One of my pet peeves about people and math is that most
people don't really have a clue of what math is. I've been writing
this blog for something over three years, and by the standards of
a lot of people, I've almost never written about math.
Yesterday, my son's kindergarten class had a picnic. On my way home,
I was listening to the local NPR station, which was interviewing some
crossword puzzle writer whose name I cannot remember; I will therefore refer
to him as "crossword-boy". (It was not Will Shortz; Shortz is much smarter than the
guy they were interviewing.) At one point, they asked him something about
His response was a bit disjointed - he couldn't decide whether to talk about
the history of Sudoku or about his opinion of it. His opinion is that it's
incredibly dull and pointless, and that designing good Sudoku doesn't require as
much creativity as designing good crosswords. (Just that much is annoying: I'm
a Sudoku addict, and I've definitely noticed dramatic differences in Sudokus
from different places. Will Shortz's Sudoku books have great ones; most computerized
Sudoku games generate rather boring ones; the ones in most newspapers are
obviously computer generated.)
In the course of babbling about how uninteresting, non-creative, and
unsatisfying Sudoko puzzles are, he let loose with the real stupidity: "You know,
Sudoku doesn't even have to use numbers, it can use any 9 symbols. It's not a
mathematical puzzle at all.
Because it doesn't rely on arithmetic, according to crossword-boy,
it's not mathematical at all. He went on to say that it's
just a logic puzzle, not a math puzzle at all.
Sorry pal, but logic is math.
Sudoku is an incredibly mathematical puzzle. It's not an
arithmetic puzzle, but it's a highly mathematical one.
In computer science terms, it's a moderately
complex constraint-solving puzzle.
Math is more than arithmetic. It's more than numbers. Math
is really the formal study of logic and structure. Numbers and arithmetic
are one kind of structured system described using logic which can be
studied and understood using math. But pretty much everything
with a precise, formal structure to it has at least an element of
mathematics. The structure of crossword-boy's crossword puzzles
is fundamentally mathematical.
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Ugh. The basic reason I took an interest in math, was in about the second grade I was working through my math work book. It was mostly if not entirely arithmetic. It occurred to me that a number wasn't a name for a pile of buttons or number of fingers, but how many things -- any *thing* -- you had, and this was the way to keep track of them. It was useful.
So many people, like this crossword puzzle writer, don't want to realize how useful math is for everyday problems.
That'd be like saying that the card game SET isn't a math game because it just has shapes and colors.
Slightly off-topic, but I thought I'd mention it and get it off my chest:
Sudoku is an interesting example for an intro to information theory. The amount of information in a unsolved Sudoku puzzle and the amount of information in the solved puzzle is the same (assuming there is only one solution). The unsolved puzzle is an encoding of the completed one.
Just a neat thought I had a few months ago, and I was curious about your thoughts on the analogy, Mark.
You've articulated one of the things that I've been thinking about lately. It really is bothering, peoples misconception of what math is.
Most people go through the K12 system (or equivalents), in which math is mostly about memorizing formulas that have numbers in them. Therefore, for most people math is numbers. Don't even get me started on what people think logic is.
PS. This is my first post here. I've stumbled upon your blog rather by chance; I'm glad I did.
The other side of this is when people say they don't do sudoku puzzles because they're "not good with numbers." I've given up; now I just laugh.
And when I occasionally have a brain-fumble in splitting a check or computing a tip, I often excuse myself with, "I majored in math... not arithmetic."
I agree. Math, in its heart of hearts, is only about THREE things:
(1) Quantity (most people only think of this);
(2) Structure (some Geometry is badly taught in school);
(3) Change (pre-Calculus, schools only teach Motion and Compound Interest).
And all permutations (Enumeration of Structures; Structure of Change; ...)
What my Physics Professor crossword-puzzle-genius wife can't understand is why the Sudoku listed as easy are hard, and vice versa, in our Los Angeles Times. Is her mental algorithm so very different from everyone else's?
I just started university studies this year, studying computer science, and in the first year among the courses given is calculus and group theory.
I was amazed and delighted right on the first week on how they rebuild math on a set of logic principles, making all of the math we learn from that point on, not a separate entity from logic, but a natural conclusion of logic itself. It was stunning, and I am still very delighted by it.
