LONDON — In the late 1940s, the British magician David Berglas started refining a trick that came to be known as “the holy grail of card magic.” To this day, nobody is certain how he did it.

Decades into his retirement, he has revealed just about every secret in his long and storied career. This includes the time, in 1954, that he made a grand piano vanish in a London hotel banquet room filled with guests. (Distracted, the audience turned their focus in time to see a pianist, who seemed to be playing the instrument a moment earlier, fall facedown to the floor). But even now, when the subject of Mr. Berglas’s famous effect is raised, he remains as cryptic as ever.

“It’s not a secret I can give to anyone because it’s not a secret as such,” said Mr. Berglas, a formal and intense 94-year-old, at his home in North London. “It’s like asking a musician who can improvise to teach you his improvisation, which of course he can’t.”

The trick is a version of a classic plot of magic, called Any Card at Any Number. These tricks are called ACAAN in the business.

ACAAN has been around since the 1700s, and every iteration unfolds in roughly the same way: A spectator is asked to name any card in a deck — let’s say the nine of clubs. Another is asked to name any number between one and 52 — let’s say 31.

The cards are dealt face up, one by one. The 31st card revealed is, of course, the nine of clubs. Cue the gasps.

There are hundreds of ACAAN variations, and you’d be hard pressed to find a professional card magician without at least one in his or her repertoire. (A Buddha-like maestro in Spain, Dani DaOrtiz, knows about 60.) There are ACAANs in which the card-choosing spectator writes down the named card in secrecy; ACAANs in which the spectator shuffles the deck; ACAANs in which every other card turns out to be blank.

For all their differences, every ACAAN has one feature in common: At some point, the magician touches the cards. The touch might be imperceptible, it might appear entirely innocent. But the cards are always touched.

With one exception: David Berglas’s ACAAN. He would place the cards on a table and he didn’t handle them again until after the revelation and during the applause. There was no sleight of hand, no hint of shenanigans. It was both effortless and boggling.

Among magicians around the world, his touchless ACAAN is one of the most talked-about and puzzled-over tricks in history. It was eventually labeled “The Berglas Effect,” and helped make its creator’s reputation in a career that spanned six decades.

Mr. Berglas — or more specifically, his effect — has its share of skeptics. They contend that the secret to his ACAAN is both simple and vulgar. They say he uses a confederate masquerading as a spectator — a stooge. All he needed, detractors note, was an ally with a hidden crib sheet listing the order of the cards. When the nine of clubs was named by one member of the audience, the stooge would consult the list and call out 31, in our example, because she or he knows the order of the cards.

Presto. Instant miracle.

Mr. Berglas and many other magicians have used allies for a wide variety of effects. A crew of accomplices, including a very game pianist, helped make that piano vanish. Collaborators are part of what makes the method behind this stunt, which Mr. Berglas detailed in a video, so brilliant.

Accusing a magician of using a stooge for a trick like ACAAN, on the other hand, is a bit like accusing an athlete of taking steroids. It’s considered a form of cheating.

Mr. Berglas has always and categorically denied using a stooge for this effect, and no one has ever stepped forward to say, “I stooged for the guy.”

Further, over the years, a number of magicians have reported private, one-on-one performances of the Berglas Effect that left them stupefied. The magician and mentalist Barrie Richardson, for instance, described a 1977 visit to Mr. Berglas’s home in his book for magicians, “Theater of the Mind.” Asked for a card and a number, Mr. Richardson settled on the seven of hearts and 42.

After that: “He motioned me into his study and pointed to a deck of cards on his desk,” Mr. Richardson wrote. “When I counted down to the 42nd card, I discovered the seven of hearts. The experience was chilling!”

Magicians lie to spectators constantly. Lying to each other is a no-no, so these testimonials are potent. Mr. Berglas points to them whenever the stooge question is raised.

“If people can’t find an explanation for something, they always say it must be a stooge,” Mr. Berglas said, sitting at his dining room table. “It used to annoy me, but it’s been seen and written up by too many well-known magicians, respected magicians. Now I laugh it off.”

I had contacted Mr. Berglas to ask if he would talk specifically about his most famous trick, and about his life. He managed to say yes while sounding not particularly interested, even a little reluctant. It was as if he couldn’t figure out what was in it for him.

A few days later, I knocked on the door of his house in the London suburb of High Barnet. Ruth, his wife of 64 years, answered and led me past a staircase festooned with vintage posters from variety shows in the 1950s at British clubs where Mr. Berglas performed. At the end of the hall stood Mr. Berglas.

He walks aided by a cane and speaks in a refined British accent that carries no hint of his pre-World War II childhood in Germany. He has a white goatee and a pair of dark eyebrows hovering like hawk wings over his eyes. If he had a cameo in a film, he’d be credited as “Senior Wizard.”

“My career ended more than 20 years ago,” he said. “Though I always say I didn’t retire. I retreated.”

Before that retreat, he led an event-stuffed life. At 10, he sat not far from Adolf Hitler during the 1936 Olympics and watched Jesse Owens win three of his four gold medals. He and his family fled Germany two years later, and he met Sigmund Freud in a London hotel popular with Europe’s refugees. After a stint in the U.S. military, where he served in Germany, helping to censor Nazi-sympathizing media and communications, he discovered magic in London in his early 20s.

