In so many ways the FAI are damned whatever they do in the wake of Old Trap. But then they might just do something which from time to time redeems the most desperate of situations.
They might speculate on what they have to lose by appointing the most divisive, abrasive and, perhaps they should not forget, one of the most inspiring figures in Irish football history.
In other words, they might simply turn to Roy Keane.
Of course, it would not be something to do with a light heart and uncomplicated emotions.
The backlash would be formidable, no doubt. From one side of the great divide which opened up back in Saipan in the World Cup of 2002 there would be old cries of treachery. There would also be hard questions about the working track record of the man many once saw as the natural-born successor to Alex Ferguson at Manchester United.
There would be charges that once again the mythology of Keano the Terrible had been re-instated in all its old and glowering force.
Yet there would be more than a mere frisson of drama in the Irish game, which has suffered such grievous blows to its self-regard in the last year or so.
The inevitable polarisation would be sharp and bitter, but then it could prove extremely stimulating.
Certainly it could be balanced against a set of options which are not overwhelming in their compulsion.
Even the people's favourite Martin O'Neill would draw some heavy-duty doubt despite his enduring charisma as a football man of weight and passion. He failed at Sunderland, in the end quite profoundly, and tactically it is hard to refute the charge that basically he amounts to Trapattoni minus the pedigree.
Given the desperately slender nature of Ireland's talent pool, these may not be the worst of credentials – certainly they are likely to carry him beyond the claims of such new contenders as Brian McDermott and Owen Coyle and the more familiar Chris Hughton and Mick McCarthy.
All of them have proven qualities, not least in the highly relevant matter of resolve in the face of the most serious odds, but where is the sense of a galvanising force, the idea that there might just be someone around capable of creating if not a new side of unsuspected accomplishment but one which might just fight at another, superior level?
At 42, and with his failed opportunities at Sunderland and Ipswich no doubt still tugging at the certainties which once seemed so implicit in everything he said and did, it might just be his last chance to impose the professional intensity which he always promised to inflict when his playing days were over.
The FAI might say that the challenge they most need to meet is the one outlined earlier this week by Keane's former patron, Niall Quinn. They may believe that indeed they have to pursue some long-term reseeding of Irish football resources and concede that if Trapattoni represented an expensive and not entirely unsuccessful attempt to get the best from what was available, the new need is for both patience and a sense of reality.
However, football fans, and not least those who fill the Aviva stadium with life-blood revenue when they have been sufficiently encouraged, or intrigued, have rarely been known to live in the future. They need an edge of excitement, and some kind of hope, every time they trek to the stadium.
Keane, it has to be guessed, would create more of that than any rival for the Trapattoni succession.
There is in him a hunger that is plainly unappeased by financial security and a widespread conclusion that his style and his instincts will never fit smoothly with the day-by-day rhythm of today's football club. But, who knows, maybe in shorter, richer doses it could provide the kind of striving momentum that ebbed so irrecoverably in the vital games with Sweden and Austria.
In his most recent appearances as one of English TV's high-profile analysts there have certainly been hints that he may finally have grasped that the demands he set for himself, so consistently and often so gloriously in the colours of both Manchester United and Ireland, are not so easily met by the kind of team Trapattoni was required to field in Vienna this week – or the one sent out by England's Roy Hodgson in Kiev.
England had done well enough in all the circumstances, he said with a forbearance that would have been odd, say, on that day when he virtually outplayed the Netherlands on one leg in the game that carried Ireland to that ill-omened World Cup.
Yet if this was a tolerant, even mellow Keane, there is still no shortage of evidence that more withering judgments hang just below the surface.
The scowl still lurks, along, no doubt, with the anger that exploded in Mick McCarthy's face on that desperate, scorn-filled day on the Far Eastern training pitch.
The most vital point is that Keane, whatever the turbulence of his last few years, retains a certain meaning, a spread of attitude that might be absurd in many men but not one of the Corkman's extraordinary achievements.
Of course he is a gamble. Of course the FAI has half a dozen options less guaranteed to cause a storm of controversy. But what is so bad about such ferment when the greatest challenge is to make the people, and not to mention the players, still care?
Keane is, after all, still a most rumbustious force of Irish nature. If it isn't his time, he may never have another. At the very least it is not something to dismiss too lightly.