Alienware m17 r1 shuts down and reboots

I just bought a used Alienware M17 R1 for my son. The guy who sold it to me didn't have a password, but id did boot to the password screen, so I assumed that once I cracked the password it would work fine. I did crack the password but now, when I tried to boot the system I couldn't get as far as the login screen. After the windows flags assembled themselves the laptop shut down. The light on the power adapter went out. Unplugging and replugging the adapter seemed to reset it, but the same thing happened again. I took a wild guess that the adapter was defective and was shutting down when the system drew to much current and to reduce the load I opened up the CD drive door. This probably seems stupid to you folks who know more, but it worked. The laptop booted into windows and seemed to work fine. I streamed videos on netflix, surfed the web and all was fine. But then I tried to load a heavy-duty game and right after the splash screen the system once again shut down, the power adapter died. I found that I had to wait a while for it to cool down or whatever before I could restart it.

I figured that the problem could be a power adapter so I bought a replacement. The new adapter did not die. Instead the laptop got as far as the introductory windows screen and rebooted. Over and over. I haven't been able to get the system to boot at all with the new adapter. I can still sometimes boot it with the old one.

I fear the worst--a motherboard problem. Before I give up and try to convince the seller to accept a return I want to try everything I can. What would you folks try next?

I posted more, possibly useful, information below.