Here's an impossibility in "Pre-calculus workbook for dummies" by Yuang Kuang:
"Solve 3-4(2-3x) = 2(6x+2). The answer is no solution.
Distribute over the parentheses on each side: 3-8+12x = 12x+4. Combine like terms to get -5+12x = 12x+4 and subtract 12x from each side. The result is -5 = 4, which is false. Consequently there is no solution."
But what if we were to interpret the "impossible" equality of -5 to 4 to mean carry on with the exercise by performing the operation of substituting -5 for 4 for in the original equation to get 3--5(2-3x) = 2(6x+2) with the solution x = 11/3
This would appear to violate at least two fundamental conventions. One is that we aren't allowed to change the question. The other is that two different numbers can't be equal, otherwise what would be the point of trying to come to any numerical answer if any other were correct? As against that I've used the same kind of extrapolatory logic by which mathematicians make sense of other initially strange expressions like 2^0 or 5-9. In fact I'd call the inference I suggested here Modus Corrigens, since it corrects the question, and compares with Modus Tollens of medieval logic.
Logic, in its creative, interrogative and exploratory incarnation, characterises challenges to declarations of impossibility in the physical sciences too. ("Impossible reaction" , Philip Ball, in "Nothing: from absolute zero to cosmic oblivion" New Scientist publications)
To echo that old joke pinned up in workplaces, mathematicians can do the difficult, the impossible takes a little longer. Centuries longer sometimes.