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Say for example, one of the cores in a quad core computer overheated and died, would the system compensate by routing all operations to the remaining 3 cores, or would the computer be inoperable? If the system compensates, could it compensate for 3 of 4 cores failing?

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If you have a 4 cylinder car and one of the cylinders breaks, do you expect it to run fine on the remaining 3? –  Hennes Jan 17 '13 at 19:43
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@Hennes: Cylinders cannot be compared to CPU cores, given that it's possible to turn off any given core manually. It's more similar to four engines in a single car. –  grawity Jan 17 '13 at 20:57

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It largely depends on the exact failure (and the system architecture, but I'll assume you're talking about a standard x86 based system).

In short, the system will not function properly - with exact symptoms ranging from appearing to be okay "most of the time" to a complete failure to boot. There is no mechanism to ignore the bad core at run time, and as such anything that attempts to execute there risks failing/corruption/crashing.

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No. Your computer is not supposed to start-up at all if any core has failed.

If any core failure is detected at testing and packaging stage, it can be disabled like those AMD 3-core processors.

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A CPU core cannot be compared to a car engine. Depending on the system BIOS, it may or may not detect a core failure. Most probably it will still route operations to the core and fail. The BIOS may allow for discrete core disabling (trial and error on each core), in which case you will be able to operate normally. At the end of the day, time to replace the CPU

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