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An Xbox 360 console may often show an E68 error when booting with an on-screen prompt instructing a user to contact Microsoft Customer Support, while a blinking red light will be present in section 4 of the ring of light on the front of the Xbox 360 console, as shown in the picture beside this text. This is a hardware failure error.
E68 is similar to E67, except that E67 appears to be caused more by hard drive errors. E68 is also caused by hard drive problems, such as the HDD not being DMA configured. The HDD could also be defective or the connection between the HDD and the motherboard could be damaged.
E68 also seems to occur because of other components too however. Some reports put the blame on external fans that attach to the back of the console, such as the Intercooler. Microsoft itself has warned users not to attach these fans to the console (they are relatively ineffective anyway). The problem could also be the result of the fans, for some reason, drawing too much power from the unit.
Removing extra peripherals or the HDD bay from the console and booting it might give you an idea what is causing the problem. This problem might be covered by your warranty with Microsoft or the store you bought it from, depending on your location.
E68 is often associated with the error code 1010. To get an error code from an Xbox 360 console that is displaying an E68 error (or 3ROL) you simply press and hold the "sync up" button (used to connect wireless controllers) and then press the Eject button for the DVD drive. This will bring up the first digit of the error code (usually it is one light blinking red, which represents 1, please note that four lights blinking red represents a 0, not 4). With the sync up button still held, press the Eject button again and you get the second digit. Do this twice more to get the full 4 digits of the error code.
For more information on the error code you receive, or on Xbox 360 hardware problems in general, read this Xbox-Scene forum thread.