То ли я торможу, то ли лыжи не едут. Вроде все логично раскидал, а валидатор ни один метод, кроме close(), не принял. По конструкторам все норм - принимает все.
package com.javarush.task.task19.task1917;

/*
Свой FileWriter
*/

import java.io.File;
import java.io.FileDescriptor;
import java.io.FileWriter;
import java.io.IOException;

public class FileConsoleWriter {
    private FileWriter fileWriter;

    FileConsoleWriter(File file) throws IOException {
        fileWriter = new FileWriter(file);
    }

    FileConsoleWriter(File file, boolean append) throws IOException {
        fileWriter = new FileWriter(file, append);
    }

    FileConsoleWriter(FileDescriptor fd) throws IOException {
        fileWriter = new FileWriter(fd);
    }

    FileConsoleWriter(String fileName) throws IOException {
        fileWriter = new FileWriter(fileName);
    }

    FileConsoleWriter(String fileName, boolean append) throws IOException {
        fileWriter = new FileWriter(fileName, append);
    }


    public void write(char[] cbuf, int off, int len) throws IOException {
        for (int i = off; i < len; i++) write(cbuf[i]);
    }

    public void write(int c) throws IOException {
        char chr = (char) c;
        fileWriter.write(chr);
        System.out.print(chr);
    }

    public void write(String str) throws IOException {
        write(str.toCharArray());
    }

    public void write(String str, int off, int len) throws IOException {
        write(str.toCharArray(), off, len);
    }

    public void write(char[] cbuf) throws IOException {
        for (int i = 0; i < cbuf.length; i++) write(cbuf[i]);
    }

    public void close() throws IOException {
        this.fileWriter.close();
    }

    public static void main(String[] args) {

    }

}