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 top 20.
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 top 20.
I'm not familiar with anything current but I'm sure it exists. When I was doing the bulk of my learning we were still carving holes in strips of cardboard to produce code. Someone younger would probably give better, more current advice.
In general, refining your problem solving skills involves a great deal of introspection. Everything you complete you should go back and analyze the stumbles you had along the way. What caused delays, what produced bugs, what just didn't work very well. Look at these things and try to determine what you could have done differently. No better teacher than failure.
Two very old books that got me started: Aha: Gotcha and Aha:Insight. They are amazing puzzle books written by the master of puzzles, Martin Gardner. They have a bit of a math slant, but not too much. Read the reviews to see if it floats your boat.
Math, imo, is the basis of solid problem solving. It's the reason we learn math from pre-K all through university. You're not doing it so you can do calculus at the grocery store, and I've never used a lick of it in my career, but it does teach you how to think in a logical manner, breaking big problems down into little ones.
Another book that had some impact on my career was Design of Everyday Things. Good read for usability.
Misuse of Godel's work is a substantial topic. I highly recommend Torkel Franzen's book, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.
You sound like the kind of person who would enjoy this. This book pretty much melted my brain after chapter 2. Written for game theory enthusiasts.
Your post has too little context/content for anyone to give you particularly relevant or specific advice. You should list what you know already and what you’re trying to learn. I find it’s easiest to research a new subject when I have a concrete problem I’m trying to solve.
But anyway, I’m going to assume you studied up through single variable calculus and are reasonably motivated to put some effort in with your reading. Here are some books which you might enjoy, depending on your interests. All should be reasonably accessible (to, say, a sharp and motivated undergraduate), but they’ll all take some work:
(in no particular order)
Gödel, Escher, Bach: An Eternal Golden Braid (wikipedia)
To Mock a Mockingbird (wikipedia)
Structure in Nature is a Strategy for Design
Geometry and the Imagination
Visual Group Theory (website)
The Little Schemer (website)
Visual Complex Analysis (website)
Nonlinear Dynamics and Chaos (website)
Music, a Mathematical Offering (website)
QED
Mathematics and its History
The Nature and Growth of Modern Mathematics
Proofs from THE BOOK (wikipedia)
Concrete Mathematics (website, wikipedia)
The Symmetries of Things
Quantum Computing Since Democritus (website)
Solid Shape
On Numbers and Games (wikipedia)
Street-Fighting Mathematics (website)
But also, you’ll probably get more useful response somewhere else, e.g. /r/learnmath. (On /r/math you’re likely to attract downvotes with a question like this.)
You might enjoy:
https://www.reddit.com/r/math/comments/2mkmk0/a_compilation_of_useful_free_online_math_resources/
https://www.reddit.com/r/mathbooks/top/?sort=top&t=all
Linear programming:
Combinatorial optimization:
Game theory:
Combinatorial game theory:
Robert Lang is the guy who pretty much pioneered mathematical/computational origami design. I'd look into some of his papers and books.
YouTuber Sarah Adams has a couple of videos on tessellations. Here's a playlist of her videos: https://www.youtube.com/playlist?list=PL13A44D22E042BB7F
Book-wise, I am personally very fond of Eric Gjerde's Origami Tessellations: Awe-Inspiring Geometric Designs.
You can buy that here: https://www.amazon.com/Origami-Tessellations-Awe-Inspiring-Geometric-Designs/dp/1568814518
Good luck and happy folding!
oh shoot I've waited to answer something like this, I used to do stuff in here
I'd say you're looking mostly at things in combinatorial geometry when it comes to understanding this area, although this paper involves some abstract algebra as well.
As the LovepeaceandStarTrek mentioned, Erik Demaine's a wizard at this stuff. He's a guru at MIT who I think taught a course or two on designing the kinds of things he comes up with - WAIT! I found it, here
Aside from all of that, there's this and this
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!
Things to Make and Do in the Fourth Dimension.
Matt Parker is a public speaker, mathemetician, and YouTuber.
> There is no way in hell bus crunching should be a problem these days.
Buses "bunching" is a widely-known statistical problem. There's even a book with the title "Why Buses Come in 3" [0]. The CTA already collects GPS data (CTA buses have GPSes--one can track where buses are in real-time with the Transit iOS App. I use it and it's mostly accurate on my routes, where the GPS has line-of-sight -- which isn't always the case in the Loop).
One of the major causes of the bunching phenomenon is people taking too long to board, or people trying to board an already full bus [1].
People need to either board quickly, or wait for the next bus if the current one is full (to avoid delaying it further). Now, if we can figure out how to incentivize people to do this naturally, we would have mitigated one source of stochasticity, which in turn makes the system dynamics more tractable and predictable. Good drivers already enforce this (at the expense of coming across as being pushy).
The other big source of stochasticity is traffic, and Google Maps already does a good job of predicting ETA in real time in current traffic (plus Google collects real-time traffic data from other Google Maps users on the road).
Other solutions that I've seen in other cities is to have the driver wait a couple of minutes at certain reset points when they are running ahead of schedule. Unfortunately, this is hard to do on a busy artery in Chicago at rush hour.
[0] https://www.amazon.com/Buses-Threes-Hidden-Mathematics-Everyday/dp/0471379077
[1] http://www.chicagomag.com/Chicago-Magazine/The-312/July-2013/Chicagos-Worst-Buses-for-Bunching-and-Why-It-Happens/
I did a 3 month historical and research analysis on it for one of my mathematics course :P
I read a lot of books and talked to pretty much every one of the my Mathematical Logic and Set Theory professors.
