Limiting Number Of Form Submissions?

I'm using the script below to limit a visitor at my site to only be able to post a form 5 times. I believe the reason I've received more e-mails is because the visitor closes their browser then opens a new one and they can post two times each time they do this.

<%
count = Request("count")
if count = "" Then
count = 0
Else
count = count + 1
End If
%>

<%
If count < 5 Then
%>
<a name="step5"></a><input type="submit" name="Submit" value="Submit">
<%
Else
%>
This form has been submitted 5 times. Please contact me for more information.

Can somebody show me how I can do this so it will only work twice within a 24 hour period even if the visitor closes their browser and opens a new one? I would think cookies be the answer to this but what happens if the visitor has cookies turned off? If this would be a good solution can somebody please help me out? I'm not familiar with cookies.