(Part 2) Best math games books according to redditors
We found 271 Reddit comments discussing the best math games books. We ranked the 104 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
My take was that we should have probably seen the 71 bombs coming. I remember watching Elliot try to re-route the trucks and thinking how the DA would also see this and could work around it by bombing the trucks en route if need be. They have the destinations and transit paths and everything Elliot has. Or just carjack them and burn the records inside. Lots of options here for the DA, but I didnt see bombing every building they stopped at coming, but someone did. In fact, a redditor figured it out weeks ago and his comment is linked on the front page, but the comment was fairly buried so maybe Sam and his team missed it.
On the more sci-fi/future tech front, the Congo is the world's source of cobalt which has more applications than just weapons and batteries. It has been discovered in the lab to facilitate quantum tunneling. My layperson's understanding is that this is all possible due to cobalt's unique magnetization properties and that QT is pretty much teleportation. It allows particles to move through a barrier without actually going through the barrier. Sam may be hinting that China has figured out how to make this work in human scale and with huge amounts of low-cost cobalt could have the key to things like dimensional travel (where do you go if you don't pass through the barrier? You could bypass it via a higher spatial dimension). If you can enter higher spatial dimensions then you more or less have time travel. You can look back on our entire timeline from the 4th dimension, see everything, and even interact with it somehow. No one really 'dies' because everyone's life can be seen from a higher spatial dimension like running through a nonstop video of their lives. Cliff Pickover has a fun and easy to read book on the subject for those interested.
On a more ridiculous sci-fi front, perhaps large scale quantum tunneling did actually happen at the WTP. Edward, and perhaps others, were briefly tunneled into a higher spatial dimension, essentially making them trans-dimensional beings whether they realize it or not. Edward's consciousness is safely in this higher dimension and intervenes in the modern world through Elliot's body as he sees fit. WR may have shown Angela that her mother is also a trans-dimensional being due to being at the WPT and could be accessed similarly. Zhang/WR then would also have this connection, perhaps granted by a similar accident in China. Perhaps WR was an WTP employee too, not Zhang himself, but a once alive physical woman that now works through Zhang the same way Edward works through Elliot. Maybe this woman was literally his sister, or other relative, who migrated to the US to study physics and landed a job at E-corp.
This can also explain WR's obsession with time. Imagine being able to access the 4th dimension where 3rd dimension time is just a property you can manipulate as you like, a bit like running the slider on a youtube video or playing around with scenes and objects in a game engine and being able to move the physics forward or backwards as you like. You'd be able to see our future but the future, I imagine, would be an infinite range of possibilities but you can narrow it down to a 'if x happens then what happens next' system and have limited but powerful precognition powers. It must feel oppressive to be in the 3rd dimension and to be 'locked' into real time. That's on top of knowing what a plausible timeline in our future could be and trying to make those future outcomes happen here, which would require perfect timing unless you want your probability wave to go someplace you didn't plan on. Everything must be perfect to the split second or your predictive information from the 4th dimension won't work.
WR's big plan may be to recreate the WTP accident globally and on a much larger scale so everyone can tunnel to a higher spatial dimension after death. Consciousness just finds its way to a higher dimension where 3rd dimension limitations like death don't matter. This also explains why the DA is so casual about dying. If WR's plan works out, they will become immortal and accessible to us regardless of when or how they died.
You need to develop an "intuition" for proofs, in a crude sense.
I would suggest these books to do that:
Proof, Logic, and Conjecture: The Mathematician's Toolbox by Robert Wolf. This was the book I used for my own proof class at Stony Brook - (edit: when I was a student.) This book goes down to the logic level. It is superbly well written and was of an immense use to me. It's one of those books I've actually re-read entirely, in a very Wax-on Wax-off Mr. Miyagi type way.
How to Read and Do Proofs by Daniel Slow. I bought this little book for my own self study. Slow wrote a really excellent, really concise, "this is how you do a proof" book. Teaching you when to look to try a certain technique of proof before another. This little book is a quick way to answer your TL:DR.
How to Solve it by G. Polya is a classic text in mathematical thinking. Another one I bought for personal collection.