The best book that helped me was this:
Gödel’s Theorem: An Incomplete Guide To Its Use and Abuse
Seriously worth a read and would clear up everyone's misconceptions in this topic.
Really anything will work at first, but it is a lot more rewarding to use decent paper. I've found Tant to be a great compromise between price and quality; here's where I got mine.
As for instructions, it really depends what you're interested in making.
I would recommend YouTube videos since you're just starting out; it will pretty much eliminate the confusion that comes along with diagrams and crease patterns.
Sara Adams has a great channel, as does Jo Nakashima. There are a bunch more, but those two are what I remember off the top of my head.
Just searching around I found one for a hummingbird - I haven't folded it or watched it before, but it looks decent.
I almost exclusively fold tessellations; I'm not sure if that would interest you or not. Shuzo Fujimoto's Hydrangea might be a good place to start. Or Eric Gjerde's Tiled Hexagons, which is a more traditional tessellations, although it isn't diagrammed as well here as in his book (which you should definitely get if you are at all interested in tessellations).
Nope. Yoshizawa Akira used wet-folding techniques, and there probably isn’t a more important person in early 20th century origami history. The rules of traditional origami are that one uses one piece of paper (any shape, although a quadrangle is preferred) and that the paper isn’t cut (that would be kirikami), and this still is within those guidelines. Robert Lang has written about him in Origami Design Secrets: Mathematical Methods for an Ancient Art, which has my highest recommendation. The paper is two-sided, which is how the contrast is achieved. Also not cheating.
Here's a nice book with a set of Algorithmic Puzzles that you can practice on. The book actually touches on an important point. Algorithmic thinking is not solely the domain of computer science or programming; it's a general problem solving topic.
Also, to echo some of the other comments here, it's not the CS degree that will make you better, it's the practice. The four years spent getting a CS degree merely forces you to practice a lot. The key point here though is that you can practice regardless of the CS degree (or pursuit of) which means if you've got some spare time, practice!
It's difficult to make recommendations without being certain of what you actually know and what you imagine mathematics to be like. A lot of university-level mathematics is technical and requires familiarity of high-level concepts. This is in contrast to softer popular mathematics, which is more related to solving problems and contest questions. One of the things I've noted about pre-university students passionate about mathematics is that they assume that the subject is only about problem-solving and fail to take into mind the level of technical knowledge that must be learnt and memorized to be a mathematician.
If you're simply looking for problems to solve, try The Art and Craft of Problem Solving by Zeitz or Problem Solving Strategies by Engel. Generally any book geared to the Olympiad or regional competitions will be alright. Here, you're not looking for a specific body of knowledge, but rather an approach to thinking and persevering when handling tough problems.
But if you're looking to learn more about 'technical' mathematics, you'll need to know the basics of numbers and sets. Numbers & Proofs by Allenby is a good introduction, using an approach that gets you to actively solve problems. Once you get past that, then you can try your hand on analysis or group theory or linear algebra or even basic graph theory. But keep in mind that with 'technical' mathematics, all knowledge is built on understanding of previous fields, so don't rush through it or you'll get discouraged by any difficulty or unfamiliarity you'll encounter.
I wrote a paper on a topic tangentially related to this and used The Nothing that Is: A Natural History of Zero as a kind of case study of one particular mathematical object.
It covers the historical development of Zero, why we use the symbol 0 for zero, different arguments pertaining to zero, etc. Very well-written and engaging.
Your particular take on the ontological status of mathematical objects will determine what philosophers / books you use. But regardless of your take, you can use the Zero book as a concrete history of a particular mathematical object that you can then assess through the philosophical lens.
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!
> Kenneth G. Wilson remarks, "If we do end by casting aside the A.D./B.C. convention, almost certainly some will argue that we ought to cast aside as well the conventional numbering system itself, given its Christian basis." [emphasis added]
That's an awful lot of wrong packed into 3 little words - zero itself occurs in the works of the Sumerians and the ancient Greeks, and place value arithmetic was considerably refined in China, India and the Arab world LONG before it was used in the Christian West. Check out The Nothing That Is for a history of the conventional number system.
I'm not a fan of string theory, so I'm biased. If you want to read a source from an advocate of string theory, I would suggest Why String Theory? by Joseph Conlon.
Yes string theory produces elegant mathematics, but it's just math. It hasn't made its connection to reality. A lot of what string theory was designed to do is not working (i.e. quantize gravity). Especially since supersymmetric particles weren't found at the LHC.
Instead of a "theory of everything," it's now thought of as a set of possible theories where quantum gravity might work. The best thing going for string theory is probably the AdS/CFT correspondence. The problem is that AdS is a spacetime with a negative cosmological constant, but our universe has a positive cosmological constant. However, there could be applications of the AdS/CFT corrsepondence to condense matter physics which is discussed in Joseph's book.
Here is a quote I took from this page where people discuss Penrose's recent book Fashion, Faith, and Fantasy.
