First off, you are right. The power cable was never at fault. It's a problem with the PSU. They should have recalled all the XBoxes and fixed them, but they are complete and udder tight wads. Don't get me started.
But back to your problem:
Hmmm, that IS interesting. Here's my question: what are the temps on you XBox? Is it running hot? The XBox will shut itself down if it's temperatures climb too high.
If the temperature are okay, then here's my next question:
How long does it run before shutting down, and what does it do when it shuts down? If it takes a few minutes before it shuts downs, then FRAGs (flash red and green), then the most likely candidate is a bad solder joint somewhere. Sometimes, if there is not enough solder, it will remain connected fine while cool, but as the XBox heats up, the metal expands and looses connection. That's the next thing I would check.
Tell me if that helps, so if it doesn't I can brainstorm some more. Right now, I'm just trying to find an excuse to NOT write my report about Gauss (due tomorrow of course

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- Greg