Crazy Shutting Down Problem!

In my 40 odd years of owning Macs, going back to the Mac Classic with a 13 inch black and white monitor, I've never seen this. It started about 2 weeks ago, when I got home from work and my Mac had shut down. I thought there might have been a power outage, so I just re-booted. No problem. But then it started happening more frequently. A few times it shut down while I was in the middle of doing something, like getting emails or surfing the web. Skipping ahead to the present... now, when I power it on I get the bong, the gray apple, and the gray progress bar, but it shuts down almost exactly when the progress bar gets half way across! Last night it did it about 10 times in a row. Today, same thing. Also, when you power up using modifier keys, like resetting the PRAM or booting up in Safe mode, or holding CMD plus D to run a hardware test, it shuts down before the progress bar even appears on the screen. Here's what I've done so far... tried booting from different hard drives, including an external... tried the boot drive in a different bay... removed the RAM cards and only used one 16GB to boot... swapped RAM slots... disconnected every peripheral, including my monitor, keyboard, and mouse... even removed the ATI Radeon video card completely from the machine... removed the optical drive... removed the processor tray to clean and inspect it. Cleaned all fans and insured that they were working. And that's about it. One thing I've noticed... sometimes the red LED that's next to each RAM slot is lit on two of them. They're the ones on the two slots that are closest to the processor. In those slots I have a 2GB card in each. If I remove them, the LEDs go out, but it doesn't help with the problem. Btw, I have two 16GB RAM cards in the two front slots. I'm thinking that my logic board is bad. I'm stumped!