Not familar with hamachi, but it would seem to me that either some third party software firewall or perhaps even the built in windows firewall is blocking pings, of course it's always possible that the router itself is configured not to reply to ICMP Echo requests so I guess you need to check your router documentation as well as determine if you have any third party software that is acting as a firewall and preventing your network from functioning properly. I might even suggest as part of a temporary test, disabling the Built In Windows Firewall for a brief period to see if you can ping then. If you have another machine on the network, determine its' IP address and see if you can ping it and see if it can ping the router or your machine.
The ping utility is one of the most fundamental and broadly used functions for testing a network to see what it can connect to and how fast it performs, without being able to use it successfully at all it will be difficult to proceed diagnostically.
Try a couple other things from an elevated command prompt
type ipconfig /release
and hit enter
then type
ipconfig /renew
and hit enter
see if that accomplishes anything.
If not try resetting the Winsock catalog and the TCP/IP Stack
again from an elevated command prompt type
netsh int ip reset c:\users\YourUserName\Desktop\ipreset.txt (hit enter) (Substitute your actual username)
and then
netsh Winsock reset (hit enter)
Reboot your computer.
As far as hamachi is concerned, I believe it comes bundled as part of the LogMeIn software, neither of which do I have much experience with, except to say that in some instances depending on how your VPNs are configured they may be preventing some things from working on your real local network as sometimes they will grab a default gateway from the network you are attaching to, but that still shouldn't prevent you from pinging another node on your local subnet. The trick would probably be to unistall any third party products that you suspect might be interfering with the proper operation of your network and only add them back once your network was functioning properly. That would include any third party firewall products like ZoneAlarm and or Comodo, and AntiVirus / Internet Security Suites and hamachi.
If you cannot ping another node on your own subnet by IP address then that would indicate that you are not connect to it or something is inhibiting the full fuctionality of that connection. You need to try a get back to the basics and start from there.
Regards and keep us posted
Randy