Using time-dilation to our advantage? Assume we have a super-computer working on an important problem, and we estimate it will take about $200$ years to solve. But, we need the answer in $20$ years.
Could we ship the super-computer into space at $1/10$ of our speed to get the answer in $20$ years?
Edit We want the super-computer to age, so it gets $200$ years of production while we age $20$ years. Is this not possible? This would be the opposite of the twin paradox: the traveling twin would be traveling much slower.
Edit2 The goal is to over-clock the computer by sending it on a slower journey. The $200$-year solution takes $x$ computer cycles, so the check-points should send results from more cycles, regardless of how its clock compares to ours.
 A: As mentioned in the comments, your conclusion is backwards (haven't checked the validity of your calculations. I am not sure what you mean by 1/10 of our speed). We know from the twin paradox that the traveling twin is the twin who has aged less when they return to Earth due to changing inertial reference frames during the trip.
Therefore, if you wanted a calculation done "quickly" you would have to be the one who hops on a spaceship, goes on a long journey, turns around, and then comes back.
I suppose there is nothing wrong with this in theory. I guess you would need to have sufficient justification that the use of resources to do this trip outweighs the time it would take to just wait on Earth.
Or you could go hang out by a black hole for a little bit.

Based on your edits I think I understand what is going on here. You are taking "the universe" to be some "absolute reference frame", and you are thinking that the Earth has some speed relative to this frame. Then you send the computer at a speed relative to the universe that is less than the Earth's speed relative to the universe. So you say that the time dilation of the Earth relative to the universe is then greater than the time dilation of the computer relative to the universe, so we get faster calculations relative to us.
This is not how relativity works. There is no absolute reference frame that has the "true time", and there is no moving relative to this absolute reference frame that changes your own experience of time. All that matters in this scenario is the relativity between you and the computer. This is why saying "the computer is at 1/10 our speed" is meaningless. The computer can only have some speed relative to us.
A: No.
First of all, $1/10$ of your speed doesn't result in $10$ times faster clock — the Lorentz's transformations are a bit more complex.
Second, the true effect of such great time dilatation is able only in speed very very close to speed of light, so it's practically impossible.
Third, to obtain the result, you have to meet supercomputer again, and the duty of changing velocity (to accomplish this) would (mainly) be on you — in the opposite situation the supercomputer would appear as a slower one, and not a faster than a supercomputer placed just next to you (in your reference frame).
A: Adding to the previous answers - it is correct that you got the twin paradox the wrong way around - it is the travelling twin that ages less. Nonetheless there is a bizarre use for this, it is not in making calculations more efficient, but one could build a relativistic freezer.
Does your produce go off before you eat it? Easy, just ship it on a rocket moving close to the speed of light away from the earth and then back - the produce will be still fresh even after a long time has passed in your rest frame. Of course, the timing of the delivery and the energetic requirements make this use a little bit far-fetched.