> String Theory arose out of particle physicists attempting to quantise the linearised version of GR, normally used in make perturbations to Newton’s laws in the solar system. This produces a free spin-2 graviton at 0th-level, and then they attempt to use first order interactions to construct a full interacting quantum field theory of gravitons - in FLAT space - which has none of the full symmetries of GR, and it fails miserably by throwing up infinities at all orders. Not to be daunted, they added supersymmetry to cancel the infinities, but that did not work as expected, so they invented Strings instead of points, to remove the short distance interactions. Why would this ad-hoc procedure produce anything like Quantum GR? It gets worse, they quantise this strings in FLAT spacetime, but find that they need 10 dimensions instead of 4, so they then ‘compactify’ 6 to get something like our world. Problem is, there is about 10\^500 different ways to do this - hence the multiverse. Never before has such a failed theory taken on such a life of its own, at least the Aristotelian epicycles produced decent predictions.
https://www.amazon.com/Buses-Threes-Hidden-Mathematics-Everyday/dp/0471379077
The easy answer is Origami Design Secrets: Mathematical Methods for an Ancient Art, Second Edition https://www.amazon.com/dp/1568814364/ref=cm_sw_r_awd_r46Hub00PYXXZ
Great book, well worth the price
I really like this book.
https://www.amazon.com/Symmetries-Things-John-H-Conway/dp/1568812205/ref=sr_1_1?s=books&ie=UTF8&qid=1493910460&sr=1-1&keywords=symmetries+of+things
https://www.amazon.co.uk/Why-String-Theory-Joseph-Conlon/dp/1482242478
My 2c : How about just asking the question "why do you subscribe to the PhilosophyofScience reddit?" and then give the prize to the comment with the most upvotes? Ties being decided by you.
EDIT : I will throw in a copy of Godel's Theorem : An Incomplete Guide to its Use and Abuse as another prize.
"Origami Design Secrets" by Robert Lang is a fantastic place to start. The basic idea is that most object can be represented as a tree (in the computer science sense), with some extra tricks to make things more efficient. The book details how a tree can be converted into a crease pattern, which you would then collapse and shape to get your desired result.
No, it wouldn't. It doesn't say anything at all about the physical universe. See Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.
Gödel's incompleteness theorem is a technical statement concerning a possible formalization of mathematical reasoning known as first-order logic. There are a million variations, but basically it states that if you start with a set of axioms which is finite or even merely enumerable by some mechanical process (Turing machine), and if these axioms are consistent and contain a very minimal subset of arithmetic, then there is a statement which is "true" but you cannot prove from those axioms with first-order logic (and, in fact, it gives you an explicit such statement which, albeit "true", cannot be proven; if your axioms contain a not too minimal subset of arithmetic, one such statement is the very fact that the axioms are consistent, suitably formalized; another variant, due to Rosser, is that even if you allow for your axioms to contain false statements, there is still going to be some statement P such that neither P nor its negation ¬P follow from your axioms).
So, in essence, no matter what axioms you use to formalize arithmetic or any decent subset of mathematics, unless your axioms are useless (because they cannot be enumerated, or because they are inconsistent), or the axioms aren't sufficiently powerful to prove that they are consistent. Even if you add that as an axiom, there is still something missing (namely that with that extra axiom, the axioms are still consistent; and if you add that, then again, etc.). Interestingly, a theory cannot even postulate its own consistency (one can use a quinean trick to form a theory T consisting of usual axioms of arithmetic + the statement that T itself is consistent, but then the theory T is wrong, and inconsistent).
This is all really a technical statement concerning first-order logic. But trying a different logic will not help: another variant of Gödel's theorem (due to Church or Turing) tells us, essentially, that there can be no mechanical process (again, Turing machine) to determine whether a mathematical statement is true or false; so there can be no mechanizable, coherent and complete, logic which attains all mathematical truths, because if there were, one could simply enumerate all possible proofs according to the rules of that logic, and obtain all possible truths. All these variants of Gödel's theorem are variations around Cantor's diagonal argument: in the original variant, one constructs a statement which says something like "I am not provable" (intuitively speaking, at least), whereas in the Church/Turing version I just mentioned one would appeal to the undecidability of the halting problem.
But one thing to keep in mind is that almost every attempt to draw philosophical or epistemological consequences from Gödel's theorem has been sheer nonsense. Explanations àla handwaving such as "every formal mathematical system is necessarily incomplete" or "formalization of mathematics is inherently impossible" or whatever, are perhaps nice for giving a vague intuitive idea of what it's all about, but the actual mathematical theorem is not so lyrical. (For example, I have used the word "truth" in quotation marks once or twice in the above. This is because I don't have the patience to write down all the caveats about the meaning of "truth" in the mathematical sense in this context. So while what I have written is correct, attempts to draw metaphysical consequences from it will not be. :-)
For further reading, besides what others have already suggested, I believe Torkel Franzén's book on Gödel's theorem (destined for a general audience) is excellent.
The incompleteness theorem had incredible impact on mathematical and scientific thinking. Philosophy and Mathematics at the time was OBSESSED with grounding all of mathematics in Set Theory. See Principia Mathematica:
http://en.wikipedia.org/wiki/Principia_Mathematica
The incompleteness theorem said that this task was essentially futile. With this, objectivity of mathematics was thrown into the fire.
You're right to say completeness isn't understanding. Completeness is a mathematical property, and understanding refers to a human belief in explanatory satisfaction.
Before we get too carried away, the incompleteness theorem is also often incorrectly used by those trying to use it in arguments. In fact, books have been written about its misuse.
http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/dp/1568812388
Part of the shock regarding the incompleteness theorem comes from a field-wide belief that was proven false. It continues to be shocking because it is a belief that many mathematicians still have until they learn about it.
This is just a giant (one might even call it colossal) book of all the things that might make one love math.
Martin Gardner's The Colossal Book of Mathematics.
https://www.amazon.com/Colossal-Book-Mathematics-Paradoxes-Problems/dp/0393020231
Also look at the "frequently bought together" section on the Amazon listing. The Colossal Book of Short Puzzles and Problems is also a masterpiece. I hadn't seen My Best Mathematical and Logical Puzzles before, but now I'm getting it...
Professor Stewart's Cabinet of Mathematical Curiosities is also a lovely book of cool things in math.
https://www.amazon.com/Professor-Stewarts-Cabinet-Mathematical-Curiosities/dp/0465013023
In fact, I'd recommend all of Ian Stewart's books.
