I’ll bite - may I ask the name of the company, and what was invented that most of us use every day?
The SAT works like this, too, so turning in the paper blank won’t actually give you the lowest possible score. The penalty isn’t actually all that large, though, so guessing completely randomly will, on average, give you approximately the same score as turning it in blank (and guessing not-completely-randomly, like ruling out an obvious wrong answer on each question, will on average still be better than nothing). The ACT, on the other hand, does not have a penalty for guessing, so if you’re down to a minute left and aren’t finished, you might as well just bubble in C for all remaining answers.
And there was one math test I took in high school (that didn’t count for anything but prestige, unless you scored really well) where if you turned in the answer sheet completely blank, you would get 60 points out of a maximum of 150 possible… But the national average was only 58. This even after the teachers all warning their students of this fact.
In a huff? In a minute and a huff?
http://www.marx-brothers.org/whyaduck/info/movies/scenes/firefly.htm
Uhm, you don’t even have to think about this to know that it is impossible.
Yes, there have ben a lot of experiments in which the participants are told that they’re being tested on, say, math ability or spatial reasoning, but the test is actually set up so that they can’t succeed, and what they’re really being tested on is generally their response to the stressor. Here is an example using an impossible anagrams task.
Are you assuming each die has numbers 1-6 on the faces? You could have no ‘1’ and multiple '2’s on one die and a ‘0’ on the other.
In consulting, we have what is called the “case interview”. Basically it’s a question like “how many flights land at JFK in a week” or “how many manhole covers are there in Manhattan”.
Now the typical SDMB response is “those questions are stupid I would never work for such a stupid company that tried to trick you with something like that blah blah blah.” The thing is, the answer isn’t important. The case interview is designed to see how you take a complex problem and break it down. We look at your reasoning and assumptions.
It’s my understanding tha the Kobayashi Maru served a similar purpose. It wasn’t so much “winning” was the objective (since you can’t). It was to get a sense for how the captain led under difficult circumstances.
Wasn’t there another captain test in ST:TNG where Diana kept failing because she wouldn’t send holographic Jordi into the irradiated air ducts to pull some stupid lever that will save the ship?
But if you are allowed to use an algorithm for getting the result that’s different from “Add the result of each die together”, (and can distinguish the dice, and are allowed to re-roll if necessary or, equivalently, allowed to return a ‘no answer this time’), it’s trivial to do. It’s not like adding A to B is the only way (or even the only natural obvious) way to get a third number from A and B.
… and that’s why I’m not qualified to work in engineering. 
Obligatory XKCD reference, showing the execution of a Fermi estimate.
Very possible and simple:
Design = one red and one blue:
1 -Role them
2 - Take the number rolled on the red die, add 6 if the number on the blue die is 1,2,3
3- reroll if the answer comes out to 1 (not really required given the OP’s guidelines)
Or correspondingly one die standard numbered and the other numbered 6,6,6,0,0,0 and add them together
Another more complex answer may be to design dice with uneven sides for instance where 1+6 have a larger probability of coming up and 4 and 3 have the lowest.
Maybe 
but that would be a much more complex solution and I’m down with my simple one above 
Except that your post #50 doesn’t actually accomplish the objective of having the two dice roll two 9% of the time; three 9% of the time, four 9% of the time, etc. Under your numbered scenario, it would be impossible to come up with the result of two, for example (because rolling 1 on the red die and 1 on the blue die would result in eight being the lowest possible total).
My first reaction was 2, 4, 6, 8, 10, 12, and 0, 0, 0, -1, -1, -1. The result is the same. It would work depending on the precise wording of the prompt; so would two dice labeled 13, 14, 15, 16, 17, 18. It’s only if the prompt is constructed very strictly to permit only the eleven unique results from 2 to 12, with no results out of that range, that this becomes a difficult task.
I assume you mean post 51 since you state red and blue dice.
Let’s use my second example, it’s easier but the same
1 die, numbered 1,2,3,4,5,6
1 (the 2nd) die numbered 6,6,6,0,0,0 (basically a coin flip, so I will simplify this to 2 possible outcomes 0 and 6)
Role possibilities (1st die, 2nd die):
1,0=1 (can be discarded if you wish, take as the die rolled off the table, invalid role)
2,0=2
3,0=3
4,0=4
5,0=5
6,0=6
1,6=7
2,6=8
3,6=9
4,6=10
5,6=11
6,6=12
Those are the only possible outcomes, and if you include the role of 1 or throw it out (as dice sometimes have to be rerolled) the probability of all outcomes 2-12 are exactly even at 0.083333- if we allow 1 as a valid role or 0.9090- if we take a role of 1 as invalid.
Roll.
If you have to adjust the data to come up with the outcomes you want, then the dice quite literally are not designed to achieve the outcome.
For example, if I have blue and red socks in by sock drawer, and I want to pick out blue socks 100% of the time; setting up a rule that discards any pick that turns up a red sock means my system doesn’t work.
I recommend a Star Trek novel The Kobayashi Maru’ by Julia Ecklar, which covers how several Enterprise officers tackle the test.
It wasn’t an actual test, but when one young pupil applied for a Chess Scholarship at my School, I managed to get him into a clock simultaneous display versus Kasparov!
Sure, but if someone is asked the question and says “well, here’s a set of dice faces and slightly modified rules that comes very close, depending on how the question was phrased”, that’s presumably a more-likely-to-result-in-getting-hired answer than someone else who just tries sets of numbers for a while and then says “gee, seems impossible to me”, even if it is in fact impossible.
Actually there were no slightly modified rules. I was asked to design a set of dice that would give equal probability for outcomes 2-12
I did exactly that, with throwing out the 1’s or including them, your choice, given the parameters I made every single one of them and every aspect of them exactly and without any compromise.
I also feel like the poster who stated the solution as impossible should be thankful for getting the job despite the incorrect answers but with kudos points for fully explaining it.
I also feel like the dimensions of the dice or weighting of them may still be able to be used (since we are allowed to ‘design the dice’) can be done that may allow standard rolling to achieve this.
So in short
1 - if one can design the dice to roll all equally via weighting or non-cubic design, that would be the people who get man to walk on the moon
2 - if one can come up with the simple workable solution of renumbering one die 6,6,6,0,0,0 (or likewise), these are the people who can get Apollo 13 back to earth safely after the first group screwed up
3- if the person can only prove it is impossible then they should be thankful they were given the job if it is proven to be possible by group 1 or 2 or both.