Q:

Why can't they use super computers to mine all the bitcoins?

I have very limited knowledge of bitcoin, but, I hope you don't mind my asking this question:
As I understand it, "mining bitcoin" requires "significant" computational prowess. A lot of dedicated mining rigs have been built out of FPGAs, ASICs and GPUs. But, I have always wondered: would it not be possible to mine ALL the bitcoins if you had access to some of the most powerful computers? So, for argument sake, if the NSA decided to stop snooping upon people and instead decided to, you know, mine bitcoins… would they be able to mine all the bitcoins in just a few hours if they redirected all their super-computational-power towards this one problem: mining bitcoins? Other potential candidates for this game include, but are not limited to: NASA, DoE, etc.
I realize that running a supercomputer to mine bitcoins may NOT be efficient (because a super computer consumes too much power, costs a lot of money, etc) but for a govt agency all that money comes from taxes so ... 
Now, hypothetically, could "they" mine all the bitcoins in a few hours, or even in a few minutes if they redirected all their super-computational-power towards this one problem: mining bitcoins?

A:

A supercomputer is way slower than mining with ASICs. A supercomputer only has much CPU power, not even GPU power and ASICs are way more powerfull than GPUs. ASICs represent the hashing algorithm as hardware which means they can't do anything else, that's why they are so fast.
At http://bitcoinwatch.com/ you can see the current network hashrate in PetaFLOPS, which is 1534782.24 at the moment. Currently the fastes Supercomputer Tianhe-2 has 33.86 petaFLOPS.

but for a govt agency all that money comes from taxes so ...

If they would use all the money to buy ASICs and start mining bitcoins, it still would not be enough because of the difficulty factor. But they would be able to get a lot of blocks because the difficulty only changes every 2016 blocks.

A:

There are two reasons why they can't flash mine all bitcoins: Difficulty and specialized hardware.
First, the pace of Bitcoin creation is limited. The protocol defines how difficult it is to find more bitcoins. At normal pace, the difficulty is evaluated and readjusted about every 10-14 days. However, because the difficulty adjustment happens after a fixed number of blocks, the time until readjustment is shorter when blocks are found at a higher rate.
When blocks are discovered quicker than the target 10 minute interval, the protocol requires a readjustment to a higher difficulty, which in turn causes the pace to slow down. For a more detailed explanation see: How does the network adjust the rate at which the coins are created?
Mining in Bitcoin is done with application specific integrated circuits. ASICs are chips that can only execute one algorithm because it is hardwired into the chip. As these Bitcoin ASICs are single-purpose designed to solve only the calculations required for mining, they are highly optimized to do so. This lack in flexibility allows to increase their efficiency over general purpose hardware immensely, and has caused the hashrate of the Bitcoin network (the total "mining power") to grow about 20,000x (twenty thousand fold) since the first Bitcoin ASICs have shipped.
General purpose chips are many magnitudes less efficient at mining, and therefore don't stand a chance at competing.