The classification of wallpaper patterns:
http://www.amazon.com/The-Symmetries-Things-John-Conway/dp/1568812205
Thanks for your reply! I bought https://www.amazon.com/gp/product/1568812019/ , but it hasn't come in yet. I'll have to make sure there isn't too much overlap, because I don't want to spend money on problems I already have.
In fact, there's literally a book about it called Why Do Buses Come in Threes?.
Algorithmic Puzzles
First thing first: If you're american, I would suggest joining Origami USA for the lending library. I've never used it (not american) but it seems useful.
If you're looking to get past the youtube videos and simple models, I would recommend some of the classics (though they might not be the newest). They are also more likely to be available. [Origami Omnibus] (http://www.giladorigami.com/BO_Omnibus.html) gives a good overview of the field though it was written before the Tessellators made it big. It should help you decide what kinds of origami you're interested in.
Origami from Angelfish to Zen has a nice overview of the history of origami.
[Origami design secrets] (http://www.giladorigami.com/BO_DesignSecrets.html) is a newer classic and covers a lot of the technical advances in the latter half of the century.
As for intermediate/advanced books, the best ones are "boutique" books from special publishers.
origami house does all the hard core japanese designs like Kamiya, Komatsu, and Nishikawa. They also publish the annual tanteidan convention book which is hands-down the best collection of diagrams each year. I almost always buy it (though sometimes I wait and buy several at once)
[Passion Origami] (http://www.passion-origami.com/marques.php) is the other major publisher and has the books by Roman Diaz, Quentin Trollip, and the VOG.
If you don't want to pay for shipping dead trees around, [Origami USA] (https://origamiusa.org/catalog/newest-downloads) has some diagrams for online purchase, I haven't looked at them all, but there are some good names there.
Modular origami is actually kinda diverse. Are you interested in pure geometrics (phizz, sonobe), clever decorative (Miyuki Kawamura), kusudama balls (glue!). Try the flickr group or browsing.
For Tessellations, it's Eric's book Origami Tessellations, and the Origami tessllations flickr group.
Other books I like (and can name off the top of my head right now):[Origami Dream world] (http://www.giladorigami.com/BO_Dreamworld.html) and origami dream world 2, Brilliant Origami is a classic for animals and has many clever models, Origami for the Connoisseur is a mix of good models from other sources, Fuse's Spirals a gorgeous art book. I have a soft spot for Origami in action .
Hopefully that's enough so that you can get a feel for what's out there right now. If you can tell us more about what you like, then we can give more specific suggestions.
Last comment: Go to an origami convention! That's really where the new and exciting stuff happens and you can meet all the designers.
Things to Make and Do in the Fourth Dimension by Matt Parker is an excellent read.
The best place to start is probably with Eric Gjerde's wonderful book, which details commonly used techniques, stages instructions for a sequence of 25 patterns in increasing complexity, and includes a bunch of photos of other works to get you thinking about even more projects.
Here's an even more basic place to get started: a pair of PDFs detailing his Spread Hexagons tiling.
Happy folding!
Edit: I forgot to include pictures! A simple google search for "spread hexagons" yields some handsome previews. Here's one of my favorites
Check out this list. It includes most if not all books mentioned here.
From more to less advanced, generally:
A Mathematical Gift, II
A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 3)
Note the first three volumes are cheaper as a set.
Intuitive Topology (Mathematical World, Vol 4)
Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5)
Knots and Surfaces: A Guide to Discovering Mathematics (Mathematical World, Vol. 6)
and 5 activity books in the same style
I'd also like to suggest Algorithmic Puzzles.
https://www.amazon.com/Origami-Tessellations-Awe-Inspiring-Geometric-Designs/dp/1568814518
This book came recommended during a conversation about learning origami design. However, if you're not yet an advanced folder, I recommend you work on building your repertoire and progressing to more advanced models before investing in a book, since this is a great way to see the methods designers use.
This model is one of my favourites and illustrates how designers use the potential of folded bases: The tail, head, and wings are all made of identical flaps!
Perhaps this or this.
Peter Smith is a philosopher and his Godel's Theorems book seems to me like what you are looking for. It considers the implications of the theorems and comes with a careful bibliography. Most university libraries would have it.
A shorter book is Torkel Franzén's.
My favorite is that there are four important mathematical theorems that make you increasingly better at the childhood game Dots and Boxes, but that the complete winning strategy is still an open problem and a subject of research.
I skimmed the comments to see if anyone mentioned this related book:
Gödel's Theorem: An Incomplete Guide to Its Use and Abuse
I recently checked it out of our math library. Pretty heady stuff for a small fry like me!
There are two theorems. Your intuition captures very bluntly the first theorem. Your intuition is nowhere near the second one. The theorem applies to certain formal systems.
A good book:
Godel's Theorem: An Incomplete Guide to Its Use and Abuse
http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/dp/1568812388
You mean this one, Mathematical Puzzles: A Connoisseur's Collection?
It's been said before, but programming and math puzzles.
Also, interactive games are great such as card tricks that have mathematical rational.
Books by martin gardner helped me enjoy mathematics when I was younger. I recommend Aha! Gotcha, http://www.amazon.com/Aha-Gotcha-Paradoxes-Puzzle-Delight/dp/0716713616
It is a good mix of puzzles, comics, games, and math.
This might be the book for you:
https://www.amazon.com/G%C3%B6dels-Theorem-Incomplete-Guide-Abuse/dp/1568812388
Not really teaching resources, but maybe good for general interest:
Books:
​
Podcasts:
​
Blogs:
​
I've also heard that "The joy of winning" is a good documentary, but I can't find it online anywhere.