Mathematics and Plausible Reasoning, Vol 1 and Mathematics and Plausible Reasoning, Vol 2 also by G. Polya, and equally classic, are two other books on my shelf of "proof and mathematical thinking."
My brother gave me Crocheting Adventure with Hyperbolic Planes, with yarn, for Christmas a few years ago. He wanted me to make him one.
I came in to recommend that book, too! It's a great book and well worth reading.
The other book about the number zero that I loved is The Nothing That Is: A Natural History of Zero.
Assorted ideas, in no particular order, but numbered in case anyone wants to refer to a particular one. These fall into three categories. Some are just to have fun and play with mathematics. You say you love math--so play with it. Some are to broaden you. Some are to prepare you or get you started on things you'll be doing when you get to college. I'm not going to say which category each falls into.
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1. Get some Martin Gardner books, particularly those that are collections from the old "Mathematical Games" column he wrote for "Scientific American". I think the earlier columns, collected in the first say five or so books, were the best. The whole set is available on CD-ROM for only about $40.
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2. Relearn the calculus you have already learned, but rigorously. Either Spivak's Calculus, or Apostol's Calculus volume I would be good for this. (Apostol is available in a paperback edition that isn't too expensive. You can usually find it on abebooks or biblio.com).
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3. There is an unwritten rule on /r/math that whenever someone in high school asks about math, someone has to tell them to read Rudin's Principles of Mathematical Analysis. Ignore them.
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4. You mentioned applied math, but if pure math also interests you, the How to Become a Pure Mathematician list is great. It lays out suggestions, usually giving several alternatives at each step, of books (traditional and free online) to take one from beginner to ready for graduate school.
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5. Gerard 't Hooft has a similar list for How to Become a Good Theoretical Physicist.
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6. Learn linear algebra. I'll leave it to others to suggest specific books or online resources. Linear algebra is the mathematical equivalent of a photobomber. You'll just be innocently doing something in math or physics that seemingly has nothing to do with linear algebra, and then there it is.
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7. Keith Devlin is giving a free online course through Coursera called "Introduction to Mathematical Thinking" that looks promising.
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8. How are you at computer programming? If you have a decent beginner level grasp of Python, David Evans' Applied Cryptography course at Udacity is interesting. You'll have to learn a little number theory to understand it, but nothing that should be beyond you. This course has some extremely engaging and fun challenge problems.
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9. If you are interested in applied math or physics, you should know some computer programming language and environment. Python is a good choice. MATLAB, or its free open source clone, Octave, is also a good choice. R is yet another good choice.
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10. A classic that has inspired many around your age is Robbins' book, "What is Mathematics?". Here's a review.
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11. Do problems! The two volume set by Yaglom and Yaglom, Challenging Mathematical Problems with Elementary Solutions, Volume I and Challenging Mathematical Problems with Elementary Solutions, Volume II is good and inexpensive, and, as the title suggests, elementary.
Volume I has problems on representing numbers as sums and products; combinatorial problems on the chessboard; geometric problems on combinatorial analysis; probability problems; problems on binomial coefficients; problems on experiments with infinitely many outcomes; and problems on experiments with a continuum of possible outcomes.
Volume II has problems on points and lines; lattice points in the plane; topology; convex polygons; sequences of integers; Tchebychev polynomials; four interesting formulas for pi; some interesting limits; and some problems on the distribution of primes.
In the back of each volume there is a section of hints for all the problems, and another section with compete solutions.
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12. The Mathematical Association of America publishes an excellent series of books, the Anneli Lax New Mathematical Library, which cover a variety of often interesting topics and are aimed at high school students and students in the first two years of college. It looks like most of these are available as PDFs for around $12-14 (although some are a bit more), with a few also being available for more in print-on-demand editions.
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13. Learn the human side of mathematics. For instance, read a biography of an interesting mathematician, such as Erdös. There are also books that cover several mathematicians. The book "[Mathematical People]"(http://www.amazon.com/Mathematical-People-Profiles-Interviews-Albers/dp/0817631917) and its several sequels are good for this.
This is actually taken without context from a book by Professor Ian Stewart. It's really good even if you typically aren't into those types of books.
Hey! This comment ended up being a lot longer than I anticipated, oops.
