php not writing data into database

First, your query can produce SQL Injection. Use Mysqli Prepared Statement :

    $servername = "localhost";
    $username = "root";
    $password = "";
    $dbname = "testdb";

    // Create connection
    $conn = new mysqli($servername, $username, $password, $dbname);

 if(isset($_POST["btnSubmit"]))
 {
    $num1 = $_POST["num1"];
    $num2 = $_POST["num2"];
    $num3 = $_POST["num3"];
    $num4 = $_POST["num4"];

    // prepare and bind
    $query = $conn->prepare("INSERT INTO table1 (num1, num2, num3, num4) VALUES (?, ?, ?, ?)");
    $stmt->bind_param("ssss", $num1, $num2, $num3, $num4);
}

This function binds the parameters to the SQL query and tells the database what the parameters are. The "ssss" argument lists the types of data that the parameters are. The s character tells mysql that the parameter is a string.

The argument may be one of four types:

i - integer
d - double
s - string

Second, your if statement misses a closing bracket }

Third, your variable $num1 is never used. You use num1, num2, but you miss the '$'

First, your if statement is missing a closing }.

Second, your SQL query is not inserting the variables you've set above. You've got variables like $num1, but then you are inserting the value just 'num' in your SQL insert. You have to change 'num1', 'num2'... to '$num1', '$num2'...

Third, please do some research on PHP Data Objects (PDO) or MYSQLi (reference links at bottom of post). mysql_ is deprecated and completely vulnerable to malicious injection.

Edit: In addition, please see fred -ii-'s comments below for some sound advice on better INSERT queries. It's safe practice to verify that the values are of the type you're expecting prior to running them against your database.

fred -ii- says:

What if one of those values happens to contain an injection such as '123?

[Use]... (int)$_POST["num1"] and check to see firsthand if the input entered is an integer. There are a few functions that will do that.

Use error reporting and error checking against your query during testing and assuming that you are able to use the MySQL_ API.

References:

• http://php.net/manual/en/function.error-reporting.php
• http://php.net/manual/en/function.mysql-error.php

Otherwise, you will need to resort to either using the MySQLi_ or PDO API.

References:

• http://php.net/manual/en/book.mysqli.php
• http://php.net/manual/en/book.pdo.php

.Change your query to;

mysql_query("insert into table1(`num1`,`num2`,`num3`,`num4`) values ('".$num1."','".$num2."','".$num3."','".$num4."')");

followed by the closing bracket ( } ) for your if statement.

<?php
   session_start();
   // always start your session before any other code

   $host = "localhost";
   $user =  "root";
   $password = "";
   mysql_connect($host,$user,$password);
   mysql_select_db("testdb") or die(mysql_error());


   if(isset($_POST["btnSubmit"]))
   {
      $num1 = mysql_real_escape_string($_POST["num1"]);
      $num2 = mysql_real_escape_string($_POST["num2"]);
      $num3 = mysql_real_escape_string($_POST["num3"]);
      $num4 = mysql_real_escape_string($_POST["num4"]);

      // mysql isn't the safest way to put your code out, however if you do, escape it. You may be better off by using prepared statements, but thats up to you, i am just fixing this code 

      mysql_query("insert into table1(num1,num2,num3,num4) 
                   values ('$num1','$num2','$num3','$num4')");

    }
?>

I made a few tweaks in your code and this should do it. Note my additional comments in the code, including the propper escaping your variables, because of the injection danger. Its not my place to judge you on your code, but you would be better off by using prepared statements.

This is a very good topic on this here on stack, I suggest you read it: How can I prevent SQL injection in PHP?

As you clearly mentioned in your question,

" I had do some testing on both file by calling an alert message, the alert messages was able to display in the html file, it's seen like the request from the html file cant send to the php file ,so the code for inserting data to database can't execute ~@Heart Break KID "

For That,

1) Change

<input type="button" value="enter" name="btnSubmit" onclick="show()">

To

<input type="submit" value="enter" name="btnSubmit" onclick="show()">

here, type='submit' is required to submit form data..

2) Closing curly brackets are not available. Close if condition.

if(isset($_POST["btnSubmit"]))
{
    // Your query.
}

Now, data will go to next page. But, read this question How can I prevent SQL-injection in PHP?

The mysql extension is deprecated and will be removed in the future: use mysqli or PDO instead

UPDATED CODE (using mysqli)

Html form

<form id="firstPrize" method="POST" action="firstPrize.php">
  <label> Number 1</label>
  <input type="text" name="num1"><br>
  <label> Number 2</label>
  <input type="text" name="num2"><br>
  <label> Number 3</label>
  <input type="text" name="num3"><br>
  <label> Number 4</label>
  <input type="text" name="num4"><br><br><Br>
  <input type="submit" value="enter" name="btnSubmit" onclick="show()">
</form> 

firstPrize.php

<?php
$host = "localhost";
$user =  "root";
$password = "";
$connect = mysqli_connect($host, $user, $password, "testdb");
session_start();

if(isset($_POST["btnSubmit"]))
{
  $num1 = $_POST["num1"];
  $num2 = $_POST["num2"];
  $num3 = $_POST["num3"];
  $num4 = $_POST["num4"];

  $stmt = mysqli_prepare($connect, "INSERT INTO table1(num1,num2,num3,num4) VALUES (?, ?, ?, ?)");
  mysqli_stmt_bind_param($stmt, 'ssss', $num1, $num2, $num3, $num4);

  $query123 = mysqli_stmt_execute($stmt);
}
?>