There are quite a few options, so feel free to take whichever one works for your particular setup.
If you want to use
/p/1
in links, text, whatever -- and have it rewrite to
/products/game.php
(using it as a URL shortener):
Options +FollowSymLinks
RewriteEngine On
RewriteRule ^p/1/?$ http://www.mywebsite.com/products/game.php [R=301,L]
If you want
/p/1
to display the contents of
/products/game.php
but still use the shorter URL in the address bar:
Options +FollowSymLinks
RewriteEngine On
RewriteCond %{REQUEST_URI} ^/p/1/?$ [NC]
RewriteRule .* /products/game.php [QSA, L]