My all-time favs of these kinds of books definitely has to be Prime Obsession and Unknown Quantity by John Derbyshire - Prime Obsession covers the history behind one of the most famous unsolved problems in all of math - the Riemann hypothesis, and does it while actually diving into some of the actual theory behind it. Unknown Quantity is quite similar to Prime Obsession, except it's a more general overview of the history of algebra. They're also filled with lots of interesting footnotes. (Ignore his other, more questionable political books.)
In a similar vein, Fermat's Enigma by Simon Singh also does this really well with Fermat's last theorem, an infamously hard problem that remained unsolved until 1995. The rest of his books are also excellent.
All of Ian Stewart's books are great too - my favs from him are Cabinet, Hoard, and Casebook which are each filled with lots of fun mathematical vignettes, stories, and problems, which you can pick or choose at your leisure.
When it comes to fiction, Edwin Abbott's Flatland is a classic parody of Victorian England and a visualization of what a 4th dimension would look like. (This one's in the public domain, too.) Strictly speaking, this doesn't have any equations in it, but you should definitely still read it for a good mental workout!
Lastly, the Math Girls series is a Japanese YA series all about interesting topics like Taylor series, recursive relations, Fermat's last theorem, and Godel's incompleteness theorems. (Yes, really!) Although the 3rd book actually has a pretty decent plot, they're not really that story or character driven. As an interesting and unique mathematical resource though, they're unmatched!
I'm sure there are lots of other great books I've missed, but as a high school student myself, I can say that these were the books that really introduced me to how crazy and interesting upper-level math could be, without getting too over my head. They're all highly recommended.
Good luck in your mathematical adventures, and have fun!
Linear programming:
Combinatorial optimization:
Game theory:
Combinatorial game theory:
Yes, it can be done. I took a MOOC called Paradox and Infinity and the prof presented a fairly complete proof accessible to a totally elementary audience. It was very impressive. http://ocw.mit.edu/courses/linguistics-and-philosophy/24-118-paradox-infinity-spring-2013/
The key steps are using the Axiom of Choice, for example to prove there's a nonmeasurable set. Rotations in three-space. The free group on two letters.
There's actually a pretty good outline of the proof on Wiki.
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
When you say hours of work, you'd actually have to be thinking weeks of work. A totally elementary proof can be given, but there are a lot of little parts, each one an elementary version of a more sophisticated idea. So what you want can be done, but no, not with only an evening's work. You'd need a detailed outline aimed at an elementary level, and it would take as long as it takes to work through it.
ps -- There's a book aimed at a popular level. http://www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/1568813279 I haven't looked at it but I've heard it's good.
Polya's How to Solve it is a classic.
You might prefer Housten's How to Think Like a Mathematician which is much more modern.
I found that they both had useful insights, though there was a fair bit of information which I didn't find helpful.
http://www.mathman.biz/
http://wheelof.com/whitney/index.php?var=v1
http://vihart.com/
http://www.go.kestrel.nu/
http://www.amazon.com/Escher%C2%AE-Pop-Ups-Courtney-Watson-McCarthy/dp/0500515905
http://www.amazon.com/Alexs-Adventures-Numberland-Alex-Bellos/dp/1408809591
http://www.amazon.co.uk/You-Can-Cube-Patrick-Bossert/dp/0141326506
http://www.amazon.co.uk/Godel-Escher-Bach-Eternal-Golden/dp/0465026567
http://www.amazon.com/Mathematics-Very-Short-Introduction-Introductions/dp/0192853619
http://www.amazon.com/Project-Origami-Activities-Exploring-Mathematics/dp/1568812582
http://www.amazon.com/Origami-Tessellations-Awe-inspiring-Geometric-Designs/dp/1568814518
http://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006
Once I got started ....whew. good luck!
The Beauty of Geometry: Twelve Essays, H. S. M. Coxeter
This requires a little background, and may require a few brain cells, but Coxeter is a great expositor. I find myself rereading this every couple of years.
The Pleasures of Counting, TW Korner
This is a tour through applied math. It needs almost no background, but does more than scratch the surface. It is the perfect response to "... bur what is this good for?". I'd also recommend his 'Fourier Analysis', but I'm pretty sure that it does not qualify as light.
Winning Ways, J. H. Conway, et. al.