If I remember Godel's theorems correctly, then we can not provide such examples. There are some accompanying statements to (1) and (2) above, that go something like
(1) If there is an unprovably true statement A, we can not ever prove that A is unprovably true.
and
(2) If there is a provable but false statement B, we can not ever prove that B is provable but false.
It is extremely handy to know these when faced in debate by someone who has only read very hand-wavy accounts of Godel's theorems. A fallacious argument based on Godel's theorems is "Statement A seems to be true, but I have no logical proof. But, Godel tells me that many things that are true can not be proved to be true. Therefore A." One response might be "Godel also tells us that unprovably true statements can never be known to be true, so if A is unprovably true, we can never logically conclude that it is true. Thus your statement is fallacious."
Honestly, I haven't run into this argument in use that often myself, but I don't spend a lot of time reading New Age drivel. Visit the Amazon page for the book Godel's Theorem: An Incomplete Guide to Its Use and Abuse . I have not read all of this book myself (just a selection), but it should provide a broader answer to your question than I have given here.
When I was younger, I read this book (Aha! Gotcha: Paradoxes to Puzzle and Delight), and it easily explained various examples of paradoxes while being very entertaining. If anything, I'd say it helped me become a better problem solver.
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
This book applies.
Godel's theorem: An incomplete guide to its use and abuse
Anyone who thinks Peterson's statement is anything more than pseudo-intellectual gibberish, please get a copy of this book and educate yourself. It's aimed at a non-technical audience and is very well written.
I have a book you may enjoy
Interesting, I have not heard it called that, and don't seem to be able to find other references. I do know that Conway calls the structure Hexasticks (or hexastakes if the pencils are sharpened, which changes the symmetry group). It is discussed for example in The Symmetry of Things. My understanding is that the design comes from George Hart, though I do not think he claims to have invented it.
This is what I was thinking
You mentioned that you don't see how math classes promote logical reasoning, because they're just rote memorization. That right there is what's wrong with K-12 math education in the States: mathematics is a creative activity that requires all sorts of clever thinking and conceptual understanding, and rote memorization is antithetical to that. (It's like an "art" class that makes students spend all their time painting fences white, or a "poetry" class that consists of nothing but memorizing lists of vocabulary.)
Fortunately, mathematics is usually taught much better in university classes, where you'll see problems that actually require some creative thinking in putting together a solution using broadly applicable techniques and concepts.
Until then, if you want to get a better sense of real mathematics, you might consider taking a look at popular books (not textbooks) on mathematics; a couple ones I recall enjoying are The Joy of Mathematics and The Art of the Infinite. Or, if you want to learn things more directly from a mathematical perspective, you could look at Polya's classic How to Solve It or Lang's Basic Mathematics (which presents basic algebra and geometry in the way mathematicians think about them).
I took a history of math course out of Howard Eves's Great Moments in Mathematics, and it was great. The problems can be very challenging and informative to historical methods, and I really liked the narrative. It's split into two slim volumes, before/after 1650. looks like about $16 each volume, and I would consider it money well spent.
I also enjoyed The Nothing That Is.
This is quite noob friendly.
It is pleasing to use Gödel's theorems metaphorically when speaking of the unknowable but Gödel's theorems actually make very specific statements about formal mathematical systems that are not really applicable in this context.
If you are interested, I suggest Gödel's Theorem: An Incomplete Guide to its Use and Abuse. This goes for the OP as well.
This is one of the problems featured in Mathematical Mind Benders, a great book that ramps up pretty quickly in difficulty to problems even harder than the one mentioned here. I highly recommend it.
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
there is a really good book on the subject http://www.amazon.com/The-Symmetries-Things-John-Conway/dp/1568812205
"Rock, Paper, Scissors" by Len Fisher
http://www.amazon.com/Rock-Paper-Scissors-Theory-Everyday/dp/0465009387/ref=sr_1_1?ie=UTF8&qid=1331762859&sr=8-1
Before you try to throw books at the problem, try to understand where your kids coming from. The public education system makes a huge effort to crush any enthusiasm a child may have for math; I know I didn't really like it when I was that old either. As a parent, you should read the essay A Mathematician's Lament, by Paul Lockhart. Math can be amazingly interesting and rewarding, but how would they know that? It is your job to show them, and this begins with you yourself becoming interested in and passionate about the subject.
Alright, now I'll start being helpful. Just promise me you'll read that essay.
Martin Gardner's books are kind of the classic books for getting kids into math. Check out the summaries and reviews of Aha! Gotcha, for example. You might also want to pick up The Colossal Book of Mathematics. Read it yourself and try to involve your kids in it as you do.
If they're more into hands-on stuff, check out How Round Is Your Circle?. It's all about building mathematical objects.
The Phantom Tollbooth is another great book, but not so mathematically oriented.
Basically, it's going to take a lot of work on your part, because you can't just hope they'll do it on their own. It's up to you to show them the wonder of mathematics.
Edit: I just realized they're not your kids, they're your niece and nephew. My mistake! Anyway, take from this what you will. Maybe you can convince the parents to get involved?
Algorithm Puzzles might fit your needs.
The Art of Computer Programming, as I understand it, uses assembly language for a imaginary processor to teach algoritms.
You could also use the time to practice designing programs; many schools teach programming by first teaching to design programs in psuedo-code without a computer.
> Kurt Gödel in 1931 with his incompleteness theorems demonstrated mathematically that only the simplest of arithmetic calculations can be complete [6].
Well, you sort of can say that in broad strokes so CSW lovers won't say I'm nitpicking, but it is not the calculations what matters but the system and axiomatics. For instance, presburger arithmetic is decidable, unlike peano arithmetics, and it is weaker than PA, it doesn't have multiplication operation, only addition and equality. But one can express multiplication using only addition, this is by the way essentially what a computer does!
