redirect page url with different url name

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]