Light in tone, but heavy in content. I love these books. Other people, including competent mathematicians, have declared them as impossible to learn from.
If you do not mind reading from the screen, try this collection of Martin Gardner's books.
I admit, I laughed when I first saw her book in my math department library, but now I need to head back and actually read it.
I've not read it personally (but I've read almost all Martin Gardner books), but probably "The Colossal Book of Short Puzzles and Problems" by Martin Gardner is a good start.
Note that I'm not a very good judge of puzzles: curiously, I love to read about math puzzles, but I have absolutely no interest in solving them myself.
By the way, there is also a legit collection of PDFs of about 15 of mathematical games books on CD, see this .
Note that it is advertised as "The entire collection of Martin Gardner's Scientific American columns on one searchable CD", but this is UTTERLY false.
There is not a single column. It is the scan and searchable index of about 15 books (inspired by his columns, but that's not the same thing!)
If you want something very mathy, here's a collection of cute applications of linear algebra to discrete math. You should be comfortable with linear algebra before reading this.
If you want light reading with a math flavor, I highly recommend Conned Again, Watson! It's a novel in which Sherlock Holmes and Watson use math to uncover scams and correct fallacies. Lots of unintuitive probability content.
Perhaps you need some guidance with problem solving skills in general (we all do). Some resources that I found useful are this video on problem solving here, Polya's How to Solve It, and Problem Solving Through Recreational Mathematics.
When I get around to designing my party's first mega-dungeon, I'll be drawing heavily from this book of logic puzzles. It has interesting thought experiments and variations on the classic logic puzzles that should challenge any players.
Edit: Here is an excerpt from the beginning.
Not exactly, this is a problems book. I'm looking for something about puzzles, like The Moscow Puzzles: 359 Mathematical Recreations or The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs.
And its sequel, "Professor Stewart's Hoard of Mathematical Treasures". They're both sitting on my bookshelf.
http://www.amazon.com/Professor-Stewarts-Cabinet-Mathematical-Curiosities/dp/0465013023
http://www.amazon.com/Professor-Stewarts-Hoard-Mathematical-Treasures/dp/0465017754
I tried to read Arnold's "Mathematical problems from 5 to 15" when I was 22, there are two possibilities...
To understand the joke, see the last problem of this list, it's easy to find a copy of it with Google.
But seriously, not being able to answer problems from past grades is not a big deal, we forget stuff all the time, just try to study it again. When you try to solve problems from past grades without success, it's probably because you don't remember the pragmatic way to answer it and when trying to answer it, you try to "invent" a solution. I think that this is a great opportunity to contemplate how superb mathematics is: These pragmatic solutions we learn at school, even from very basic mathematics took centuries to be developed in the form it is presented to you. Newton's "Standing on the shoulders of giants" couldn't be more fit.
You should take a look at Dörrie's "100 great problems of elementary mathematics: Their history and solution"
Check out Surfing through Hyperspace by Clifford Pickover. It does a pretty good job of explaining what it would be like to interact with a higher dimension. Unfortunately there is a weird detective story nested inside the non-fiction material that is used to sort of illustrate the concepts he discusses. I eventually just skipped those sections as they're very bad and not really necessary.
Also, if you find you like it, you can follow-up with Pickover's Sex, Drugs, Einstein and Elves.
Take a look at A Cultural History of Physics by Károly Simonyi. You can find an excerpt of the book here (pdf). I have not had the time to read it myself, but I'm very impressed by it from just looking at pages at random.
Erik Demaine, Robert Lang and Tom Hull have great ones. Also check out the OSME conference proceedings. These people are always happy to talk via email if you have any questions, but you will have to look them up.
Also I heard this one is good but I haven't seen it yet: Origamics: Mathematical Explorations through Paperfolding by Kazuo Haga, who I believe was one of the first to write academically about origami.
