            dup2(p2[0], STDIN_FILENO);
            dup2(p1[1], STDOUT_FILENO);
            int n = read(STDIN_FILENO, buff2, sizeof buff2);
            
            puts("this is a message from child 2");
            write(fd2, buff2, n);
        }
        else
        {
            printf("final parent in pid: (%d)\n", (int)getpid());
        }
        
    }
    return 0;
}

if all went correctly. I should see messages from child1 or child2 in the txt files. But nothing happened when I run this code. But if I delete any one of the read statement like this:

#include <stdio.h>
#include <unistd.h>
#include <stdlib.h>
#include <fcntl.h>
int main()
{
    printf("parent in pid: (%d)\n", (int)getpid());
    fflush(stdout);
    int p1[2], p2[2];
    pipe(p1);
     pipe(p2);
    
    char buff1[100];
    char buff2[100];
    int rc = fork();
    if (rc < 0)
    {
        fprintf(stderr, "fork failed");
        exit(1);
    }
    else if (rc == 0)
    {
        int fd = open("test1.txt", O_CREAT|O_WRONLY|O_TRUNC, S_IRWXU);
        printf("%d\n", fd);
        printf("entering child1 process at pid : (%d)\n", (int)getpid());
        dup2(p1[0], STDIN_FILENO);
        dup2(p2[1], STDOUT_FILENO);
        //int n = read(STDIN_FILENO, buff1, sizeof buff1);
        
        puts("this is a message from child 1");
        //write(fd, buff1, n);
    }
    else
    {
        int rc2 = fork();
        if (rc2 == 0)
        {
            int fd2 = open("test2.txt", O_CREAT|O_WRONLY|O_TRUNC, S_IRWXU);
            printf("%d\n", fd2);
            printf("entering child2 process at pid : (%d)\n", (int)getpid());
            dup2(p2[0], STDIN_FILENO);
            dup2(p1[1], STDOUT_FILENO);
            int n = read(STDIN_FILENO, buff2, sizeof buff2);
            
            puts("this is a message from child 2");
            write(fd2, buff2, n);
        }
        else
        {
            printf("final parent in pid: (%d)\n", (int)getpid());
        }
        
    }
    return 0;
}

or like this

#include <stdio.h>
#include <unistd.h>
#include <stdlib.h>
#include <fcntl.h>
int main()
{
    printf("parent in pid: (%d)\n", (int)getpid());
    fflush(stdout);
    int p1[2], p2[2];
    pipe(p1);
     pipe(p2);
    
    char buff1[100];
    char buff2[100];
    int rc = fork();
    if (rc < 0)
    {
        fprintf(stderr, "fork failed");
        exit(1);
    }
    else if (rc == 0)
    {
        int fd = open("test1.txt", O_CREAT|O_WRONLY|O_TRUNC, S_IRWXU);
        printf("%d\n", fd);
        printf("entering child1 process at pid : (%d)\n", (int)getpid());
        dup2(p1[0], STDIN_FILENO);
        dup2(p2[1], STDOUT_FILENO);
        int n = read(STDIN_FILENO, buff1, sizeof buff1);
        
        puts("this is a message from child 1");
        write(fd, buff1, n);
    }
    else
    {
        int rc2 = fork();
        if (rc2 == 0)
        {
            int fd2 = open("test2.txt", O_CREAT|O_WRONLY|O_TRUNC, S_IRWXU);
            printf("%d\n", fd2);
            printf("entering child2 process at pid : (%d)\n", (int)getpid());
            dup2(p2[0], STDIN_FILENO);
            dup2(p1[1], STDOUT_FILENO);
            //int n = read(STDIN_FILENO, buff2, sizeof buff2);
            
            puts("this is a message from child 2");
            //write(fd2, buff2, n);
        }
        else
        {
            printf("final parent in pid: (%d)\n", (int)getpid());
        }
        
    }
    return 0;
}

I could see correct output in the txt file from both sides. I'm wondering what happened in my code and what went wrong in the first case?