In Ender's Game, what are some quotes that show that Ender is intelligent? 

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There is no doubt that Ender is an extremely intelligent character. His intelligence and emotional stability is what got him to Battle School in the first place. What is great about Ender is that he is not only book smart, but he is also incredibly street smart. He picks up on school teachings quickly, but he is also able to see beyond the facade of what is actually being presented to him. Take the following quote, for example:

He held up a limp hand. "See the strings?"

This quote comes from chapter 13 and is said to Valentine during Ender's brief return to Earth. The quote is a complaint by Ender that shows he understands he is nothing more than a puppet to be used and manipulated by the adults in his life. The quote is something very reminiscent of what Peter told him much earlier in the book.

If you need a quote that is more focused on traditional intelligence, then I would look to chapter one. There is a great quote about Ender's school ability.

Miss Pumphrey talked about multiplication. Ender doodled on his desk, drawing contour maps of mountainous islands and then telling his desk to display them in three dimensions from every angle. The teacher would know, of course, that he wasn't paying attention, but she wouldn't bother him. He always knew the answer, even when she thought he wasn't paying attention.

Another great quote comes in chapter five. We are told that Ender benefits from practice, but we are also told that he's smart enough to figure out games and systems well enough to excel at them by doing nothing more than observing.

All he had to do was watch the game and understand how things worked, and then he could use the system, and even excel.

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Ender's intelligence is slowly revealed from the very beginning of his experience at battle school. It starts in his room, when he figures out how to use his desk almost immediately and with very few instructions. Later, in the game room, his intelligence is clear in how quickly he figures out one of the games:

Ender understood the regularities by then. Understood the rules the computer was following, so that he could always, once he mastered the controls, out maneuver the enemy (46).

When it comes to the point that the computer is only winning by being so fast he cannot physically keep up, Ender further proves his intelligence in taking on one of the older boys in a short battle. He demonstrates his intelligence, not only in the game itself, but in the set up of playing the best two out of three, so that he has a chance to figure out how to use the controls.

Later, Ender figures out how to crack the security of the individual students' desks, and wins a silent war in his launch group by anonymously attacking Bernard, a known bully.

He was not studying. He was telling his desk to keep sending a message into the interrupt queue every thirty seconds... (49)

Though he never overtly gives himself away, Alai later calls him out on figuring this out so quickly. He confronts Ender with:

Come on, who else? It sure wasn't Bernard...Doesn't matter. I found out how to fake a new student entry...I found something else out too. I can't do it with your name. (67)

The other students become aware of Ender's intelligence very quickly as well. This pattern continues throughout Battle School and into Command School.

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