Cheyenne Mountain/NORAD surviving a nuclear war is always a common trope of post-apocalyptic fiction, and I remember reading that the facility is designed specifically to withstand "some" nuclear attacks. Like maybe a 1-2 megaton warhead that strikes nearby but not directly on the mountain itself because it's made out of granite, one of the hardest rocks in the world.
I also have a vague memory of reading that despite Cheyenne Mountain's supposed survivability, the Russians likely targeted it with several hundred missiles, to ensure overkill. This also being because I remember reading that Soviet missiles were more inaccurate than American ones, and thus they compensated accuracy with megatonnage.
So with that understanding, during the Cold War and given the state of Soviet technology, how many nuclear strikes could Cheyenne Mountain (NORAD) sustain and continue to function in some capacity?
Some of the wording here is obviously vague, but I figure that would allow space for broader discussion. Whether "some capacity" means fully functional with everyone in it surviving, partially functional with some survivors, or scavengable/repairable with a few or no survivors.
That way, I can paint a proper mental picture of my post-apocalyptic fictional world and the struggles of the survivors to rebuild civilization from out of the ashes of Cheyenne Mountain.
A 5Mt nuclear device would be powerful enough to destroy the facility, if it detonated at the right place. The blast doors are capable of withstanding 600 psi overpressure before failing. That level of overpressure is produced in the area close to the epicenter of a large nuclear device. However, the circular area probable of the delivery systems was very large. For example, the R7, which was active until 1968, carried a 3 Mt warhead, and had a circular area probable of 5 km, so it would take a lot of luck to get close enough to produce the necessary load on the blast doors, even with multiple strikes.
Newer missiles that were coming online in the late 1960's packed an even larger punch, and could soon boast a CEP of only 220m while carrying as many as 10 independently targeted 1Mt warheads. A handful of such missiles would have a very high probability of knocking out the facility.
Modern Russian ICBMs have CEPs as low as 50m with 4x 550kt warheads. Devices like this made Cheyenne mountain irrelevant, and it's operations have been moved to more accessible but not well protected premises. It is no longer considered practical to protect critical facilities from sustained nuclear attack from a technologically advanced adversary.
Honestly, the comms going into and out of the facility would have been wrecked long before Cheyenne Mountain itself was destroyed. Even at the height of the Cold War there was a finite amount of deliverable warheads, and there would have been better uses for them than hitting the mountain until it was leveled.
The problems are that the ICBMs themselves tended to be less reliable than bombers as delivery platforms(I think the percentage Stuart Slade, probably the only outed nuclear targetteer on the Interwebs, used for ICBMs making their way to the target was 'round 60%), so instead of expending them again and again on Cheyenne(or SAC in Offut, or Site R, etc) they would want to work over the American delivery systems like ICBM fields, SSBN bases, and air fields. Set a airburst over the various antenna farms near the bunkers for a mission-kill, and then concentrate on what can actually hurt you.