> Science is all about models. We like to believe we can know it all, but this grasp of unbounded knowledge something that will always lie outside our grasp. Gödel proved that.
Gibberish, gödel didn't prove such thing. I suggest u/craig_s_wright reads this book: https://www.amazon.com/G%C3%B6dels-Theorem-Incomplete-Guide-Abuse/dp/1568812388
ps: TIL -- roughly speaking, gödel's formal system (or PA if you like) becomes decidable with the addition of transfinite induction.
I've been meaning to check out these books which claim to be about that exact topic: Origami^3, Origami^4 and Origami^5. They are a conferance about mathematics and Origami, with the books put together by Thomas Hull. There is most likely also an Origami^1 and Origami^2, but I am unable to find them for sale anywhere.
One of my favorite recent mathematics books - and one that offers a nice continuum between 'pure' mathematics and a specific application of it, as well as a nice spread of mathematical sophistication from pop math to some research-level depth, is The Symmetries Of Things by John Conway, Heidi Burgiel and Chaim Goodman-Strauss. It's an exploration of 'discrete' symmetries of the plane and of space - and of the tilings, polyhedra, etc. that they give rise to - as well as an introduction to some aspects of Coxeter groups and a (slightly out-of-place) chapter on the number of finite groups of various orders. I can highly recommend all of Conway's writing, but this is perhaps the finest instance available right now.
There is a nice book about it, too. It's short with some exercises as well. Beware of the reviews. One single-star review complains that it's not a smartphone game.
Dive in, number theory doesn't need any real prerequisites beyond being able to count and an eagerness to learn. As for reading, you can probably find lecture notes from any university on a first year course on number theory on the web, e.g.
http://www.pancratz.org/notes/Numbers.pdf
https://dec41.user.srcf.net/h/IA_M/numbers_and_sets/full
If you want a book, I recall that I liked Numbers & Proofs by RBJT Allenby.
http://www.amazon.com/Mathematicians-Delight-Penguin-mathematics-Sawyer/dp/0140130349
and
http://www.amazon.com/Professor-Stewarts-Cabinet-Mathematical-Curiosities/dp/0465013023
Well. there is a distribution pattern, actually.
There even is a formula for it (despite being inaccurate most of the time).
Edit: you may also enjoy this book if you're interested.
These are some good books:
Mathematical Thinking: Problem-Solving and Proofs
(DJVU)
Mathematical Puzzles: A Connoisseur's Collection
(PDF)
(DJVU)
The first is a good introduction to proofs. The second has a bunch of puzzles that are reasonably challenging and don't require advanced mathematics but do require an understanding of mathematical proofs.
That reminds me of a book that could be perfect for a course like this:
http://www.amazon.com/The-Symmetries-Things-John-Conway/dp/1568812205/
It discusses the idea of symmetry in great mathematical depth, but in a way that is much less formal and pedantic than a traditional math text. For me, there is something beautiful in the extraordinary variety available in the forms of symmetry explored in this book.
Yep. That one and this one
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.
> This is also the reason why the no-show of supersymmetry has no consequences for string theory. String theory requires supersymmetry, but it makes no requirements about the masses of supersymmetric particles either.
And what will become of supersymmetric string theory if SuSy is bumped off? String theory still needs the supersymmetry for to eliminate its wast landscape of predictions and for to keep the vacuum stable (since the vacuum energy for fermions is negative and for bosons positive the hope was that for every fermion there is a bosonic partner with about the same mass to cancel these contributions - see Conlon's book - review here). Thus, no supersymmetry would imply: no gravitons and no quantization of spacetime. Which is also reasonable, since gravity is not renormalizable anyway.
The Standard Model also "required" Higgs boson, but it made requirements about its mass neither. What the absence of Higgs boson would mean for Standard Model after then? The history is written by winners.
Higgs boson in Standard Model is based on different concept, than the Higgs-Anderson mechanism in boson condensates and its technical derivation consists in a mere reshuffling of degrees of freedom by transforming the Higgs Lagrangian in a gauge-invariant manner. A well known "hiearchy problem" implies, that quantum corrections can make the mass of the Higgs particle arbitrarily large, since virtual particles with arbitrarily large energies are allowed in quantum mechanics. During time, Higgs boson mass has been guessed from 109±12 GeV to 760±21 GeV, plus two unconventional theories with 1900 GeV and 10^18 GeV. There are so many comparably likely models - most of which contain continuous parameters whose values aren't calculable now - that the whole interval is covered almost uniformly.
The title of recent another NewScientist article "In SUSY we trust: What the LHC is really looking for" (full version) illustrates clearly which priorities were given before Higgs boson finding once its search started to take too long. The article could be interpreted like: "Umm, well, ... we actually don't believe, Higgs boson will be ever found at LHC - so we should concentrate to supersymmetry instead. ." Moving the goals seems to be the only option for the true believers.
For further reading: Fundamental physics is frustrating physicists , Massive failure of mainstream physics theories at the LHC and Why the LHC is such a disappointment: A delusion by name “naturalness”.
Discrete mathematics with ducks!
Contemporary Abstract Algebra
An Illustrated Theory of Numbers
Visual Complex Analysis
Number-Crunching
The Magic of Math: Solving for x and Figuring Out Why
The Princeton Companion to Mathematics
The Number Devil: A Mathematical Adventure
Professor Stewart's Cabinet of Mathematical Curiosities
Gödel, Escher, Bach: An Eternal Golden Braid
One Two Three . . . Infinity: Facts and Speculations of Science
Recently made a move and these are a few books on my shelf that stand out more.
