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There is a short story I read many years ago which must be from the 60s or before. It follows a young man who is selected by aliens with other humans as the smartest and taken to another planet or moon for training and education. He doesn't really fit in with the others and you think he just isn't cut out for it or selected in error. The ending is a surprise in that it turns out that he is actually much smarter and in a different way than the others and is just what the aliens have been trying to find and select for all along. I may have some of these details wrong but the main thing is the treatment of what you thought was not smart to be smarter than ever.

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You may be thinking of the 1954 novella "Immigrant" by Clifford D. Simak, first published in Astounding Science Fiction, March 1954, available at the Internet Archive). The following description is from a review of the anthology Galactic Empires Volume I at Visions of Paradise:

Three novelettes are the highlights of the book. The best story by far is Clifford D. Simak’s “The Immigrant,” one of his typically-low-key stories about an alien world which is so idyllic that only a few select humans are permitted to emigrate there, needing to pass a series of difficult IQ tests to do so. All humans who succeed send back tantalizing letters about the quality of life there, but the protagonist of the story migrates to the planet and quickly learns that what is hidden between the lines of the letters is sometimes more telling than what is actually stated. This story is typical of Simak at his best, a very thought-provoking story about the possible relationship between humans and the first aliens they encounter, a story whose protagonist thinks his way through the story rather than reacting to all circumstances physically. It reminded me of why Simak has always been one of my favorite sf writers and why I still enjoy reading his fiction as much as ever.

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Except for the aliens, it sounds very much like you're describing Isaav Asimov's "Profession", in which children are taught to read by imprinting at 8 years old on "Reading Day", and are assigned a profession (and imprinted with full professional expertise) at 18 on "Education Day".

Of course, Reading Day had been different. Partly, there was the simple fact of childhood. A boy of eight takes many extraordinary things in stride. One day you can’t read and the next day you can. That’s just the way things are. Like the sun shining.

[. . .]

Then the helmet was taken off his head, and the light was so bright that it hurt his eyes while the doctor’s voice drummed at his ears.

The doctor said, “Here’s your card, George. What does it say?”

George looked at his card again and gave out a strangled shout. The marks weren’t just marks at all. They made up words. They were words just as clearly as though something were whispering them in his ears. He could hear them being whispered as he looked at them.

[. . .]

Ellenford shrugged. “I’m sure you know how Earth runs its Educational program, George. Practically any human being can absorb practically any body of knowledge, but each individual brain pattern is better suited to receiving some types of knowledge than others. We try to match mind to knowledge as well as we can within the limits of the quota requirements for each profession.”

George nodded. “Yes, I know.”

“Every once in a while, George, we come up against a young man whose mind is not suited to receiving a superimposed knowledge of any sort.”

“You mean I can’t be Educated?”

“That is what I mean.”

“But that’s crazy. I’m intelligent. I can understand –- He looked helplessly about as though trying to find some way of proving that he had a functioning brain.

“Don’t misunderstand me, please,” said Ellenford gravely. “You’re intelligent. There’s no question about that. You’re even above average in intelligence. Unfortunately that has nothing to do with whether the mind ought to be allowed to accept superimposed knowledge or not. In fact, it is almost always the intelligent person who comes here.”

[. . .]

“Can’t you try Educating me? You haven’t even tried. I’m willing to take the risk.”

“The law forbids us to do that, George. But look, it will not be bad. We will explain matters to your family so they will not be hurt. At the place to which you’ll be taken, you’ll be allowed privileges. We’ll get you books and you can learn what you will.”

“Dab knowledge in by hand,” said George bitterly. “Shred by shred. Then, when I die I’ll know enough to be a Registered Junior Office Boy, Paper-Clip Division.”

[. . .]

“Well, anyway, it was all clear to me, as though I had known it all the time but wouldn’t listen to myself. I thought: What was it I had wanted Novia to let me do? I had wanted to go to Novia and take a batch of un-Educated youngsters and teach them out of books. I had wanted to establish a House for the Feeble-rninded – like here – and Earth already has them – many of them.”

Omani’s white teeth gleamed as he smiled. “The Institute of Higher Studies is the correct name for places like this.”

“Now I see it,” said George, “so easily I am amazed at my blindness before. After all, who invents the new instrument models that require new-model technicians? Who invented the Beeman spectrographs, for instance? A man called Beeman, I suppose, but he couldn’t have been tape-Educated or how could he have made the advance?”

“Exactly.”

[. . .]

“You can’t be told. That’s exactly it. It’s the final test. Even after we’ve thinned out the possibilities on Education Day, nine out of ten of those who come here are not quite the material of creative genius, and there’s no way we can distinguish those nine from the tenth that we want by any form of machinery. The tenth one must tell us himself.”

“How?”

“We bring you here to a House for the Feeble-minded and the man who won’t accept that is the man we want. It’s a method that can be cruel but it works. It won’t do to say to a man, ‘You can create. Do so.’ It is much safer to wait for a man to say, ‘I can create, and I will do so whether you wish it or not.’ There are ten thousand men like you, George, who support the advancing technology of fifteen hundred worlds. We can’t allow ourselves to miss one recruit to that number or waste our efforts on one member who doesn’t measure up.”

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"What Can You Say About Chocolate Covered Manhole Covers?", by Larry Niven. It was included in All the Myriad Ways. The theme of the story is that it is a great party game to come up with weird ideas, and then dare people to collaborate to come up with plausible explanations for them. The premise is that especially smart people will be especially good at it, so the game can be used as a fun intelligence test.

The surprise ending is that

the "person" asking these questions is really an alien, who wants to populate a new planet with a few really smart people.

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How about Ender's game the book :)

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    This doesn't fit the description at all well. – Valorum Mar 22 '15 at 0:01
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    The biggest difference between Ender's Game and the requested story is that the selectors in Ender's Game are humans, not aliens. But Ender's Game does include telepathic contact between Ender and aliens -- after he has been selected. Other than the date of the story, Ender's Game has the other story elements that were requested by the original poster. – Jasper Mar 29 '15 at 15:11

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