We also have the same problem here. We have two 360s, one launch, and one from Nov. 2006.
Our launch 360 started displaying what looked almost like "cel shading" (I think someone else said the same thing a couple pages back) two days ago. We rebooted the console to see if that would fix it, and we got the Black Screen of Death. We always connect using the component cables, so I tried connecting the cable to our other 360 and it worked just fine (on the same television, with ALL the same settings). So, looks like it's not the cable. Just to be sure, we connected 360 #1 (the dead one) to the t.v. using the regular RCA type cables and got the black screen with white dots. Then, we connected it to my LCD using a VGA cable and got the Black Screen of Death. There is no doubt that this problem has nothing to do with the cables (so the help articles above are no help).
We also tried rebooting repeatedly as a lot of other posters say to try, and still nothing. We removed the HDD and rebooted, and still nothing. We tried standing it up - nothing. Laying it on its side - still nothing. We tried switching between HDTV and TV on the cable, and still nothing. We tried resetting the video to default (holding Y and RT), and guess what? Still nothing but a big Black Screen of Death.
I would like to point out though, throughout this whole ordeal, the surround sound still works like a charm. I guess that's something, at least.
Sure, we can still play on the other 360, but that's not much consolation - we need our launch consoled fixed!
There was a time when a certain other of MSFT's products liked to give us the infamous Blue Screen of Death, and we were told, "every computer is a different and there's really nothing we can do about it." Well, everyone's 360 is the same, and a whole lot of us are having the EXACT same problem all within a very short timeframe. This is not just a handful of people with similar problems, this is a growing number of people with a very specific problem, with very specific symptoms. I think it would be negligent on MSFT's part to ignore this.
It's been suggested to me by a friend - who had her 360 die the same way yesterday - that maybe this is a software issue. A lot of people said they were playing Forza 2 and GOW when this happened, and there have been updates to both these games recently. What if, and this is a big (but not impossible) if, there was malicious/viral code accidentally embedded in one of these updates? If the systematic symptoms were happening on PCs, that's the first thing people would assume - virus.
I agree, it's highly unlikely that there is a virus causing this problem, but it's also highly unlikely that so many people would encounter the EXACT same problem so close together.
Anyway, that's my 25 cents.. I guess Black is the new Blue.
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