There are so many beautiful mathematical images that could be chosen though and I do wish more publishers would have more creative book covers also.
The book that I most strongly recommend to Grey, and like-minded Cortexans, is Rock Paper Scissors by Len Fisher. I also have a LOT of trouble getting into fiction books and I burned through this book because it's hilarious and informative. I think it totally works regardless of how much you know about the subject (I'm studying game theory and it didn't bore me in the slightest).
A book that I loved when I was around her age is The Joy of Mathematics
Aha! I see a Menger sponge or two and some Thomas Hull intersecting tetrahedra. Excellent!
I'd highly recommend this book (http://www.amazon.com/Origami-Tessellations-Awe-Inspiring-Geometric-Designs/dp/1568814518) for getting started in tessellations. They're similar in spirit to modular origami (to which I see you're well-informed), but I find it much easier to fold these when I travel for work (It's just one piece of paper) and make better gifts (they can be hung from a wall/window for backlighting and fit an "adults" house better).
Bonne chance!
Not a math question, but have you ever read The Nothing That Is?
No, I'm not Robert Kaplan.
rock paper scissors
http://www.amazon.com/Rock-Paper-Scissors-Theory-Everyday/dp/0465009387
For visual beauty, it's hard to beat The Symmetries of Things (2008) by Conway, Burgiel & Goodman-Strauss.
The MAA review says, "The first thing one notices when one picks up a copy of The Symmetries of Things is that it is a beautiful book."
Since you've got a math flair, I'd recommend Things to make and do in the fourth dimension by Matt Parker. I thought it was very funny and it it's great at explaining maths concepts I didn't even knew existed.
I also enjoyed Marcus Du Sautoy's Music of the Primes. I've got exponentially more excited about prime numbers since reading this! He's also done a BBC documentary on prime numbers (I think by the same title) if you really want to nerd out.
Simon Singh's books (already mentioned) are great as well. He made made a 50 minute documentary on Fermat's last theorem (non-UK link) for the BBC in '96 when Andrew Wiles finally proved it. As he writes on his website, this documentary led him to write the book. I'd definitely recommend watching it together if you're going to give this book!
This was a book I loved as a kid: The joy of mathematics by Theonni Pappas. It has tons of great puzzles, interesting stories.
Algorithmic Puzzles. Reading about programming is dull. Doing is more fun, but you have already to know something. This book introduces algorithmic way of thinking without actual programming
https://www.amazon.com/Algorithmic-Puzzles-Anany-Levitin/dp/0199740445
You should get this book Origami Tessellations: Awe-Inspiring Geometric Designs by Eric Gjerde, you can buy it from here,
http://www.amazon.co.uk/Origami-Tessellations-Awe-Inspiring-Geometric-Designs/dp/1568814518/ref=sr_1_sc_1?ie=UTF8&qid=1415997968&sr=8-1-spell&keywords=origami+tessillations
Its a very good starting place, the models start of very simple, and get slightly more difficult as you progress through the book, the diagrams are very clear, pictures are shown for most steps and there is also a coloured Mountain and Valley Crease pattern for each model which helps ALOT! :).
If you dont want to buy the book, you can find lots of examples and tutorials on youtube, ill provide a link to a few,
Five-and-Four Tessellation
https://www.youtube.com/watch?v=-2boGii3i9s
Star Puff
https://www.youtube.com/watch?v=WhyM6_ioTCE
For the paper, you can use Glassine, super thin, which you can get from the Origami shop.com
http://www.origami-shop.com/en/extra-thin-glassine-xsl-207_215_624_633.html
Tant
http://www.origami-shop.com/en/tant-papers-origami-xsl-207_215_458_625.html.
Hope this helps! :D
I've seen some cool stuff from this book http://www.amazon.com/Professor-Stewarts-Cabinet-Mathematical-Curiosities/dp/0465013023
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
hey there the bridges conference is about your research topic. Here is a really cute video displaying some of the pieces, which there are descriptions of on the site.
This youtube channel also has a lot of other maths inspired art such as this sculpture and a cute little video on symmetry in music.
Good luck with your project!
e: also thirding the mc escher suggestion :)
e2: also if you're interested here is an accessible book (pdf)on symmetry in mathematics, which as you can imagine, ends up being a relevant topic for thinking about art.
Michael Lafosse books are good, like Origami Art and Advanced Origami
http://www.origamido.com/
I like Eric Gjerdes diagrams in here
http://www.amazon.com/Origami-Tessellations-Awe-Inspiring-Geometric-Designs/dp/1568814518
Also John Montrol has lots of animal books
I'm a huge origami fan. Great books for teachers:
https://smile.amazon.com/Origamics-Mathematical-Explorations-Through-Folding/dp/9812834907/ref=mp_s_a_1_2?ie=UTF8&qid=1538344551&sr=8-2&pi=AC_SX236_SY340_FMwebp_QL65&keywords=origami+explorations+math&dpPl=1&dpID=41HUKVEXbXL&ref=plSrch
https://smile.amazon.com/Project-Origami-Activities-Exploring-Mathematics/dp/1466567910/ref=mp_s_a_1_9?ie=UTF8&qid=1538344609&sr=8-9&pi=AC_SX236_SY340_FMwebp_QL65&keywords=math+origami&dpPl=1&dpID=515Ywuc4QYL&ref=plSrch
Check out Thomas Hull, Robert Lang, or Joseph O'Rourke. Ex:
I'm a graduating computer science major and I've always had trouble conceptualizing math. So much so that I grew to hate it. One day a stack of old books was sitting next to the library entrance with a 'FREE' sign and I picked this guy up simply because I identified what I thought was a hilariously ironic title:
The Joy of Mathematics-Discovering Mathematics All Around You
IT CHANGED THE WAY I SEE EVERYTHING.