“I was completely taken over by it,” he said. Every night of the week he went to a different magic club or performance. In 1955, he was the first magician to have his own television series, “Meet David Berglas,” on the BBC. He performed for audiences as varied as Winston Churchill and fans of the Rolling Stones, for whom he once served as the opening act.

He debuted what became known as the Berglas Effect in 1953. Most magicians asked spectators to physically pick a card from a deck, but he was far more intrigued by asking them to simply think of a card and then name it out loud.

“By the way,” he said, casually, “mention a card, because I’m always interested in what people say. Mention a card now.”

“The seven of diamonds,” I said.

There was a long pause.

“And if I said give me a number?”

“Forty-four,” I said.

He nodded and the conversation returned to his biography. It seemed, in the moment, like he was simply conducting a poll to divine popular cards and numbers.

Then I remembered an anecdote related by Steve Cohen, a magician who performed for years at the Waldorf Astoria and will bring his show to the Lotte New York Palace Hotel in June. Mr. Cohen has described how, in 2002, during a visit to London, he got a ride from Mr. Berglas to a subway station after a night out of dinner and card tricks. As the two neared the train, Mr. Cohen said that the next time they met, he’d love to see Mr. Berglas’s Any Card at Any Number.

With the car parked, Mr. Berglas turned serious. Remember, he told Mr. Cohen, “that you were the one who initiated this — you asked me to show this to you.” He added that Mr. Cohen would remember what was about to happen for the rest of his life.

It turned out that the three of diamonds, Mr. Cohen’s named card, was at the bottom of a deck that Mr. Cohen was asked to fish out of Mr. Berglas’s jacket, which was draped in the back seat. (Yes, it was the only deck in the jacket.) It didn’t escape Mr. Cohen’s notice that technically, this was not Any Card at Any Number. He wasn’t asked for a number. This was more like Named Card at Bottom of the Deck.

The best part of this performance, Mr. Cohen said in a recent phone interview, was the framing. You asked me — in other words, this isn’t some setup. Well, that along with the bold assertion that what happened next would never be forgotten.

“It’s ingenious,” Mr. Cohen said. “It’s almost epoch making. You’re planting a landmark in people’s heads. That showmanship is really key to the story.”

On the chance that Mr. Berglas was waiting for an invitation to demonstrate his ACAAN, as he might have been with Mr. Cohen, I summoned the nerve to ask for a performance.

This was a mistake.

“No,” he said, flatly. “I don’t need to prove myself, to you or anyone else.”

He pivoted quickly from irked to indignant.

“For somebody to come in here and ask me to perform, at my age, I think it’s out of the question. It makes me feel strange that you need to ask that. It puts everything on a different footing.”

Then he got angrier. With evident pique, he asked me to perform a card trick for him, suggesting that this request might give me a sense of his umbrage.

I apologized and explained that I had the Steve Cohen story in mind. A few minutes later, for reasons that I can’t explain, he softened.

“There’s a drawer behind you,” he said. “There are some cards in there which I haven’t touched in a long time.”

I turned and opened the drawer. Inside I saw three decks in their cardboard boxes. Time began to slow down. I placed the decks in the middle of the table. He didn’t touch them.

“Choose one of these,” he said.

I picked one.

“Open it and place it facedown in front of you,” he said. “What was the card and number you mentioned?”

“Seven of diamonds, and 44,” I said.

“I’ll take a chance,” he said. “I might be off by two.”

I dealt the cards myself, one at a time, face up. By the time I reached the 20th card without seeing the seven of diamonds, I felt a growing incredulity. When I reached the 30th, I felt the stirrings of astonishment. By the 39th card, I was giddy.

Forty. Forty-one. Forty-two. I turned over the 43rd card.

It was the seven of diamonds. I stared at it, both gobsmacked and baffled. Gobsmacked because it seemed wildly improbable that he’d come so close. Baffled because it wasn’t spot on.

“One off,” Mr. Berglas said, evenly.

He didn’t try to sell 43 as a triumph. On the contrary, he said that in lectures to magicians he has always told his audience that one off isn’t close enough.

I left his house in a muddle, and I have returned to that muddle every time I think of this performance. Off by one seems, on some level, more perplexing than nailing it. Off by one implies that there is nothing automatic about this ACAAN, that it isn’t a contraption that simply works when deployed. It’s more like archery, which requires practice and concentration and can end with something other than a bull’s-eye.

I ran these ideas by Aaron Fisher, a highly regarded American magician who did a commentary in July on his YouTube channel of an old live show by Mr. Berglas. Mr. Fisher said he didn’t know what to make of 43 either. But he noted that Mr. Berglas is not renowned for dazzling sleight of hand.

“He messes with minds,” Mr. Fisher said, “not decks.”

None of this resolved the stooge question. Mr. Berglas may have a number of different methods, depending on the circumstances. “He never knows what he’s going to do before he does it,” Richard Kaufman writes in “The Berglas Effects” — note the plural — a lengthy book for magicians that explains every card trick in the Berglas canon, with one very notable exception.

The book suggests that Mr. Berglas is nothing if not a masterful improviser and a born gambler. What seems like a cohesive performance is actually a high-wire display of spontaneity with a heavy overlay of psychological manipulation. In hindsight, it seems likely that his anger was part of the show, a framing device. “I don’t need to prove myself” is just a different, more contentious version of “You’ll never forget what is going to happen next.”

Once you’ve been told that a demonstration is “out of the question,” you’re ready to leave empty-handed. Compared to nothing, off by one isn’t just thrillingly close. It’s amazing.