The game is called dots and boxes (http://en.wikipedia.org/wiki/Dots_and_Boxes)
It actually has a reasonably interesting theory behind it.
http://www.amazon.com/Dots-Boxes-Game-Sophisticated-Childs/dp/1568811292
Also more stuff in volume 3 of Winning Ways for your Mathematical Plays in chapter 16.
http://www.amazon.com/Winning-Ways-Your-Mathematical-Plays/dp/1568811438
Risk
Bad Science book , book and blog
PD at TED
numberwatch on the data dredge
Fun and game books Duelling Idiots and Cabinet ... there are many books on this subject but I haven't read most of them.
serious probability writing Jeffreys and Yudkowsky
This book is actually quite useful in answering your question. https://www.amazon.com/Surfing-through-Hyperspace-Understanding-Universes/dp/0195130065/ref=nodl_
The short answer is “no” — as 3rd dimensional beings, we cannot really conceptualize a 4th spatial dimension. We can, however, conceptualize the 3D intersections of 4th dimensional objects.
The actual structure of a tesseract is inconceivable to us, but we can model it’s third dimensional plane. Similarly for a 2D experience, only the 2D intersection of a 3D object could be perceived. Based on the behaviour of that 3D object intersecting with a 2D plane, a 2D creature could figure out that the 2D object was in fact an intersection of a 3D object. So we can use lower dimensional intersections to conceive of higher dimensional structures.
Imagine a cone passing through a sheet of paper vertically. You’d see an expanding circle in 2D. From that, you could try to deduce and explain the expanding of the circle and possibly conclude it’s actually a higher dimensional structure intersecting with your plane.
Put that same cone slightly on its side and pass it through vertically and it would be an expanding ellipse — you might then accidentally describe two different 3D structures from a 2D perspective, even though it’s actually the same cone.
Pass a fork through, though, and it would be just the small circles of the fork’s tines visible in the 2D intersection. It would be all but impossible to conceive of those 4 circles being one object in a higher dimension. Pass the fork through sideways and you’d see this disappearing and reappoint long rectangle. You’d have to have a lot of data to surmise that these two experience are in fact connected to each other in the third dimension in one object. But the actual shape of the fork would likely remain a total mystery, ad certainly would not be possible to create or draw in the lower dimension. At best you’d have abstract mathematical models that could simulate the behaviour of the higher dimensional object in your dimensional plane.
>Lol, I think you might be undervaluing the invention/creation of 0.
>> 0 as a number is only really needed for advanced mathematics
>is pretty incorrect.
Again, this is what I replied to. You might want to read it again, and also every comment I have made on this topic.
I agree zero is important, but in 1AD Rome, what's the practical use? Academic use maybe, but how would it be necessary for use of mathematics as a tool in vocational fields? With zero, you have elegant system where amount of symbols you need grows logarithmically in relation to how large the number is that you need to display, but that's about it? I am not sure how practical concern that is, either.
>There are thick books (rec)written on zero and I think you should read those
And you, given that you are recommending this book to me, you have probably read it already, so can give me an excellent example where a person would need to use the number zero outside advanced mathematics?
Just about any integer programming technique can solve the TSP without exhausting every result. You just stop the algorithm when it reaches 0% optimality gap (in eleventy billion years for the tougher TSPs).
Here's a fun book: http://www.amazon.com/Pursuit-Traveling-Salesman-Mathematics-Computation/dp/0691152705
A) How to Solve it: A New Aspect of Mathematical Method (Penguin Science) https://www.amazon.co.uk/dp/0140124993/ref=cm_sw_r_cp_apa_i_jmYGDbXX07R9M
B) Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences https://www.amazon.co.uk/dp/0199282013/ref=cm_sw_r_cp_apa_i_ZkYGDbWSX4JSP
These two.
Read Polya, then do the hard work of studying through the sections of Techniques you need to know.
This is a fun one.
>"A well written and witty look at hundreds of mathematical puzzles, stories and jokes. I am a maths teacher and there is so much material here, it's amazing. I have already used a few of these with my classes and the puzzles have really caught their imagination. Highly recommended."
17 equations that changed the world, by Ian Stewart. The entire book is good, but there is one or two chapters covering calculus and the story of how Newton and Leibniz went about "inventing" it.
It's super easy if you know how to knit. get really long needles (I usually use a circular needle): Cast on as many stitches as you can fit/until you get bored. K n, k2tog, repeat. (where N is whatever you want- if you want it to be over quickly, it can be 0. if you want a larger object, you can go with 7 or 20 or whatever you want).
