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Math Riddles Book

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"What is the Name of This Book" is fun if he's interested in logic.
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maybe he would like “To Mock a Mockingbird”
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Anything by Martin Gardner. I can personally vouch for The Colossal Book of Mathematics as a thoroughly engaging exploration of a variety of recreational mathematics topics, from topology to paradoxes to game theory. Lots of good stories, and some riddles, with answers usually included.
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Here are some suggestions at various levels of difficulty:

\- "The Moscow Puzzles" by Kordemsky

\- "Mathematical Puzzles and Diversions" by Martin Gardner

\- "Mathematical Amusements" by Dudeney

\- "Mathematical Recreations" by Kraitchik

\- "Mathematical Recreations and Essays" by Rouse Ball

\- "Mathematical Challenges" (easier) or "More Mathematical Challenges" (harder) by Tony Gardiner

\- "The Ultimate Mathematical Challenge" published by the UKMT.
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What mathematics does he know already?
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Have them sign up for AOPS. You will find a ton of great free material there.
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I'd actually recommend playing the video games in the professor Layton series. They are full of puzzles and riddles of all kinds (including mathematical), but they also have really good stories to go along with them. They were originally for Nintendo DS like over a decade ago (and will work on 3ds), but I think you can get a lot of them on phones and tablets now.

I played them in order (starting with the curious villiage), but the 3rd one (the lost future, or the unwound future depending on region) has by far the best story. Absolute 10/10. I wouldn't recommend the latest one (the millionaires conspiracy) - it's a spin off of small disconnected stories and I haven't bothered to finish it yet.
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how are you meant to track whether a 13 yo can get to Oxbridge?

get him to do his GCSE early if he’s that bored, and you could also expect UKMT olympiad rounds every year
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"Pillow Problems" and "A Tangled Tale" by Lewis Carroll (aka Charles Dodgson) are both pretty good. "A Tangled Tale" is fun because of the way he praises/critiques the solutions that his readers provided! Nothing quite hits like a classic 19th-century mathematical diss.

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