I don't know what the solution is, but I've had the same problem, and my computer isn't more than 6 months old. My sister also has the same problem with hers. It only seems to be when I completely shut down, and then try to immediately restart. If I choose "restart" in windows, it works just fine. Instead, when I shut down, and then turn it back on 10 seconds later, I get blinking lights and my CPU fan running on high, but nothing. It'll do that forever if I let it.
Gateway
128 MB RAM
20 GB HDD
800 MHz Intel
I have a REALLY OLD Gateway that I have had for a little while, from my sister. It is one of othose mini ones, so I have the top off so I can fit my USB 2.0 card into the PCI slot (it's too tall to fit inside, with the top on). It's really weird in that when it is off, I have to turn it on, let it sit for a while (fans spinning, drives reading... LED = orange). I can then restart it after ten minutes or so, and the LED will turn green, and it will start up just fine. It is almost like warming up a car.
For this reason, I mostly leave my computer on all the time, unless I am leaving for a while (over the weekend, or something like that). I then unplug my stuff from the wall (stereo, computer, etc), so they can get a "rest," and so they can't catch fire, or something.
Well, I turned it off last night, to see how well I could sleep with it off, and when I woke up, I turned it on to let it warm up... I did some stuff in the BIOS, but nothing really that would mess anything up - I was just trying to see if I could stop it from doing whatever it was doing with the whole start up issue of warming up. I think I was able to do it before. Well, after I did that, I decided to mess with the password settings, and then I decided I wanted to cancel them, but for some reason I couldn't disable it. Lots of the settings were now gone out of the menu, so I turned the PC off, and took the battery out, so the BIOS would be reset. When I put the battery back in, I slipped, and accidentally tapped the board with it.
I pressed the power button, and nothing.
I don't think I fried the board, but I don't know.
Everything's plugged in, everything is fine. But when I press the button, I get nothing. I even tried touching the two power contacts on the board with a screw driver, to simulate pressing the power button, thinking maybe /it/ was faulty. Nothing.
THe hard drive works fine; I booted from it from a different computer (my stepfather's), but even with faulty drives, it should start up.
No fans, no electricity running through it.
What's up? Any clues?

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