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Concrete Mathematics by Knuth and Patashnik (already mentioned for u/pmiller2) if the kid likes numbers. That's perhaps the guiding thread of the book -- it's a

Concrete Mathematics by Knuth and Patashnik (already mentioned for u/pmiller2) if the kid likes numbers. That's perhaps the guiding thread of the book -- it's about the beautiful (yet usually very elementary and natural) things you can do with numbers. Geometry Revisited by Coxeter and Greitzer and/or Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Honsberger if the kid is into plane geometry. It's an idyllic subject, great for independent exploration, and the books shouldn't take long to read. Not very deep, though (at least Honsberger). Anything by Tom Körner, just because of the writing. Seriously, he can make the axiomatic construction of the real number system read like a novel; open https://web.archive.org/web/20190813160507/https://www.dpmms.cam.ac.uk/~twk/Number.pdf https://web.archive.org/web/20190813160507/https://www.dpmms... on any page and you will see. Proofs from the BOOK by Aigner and Ziegler is a cross-section of some of the nicest proofs in reasonably elementary (read: undergrad-comprehensible) maths. Might be a bit too advanced, though (the writing is terse and a lot of ground is covered). Problems from the BOOK by Andreescu and Dospinescu (a play on the previous title, which itself is a play on an Erdös quote) is an olympiad problem book; it might be one of the best in its genre. Oystein Ore has some nice introductory books on number theory (Number Theory and its History) and on graphs (Graphs and their uses); they should be cheap now due to their age, but haven't gotten any less readable. Kvant Selecta by Serge Tabachnikov is a 3(?)-volume series of articles from the Kvant journal translated into English. These are short expositions of elementary mathematical topics written for talented (and experienced) high-schoolers. I wouldn't do Princeton Companion; it's a panorama shot from high orbit, not a book you can really read and learn from.

jacobolus 6y ago
If the kid likes plane geometry and is interested in further math, I’d highly recommend Yaglom’s books Geometric Transformations. They are a series of (hard) problem-focused books which teach the ideas of transformation geometry in service of solving various construction problems. In general transformation geometry is drastically underemphasized in American (and possibly other countries’) secondary and early undergraduate math education.
Nikhiil 6y ago
Another great book that's worth a read is "Things to make and do in the 4th dimension" by Matt Parker. It is a really interesting book as it covers multiple topics from mathematics and being that you are a young reader you might benefit from that exposure to topic. From there on you can fine the topics that interest and dive deeper and learn the nitty gritty! Thats how I personally found my passion in topology.

jacobolus 6y ago

If the kid likes plane geometry and is interested in further math, I’d highly recommend Yaglom’s books Geometric Transformations. They are a series of (hard) problem-focused books which teach the ideas of transformation geometry in service of solving various construction problems. In general transformation geometry is drastically underemphasized in American (and possibly other countries’) secondary and early undergraduate math education.

Nikhiil 6y ago

Another great book that's worth a read is "Things to make and do in the 4th dimension" by Matt Parker. It is a really interesting book as it covers multiple topics from mathematics and being that you are a young reader you might benefit from that exposure to topic. From there on you can fine the topics that interest and dive deeper and learn the nitty gritty! Thats how I personally found my passion in topology.