Right. The best thing you can do is say the numbers are an element of the set of tuples {(x, 6-x) : x \in R}. And that's only if you restrict to reals. Consider the fact that the "average" of i and 6-i is also 3. In general, you replace R by k, where k is your field (or even any unital ring where 2 = 1+1 is invertible). Thus, in the finite field F_5, the possible numbers are (0,1), (1,0), (2,4), (4,2), (3,3). More cleanly, you can take the k-points on the variety x + y - 6 = 0.
My answer to his question at the end of the essay would be "0, π and 2π". Strangely, this triple is the only one that looks nice to me, even though all of "3 and 3", "0 and 6" and "2 and 4" as well as coming up with a pair with long decimals looks OK to me for the original question.
Actually they did:
"But three other people all said "2 and 4". That is certainly another "nice" answer, but the constraints it satisfies are more subtle. They have something to do with requiring the solution to have integer components that are distinct, but near the specified average. It's harder to state and compute the solution in MATLAB without just giving the answer."
In English, when you say you have two "somethings", they're generally considered distinct. This is one of the gotchas when translating between English and math.
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[ 3.5 ms ] story [ 50.2 ms ] threadThe 2010 followup on compressed sensing: http://www.mathworks.com/company/newsletters/articles/cleves...