Without diving too deep into theories and explanations, this book enlightens even the most math-tarded people (i.e. me). It really explores origins of math and the human race's intuition for it.
The aim of the book is not to cover one large fundamental concept, but instead to briefly investigate a few dozen different ideas. Just about every page or so is a completely different concept than the previous or successive one. I still open it to some random page and start reading when I get bored.
I really hope you and your boyfriend take a look at this one. I'm positive you'll find it insanely interesting!
Also, the book is available via Amazon, and through other online booksellers, as well as local bookstores. :)
Robert Kaplan's The Nothing that Is: A Natural History of Zero
I like Matt Parker's work so much that I bought a print copy of his book. I really need to start in on it.
http://www.amazon.com/Algorithmic-Puzzles-Anany-Levitin/dp/0199740445
for practical applications of algorithms.
I agree with the Cormen book suggestion, as I hear a lot about it.
A great book on this subject is Marcus du Sautoy's Music of the Primes. I read this is the last year of sixth form (senior year of high school in the US?) and found it very understandable.
Could it be Martin Gardner's Book?
For some great comp sci style thinking puzzles without the computer, this is crazy fun. From bring up at party casual to mind meltingly difficult. Algorithmic Puzzles https://www.amazon.com/dp/0199740445/ref=cm_sw_r_cp_awd_AeMzwbE3NJARB
Example: if you can only fit two pancakes on your griddle, what's the fastest way to make three pancakes?
Yes! In Origami Tesselations by Eric Gjerde. This book is the best entrance point into origami tesselations. I would warn you though, this model was way harder than I expected, so you probably want to start with some simpler tesselations first.
There's a book about that
perhaps Gödel's Theorem: An Incomplete Guide to Its Use and Abuse?
http://oyc.yale.edu/economics
I haven't watched this myself but it is an intro course and it is from Yale so it is probably good basic material. I was going to check out Shiller's course sometime next week.
http://www.amazon.com/Rock-Paper-Scissors-Theory-Everyday/dp/0465009387/ref=sr_1_1?ie=UTF8&s=books&qid=1249068580&sr=8-1
I also thought rock, paper, scissors was a fun fairly easy reading book.
If you're looking for advanced material you're probably better off bugging professors than reddit. Hopefully another commenter will prove me wrong :)
Somewhat relevant is "The Nothing That Is: A Natural History Of Zero" by Ellen Kaplan (http://www.amazon.com/The-Nothing-that-Is-Natural/dp/0195142373). While this book focuses on the number zero, it is an interesting read about the concept of "nothingness" and how some cultures simply don't have any representation for it. It goes into the incredible value of the number zero mathematically as well. If you don't find a book that gives you everything you are looking for, I highly recommend making this book part of your research.
Reading this
http://en.wikipedia.org/wiki/Mamihlapinatapai
I also came over this book:
http://www.amazon.com/Rock-Paper-Scissors-Theory-Everyday/dp/0465009387
If it will make you feel better, you can say you are quoting Martin Gardner, rather than Sarah Palin, because if I recall correctly this is covered in Gardner's book "Aha! Gotcha".
http://www.amazon.com/Aha-Gotcha-Paradoxes-Puzzle-Delight/dp/0716713616
The Joy of Mathematics. Heard that this one is pretty good.
You may be interested in Peter Winkler's two books:
(1) Mathematical Puzzles: A Connoisseur's Collection
(2) Mathematical Mind-Benders
The questions do not require anything more than high school Mathematics.
There is a range of difficulty in the problems, although quite a number are pretty tough.
Let me give an example from the first book:
Each member of a team of n players is to be fitted with a red or blue hat; each player will be able to see the colors of the hats of his teammates, but not his own. At a signal, each player will simultaneously guess the color of his own hat.
The players are allowed to collaborate on a common strategy before the game begins. The objective is to guarantee as many correct answers as possible, assuming the worse case situation. That is, you may assume the enemy knows their strategy and will try their best to foil it. How many correct answers can you guarantee?
To be honest, I have recently started practicing proof-writing of different theorems and one book which I have found extremely helpful is Numbers and Proofs by R Allenby.
However, if you are not looking for a textbook to practice writing proofs you can just make sure and try to write down proofs of theorems in the textbooks that you use before they tell you how they have derived it. This will benefit your problem-solving skills greatly.
My favorite tessellations (including your one) comes from Eric Gjerde (Sarah Adams did many of those in her channel). Get his book and visit his website and flickr.
Start with this one. It is not a tessellation by mean, but it is fun model and good for practicing precise folds and some skills.
This one is interesting model and it will give you an idea how important is to fold a grid (basically how to "prefold" a paper before making creases).
And finally - this one followed by [multi-level version] (http://www.youtube.com/watch?v=474PmBXekK8) is your holy grail. If you stick with this one and find yourself enjoying hours spend with this bad boy you'll know that you're hooked on tessellations (here is my humble 5-hour contribution). Then just go through the book and videos and diagrams.
I just find it so relaxing. Just turning on some music, and for hours creating something complex and symmetrical. Have a great time!
Well, obviously it's a function of the size of the grid. Dots and boxes on a 2x2 dot grid would be a stupidly trivial game. Here's a paper discussing solving dots and boxes for 4x5, in which they mention that the naive search space of a 4x4 game is is 40!. So it's a game that very quickly leaves the realm of "obvious". If you're interested in the strategy of dots and boxes, this book is supposed to be an good start, though I've not read it. It's also touched upon on in chapter 16 of Winning Ways for your Mathematical Plays.