The answer astounded them.
The cadets were from Ireland, fresh-faced 18- and 19-year-olds who, just a day earlier, had been whisked from their barracks on a remote, wind-swept plain in County Kildare to travel, along with Irish President Eamon de Valera, to Washington for the funeral.
With names like McMahon, Coughlan, Sreenan and O'Donnell, they hailed from towns and villages all over Ireland. Most had never been abroad, never been on a plane. Yet there they stood, a foreign army on American soil about to give a final, silent salute to a U.S. president with an Irish name: John Fitzgerald Kennedy.
Even today, they marvel at the fact that, in her darkest hour, Jacqueline Kennedy made a special request of the U.S. State Department: that the Irish cadets who had so mesmerized her late husband with a memorial drill for the dead during his visit to Dublin just months earlier, perform that same drill by his grave.
"This is not the land of my birth but it is the land for which I hold the greatest affection," Kennedy told the cheering throngs at the end of his historic four-day visit to Ireland in June of 1963.
His trusted adviser, the late Ted Sorensen, said, "The joy never left him."
Joy consumed Ireland, too, as it welcomed home its anointed son. Kennedy's great-grandfather had emigrated from County Wexford in 1849, and the Irish took an intensely personal pride in their connection to America's first Irish Catholic president.
From the stately chambers of Dail Eireann, the parliament in Dublin, to his ancestral home on a farm in Dunganstown, where he drank tea with relatives and broke away from his bodyguards to join a children's choir in a rousing rendition of "The Boys of Wexford," Kennedy received a rapturous welcome.
"Occasionally in the history of a country, a thing happens that means more than can be put quite into words," wrote Patrick O'Donovan in The Observer, a London newspaper. "The visit of President John Kennedy to Ireland was one of those things."
Kennedy himself wrote, in a thank-you letter to de Valera, that the trip had been one of the most moving experiences of his life. And a highlight, he said, was a wreath-laying ceremony by the graves of the leaders of the 1916 Easter Rising in Dublin.
As part of the ceremony, 26 army cadets, led by an officer, performed a special, silent drill in remembrance of the dead. The slow-moving solemnity and precision of their movements captivated Kennedy. The drill concluded with the cadets bowing their heads over their rifles, a gesture of quiet contemplation for the departed warriors.
"That is the finest honor guard I have ever seen," Kennedy told the officer in charge, Lt. Frank Colclough.
Back in Washington, Kennedy requested a film of their drill: There was some suggestion that he wanted to introduce elements of it to honor guards at Arlington.