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When I first installed my Hyper 212 Evo I didn't follow the instructions closely, and did not do the above.
The machine seems to runs fine but I'm paranoid I damaged the 3770K in some way. The CPU has some scrapes on the top, but I bought it off of eBay so that may be the cause. What is the likelihood I damaged the CPU and how would I know?
Its the same reason you'd do it on a car tire when installing it.
The idea behind tightening opposite screws is to make sure that the force is even - you end up tightening a pair, and they hold down the unit nice and flat while you tighten the other two. You're unlikely to have damaged anything, and if its cooling properly, you've tightened it enough make good contact.
If it works, you probably did not damage anything. If you bent a pin or crushed a core, the CPU simply wouldn't work.
Anyway whats the likelihood I damaged the CPU
I've never knowingly done it, and never broken a cpu. As long as you don't over-tighten anything, you should be good to go. It's probably a good idea to do it in that order anyhow though (it can't hurt).
On the other hand, the vast majority of cpu fans i touch these days have plastic punch down, twist locking, or some other system to hold them down rather than directly screwing into place.
It's not very unsafe. Worst case scenario is you push some of the thermal paste to one side. That should not happen (or have a critical effect) if the amount is minimal and well spread.
The important part is to keep the cooler steady and leveled when attaching and securing it to the mainboard. The order is not very important, it's more important to gradually screw all the screws in. As opposed to screwing them all the way to the bottom one by one.
