What would your approach be to developing a solution for your boss to a problem for which a primary requirement of a solution is that there be no actual solution of the problem?

One approach might be to forthrightly declare that there can be no solution to a problem which is required to have no solution. You then take the hit to your "future", such as it is, with this company.

Another approach would be to deposit your personal and professional integrity alongside your boss's in the receptacle in which it now moulders, and confidently declare that the problem can, indeed, be solved by the application of sufficient snake oil — and you have just the snake oil needed. You then immediately start looking around for some co-worker whom you can set up to use as the fall-guy when the "solution" you will provide collapses in the face of the first real test of adversity it encounters. If you are sufficiently Machiavellian, you may, as poetic justice and for extra points, be able to set up your boss as the patsy. Be assured that your boss already has you marked down for this position!

In any event, you start prospecting for the next step on your career path, recognizing that all the nails are already solidly hammered home in the coffin of your future with this benighted company. (And who's the president of this hypothetical outfit anyway? Tom Ripley?)