Do you have an older version of flash installed now? Look on Adobe site they have a cleanup/uninstall tool. Try uninstalling what you have now and then run the uninstall tool to cleanup leftover crap.
If you are using IE 10, then flash player is included by default. It is updated through Windows Update.
Most likely the problem is related to the Facebook Game not being updated to render properly in IE 10. Most web pages are behind the curve. You can try to use IE 10 compatibility mode. Use the broken box looking icon in the address bar at the top.