They're fun to play with or for christmas ornaments or whatever, but you can also make PANTS:
Here's a picture, here's a Wolfram Demonstration that will make a pattern for you, and here's a book that contains a chapter on the hyperbolic pants. There was also a talk on hyperbolic baby pants at the Joint Meetings in 2008.
For me there are several ways that knitting is interesting. First, it's something repetitive that I can do to soothe my mind and actually have a useful product at the end (using double-pointed needles provides just the right amount of mental stimulation). Second, it's a fascinating topological exercise how a string can get turned into a complex, 3D object, like a sock. Third, cables and lace and designing custom-fitted objects are interesting engineering projects, especially when you take into account the various personalities of the different kind of fibers you can use. (This type of knitting is very mental and not at all soothing, especially when you lose track of where you are or drop a stitch.)
Scarves and hats are boring. My next project (after I finish the mittens for my daughter) is a torus from this book. The fractal shawl also looks interesting. :)
Historically, at least in some places, knitting was unisex, especially among fishermen, who had their own cable patterns in their sweaters to make identifying their bodies easier if they drowned at sea.
We need more men who knit, as well as young women, to break the stereotypes.
["In quantum mechanics, there is a common misconception that it is the mind of a conscious observer that causes the observer effect in quantum processes. It is rooted in a misunderstanding of the quantum wave function ψ and the quantum measurement process."](https://en.wikipedia.org/wiki/Observer_effect_(physics)
I'm not saying that the presence of a mind causes the observer effect. To me, All is the Mind. All objects have consciousness to some degree or another. I am saying the state of the mind causes the collapse. This goes for the state of any machine which is observing the process as well as if All is Mind it also has a mindstate as well. If you still view that as wrong due to misinterpretation of the process, if you have an ELI5 somewhere, or a book I can read on the topic, that'd be magical. Wikipedia is nice, but only if you have a degree in the matter. I'm going to be reading Giancoli and this really soon.
I think Problem Solving Through Recreational Mathematics is exactly what you're looking for. It introduces mathematical ideas disguised as puzzles that begin simply, but gradually get harder until you've discovered topics like Hamiltonian Paths or Diophantine Equations on your own. It can give you a surprisingly intuitive way to look at some mathematical ideas, particularly graph theory and number theory.
There's a pretty good book that explains the paradox as well: The Pea and the Sun: A Mathematical Paradox. It goes pretty slow and is therefore pretty approachable even with only high school level mathematics.
This is just my perspective, but . . .
I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.
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Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics
Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.
Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.
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As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.
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Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.
I personally enjoy this book. The problems are simple, but not easy. I'm sure they will be challenged in a thoughtful, and fun way.
https://www.amazon.it/Conned-Again-Watson-Cautionary-Probability/dp/0738205893
in this book in one chapter they discuss some fair techniques to divide land with mixed values distributed among it.
if you need to distribute a circle of land among 2 people you have one person split it into two as he wishes and the second person chooses first what part to take.
if you have three people you have two people dividing it and the third picking first and so on
Since he's 8, I'm assuming he's currently doing multiplication and division and stuff. Everyone in this thread is talking about a bunch of advanced stuff which is great, but I would recommend as a first step How to Count Like an Egyptian. It's a book about how the ancient Egyptians added, subtracted, and their unique numeral system. Anyway, I think the key here is not to overwhelm your kid with math; every kid has a fancy or two. You don't want to go too deep, just keep him interested! And this book is an example of something that's made for kids, but is really cool and also teaches history and culture!
Also, I feel like this would be a nice way to feel him out! If he's still interested, you can go deeper into more serious math later. There's also a bunch of random math puzzles that are fun. I'll edit this after looking for them.
This is a good book that shows you much more than the multiplication/division shown in this video:
https://www.amazon.com/Count-Like-Egyptian-Hands-Introduction/dp/0691160120
It's actually pretty fun to go through the exercises.
You may wish to check out In Pursuit of the Traveling Salesman. The first chapter is available here.
> travelling salesman is intractable
In Pursuit of the Traveling Salesman is pretty interesting historical and technical overview of advances here. With modern ILPs and cuts and other techniques, the notion of intractable today vs. 20 years ago is vastly different.