As I understand, I am suppose to call wait() on the mutex, when I want the current thread to stop working until another thread calls notify() on the same mutex object. That doesn't seem to be working.
I'm trying to make a thread print 1-10. Then wait for another thread to print 11-20. And then the first thread would again print 21-30
Main.java
public class Main {
    public static void main(String[] args) throws InterruptedException {
        Object mutex = 1;
        Thread child1 = new Thread(new Child1(mutex));
        Thread child2 = new Thread(new Child2(mutex));
        child1.start();
        child2.start();
    }
}
Child1.java
public class Child1 implements Runnable {
    Object mutex;
    public Child1(Object mutex){
        this.mutex = mutex;
    }
    public void run() {
        synchronized (mutex) {
            for(int c = 0; c < 10; c++){
                System.out.println(c+1);
            }
            try {
                wait();
            } catch (InterruptedException e) {
                // TODO Auto-generated catch block
                e.printStackTrace();
            }
        }
        for(int c = 20; c < 31; c++){
            System.out.println(c+1);
        }
    }
}
Child2.java
public class Child2 implements Runnable {
    Object mutex;
    public Child2(Object mutex) {
        this.mutex = mutex;
    }
    public void run() {
        synchronized (mutex) {
            for (int c = 11; c < 21; c++) {
                System.out.println(c);
            }
            notify();
        }
    }
}
Output
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17Exception in thread "Thread-0" 
18
19
20
Exception in thread "Thread-1" java.lang.IllegalMonitorStateException
    at java.lang.Object.wait(Native Method)
    at java.lang.Object.wait(Object.java:502)
    at task42.Child1.run(Child1.java:18)
    at java.lang.Thread.run(Thread.java:745)
java.lang.IllegalMonitorStateException
    at java.lang.Object.notify(Native Method)
    at task42.Child2.run(Child2.java:15)
    at java.lang.Thread.run(Thread.java:745)
What am I missing?
You must add the mutex reference to wait() and notify(); that is, change wait() to mutex.wait() and notify() to mutex.notify().
Without this, you are calling to wait/notify on this (method() is equivalent to this.method())
Here is your code with the appropriate changes made:
Child1.java
public class Child1 implements Runnable {
    Object mutex;
    public Child1(Object mutex){
        this.mutex = mutex;
    }
    public void run() {
        synchronized (mutex) {
            for(int c = 0; c < 10; c++){
                System.out.println(c+1);
            }
            try {
                mutex.wait(); // Changed here
            } catch (InterruptedException e) {
                // TODO Auto-generated catch block
                e.printStackTrace();
            }
        }
        for(int c = 20; c < 31; c++){
            System.out.println(c+1);
        }
    }
}
Child2.java
public class Child2 implements Runnable {
    Object mutex;
    public Child2(Object mutex) {
        this.mutex = mutex;
    }
    public void run() {
        synchronized (mutex) {
            for (int c = 11; c < 21; c++) {
                System.out.println(c);
            }
            mutex.notify(); // Changed here
        }
    }
}
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