I love how they threw away what we knew as math from high school, where we just took numbers as is, and rebuilt them from scratch. I love that I now, can prove, from an initial set of very primitive axioms, and logic, go all the way to show that the integral of cos(x) is sin(x).
I was a math major in college (back when we used Roman numerals and before Sir Isaac introduced the Fluxions), and I also enjoyed the structural aspect. Abstract algebra and topology were really fun.
Oddly, I have never developed an appreciation for Sudoku. I tried it for a while, but it never "took". I do love crosswords - I'm a word play guy. Love puns, misdirection, etc.
Maybe I'll have to try one of Shortz's Sudoku books.
Re. Sharkey (No. 3)
Puzzle as an encoding is an interesting way to look at it. Wikipedia says there's 6,670,903,752,021,072,936,960 ways to fill in a blank Sudoku. If you take the logarithm base 10 of that, you get 21.8, so 22 placed digits should be the minimum needed to get a unique solution, but that's a lower bound---it might be that you can't put clues down that efficiently. Has anyone encountered a Sudoku with a unique solution with that few digits?
In Hebrew we have a saying for such situations, "don't bother confusing ignorant people with facts".
You have a lot of patience, my friend. My brush with the world of math has been much more modest than yours, and I already grew weary of trying to explain to people what math is and what math isn't.
I guess Brown Sharpie put it best when she said that most people who say they hate math never earned the right :)
He raised a good point - the numbers are a red herring.
Also, if I were to talk about math to the guy in the street, or the guy on the radio, I wouldn't expect them to understand the full scope of mathematics. But I guess that's pandering to a misconception, rather than dealing with it.
I wonder if the speaker would have liked to have a go at designing crosswords for the japanese? Perhaps he'd getter a better understanding of why sudoku became so popular in the first place. :)
Some logic is mathematical logic, but not all.
Wikipedia also says that there have been many thousands of 17-clue puzzles found. If the clues have to be placed with rotational symmetry, 18 clues is the minimum found so far.
As few as 17 placed digits can lead to a unique solution - see: http://en.wikipedia.org/wiki/Mathematics_of_Sudoku#Minimum_number_of_gi…
Re. 13, 14
Good to know. I should have considered that 22 digits would be an average over all possible games, rather than the best you can do for any one given game (see what I get for being lazy in my assumptions---I also used log base 10, when I should have used base 9 to match the alphabet size, and furthermore, neglected to factor in that placed digits have a position as well as a value, which adds information in a way I have no idea how to account for).
I think a large part of the reason that people who want to promote Sudoku claim that it's nonmathematical -- even people who should know better -- is that they know most people are scared of math.
TV Guide does one with letters every issue; one line spells some word or name (sometimes with extra letters).
My sister loves the cryptogram but stayed far from Sudoku, thinking it had to add up somehow. She hates "math"... like the cryptogram has no math to it.
Right on, Mark. I think that the misunderstanding you're talking about is endemic in how math is perceived in the broader culture. I just finished college where I was a math major, and got a big kick out of always being asked to figure out the bill/tip whenever we went out to eat. Generally, I'd laugh it off with a quip about taking "Addition III" or being an arithmetic major.
Interestingly, the subject where I actually learned the most about how to do mental math quickly and accurately was high school physics. Those skills really do come in handy, especially in figuring out tips or adding things up. But that's a totally different skill set than mathematical (i.e. logical) reasoning, which, to my eye, has more to do with being able to pick up new structured ways of looking at things, and understanding how little bits of information or structure relate to bigger structures.
Good post. I have followed your blog for quite a long time now. It always has valuable information. About this post, I would like to point out that saying that logic is math, is like saying that biology is biostatistics.
To be pedantic, I might expand my simplification in #6:
Math, in its heart of hearts, is only about THREE things:
(1) Quantity (most people only think of this);
(2) Structure (some Geometry is badly taught in school);
(3) Change (pre-Calculus, schools only teach Motion and Compound Interest).
And all permutations
I've used that with students as early as Middle School, and it usually startles them that no teacher had ever said anything of the kind before.
Others would add to the list:
The others being as referenced below.
However, I can still explain:
(4) space, in terms of structure (for example, Euclidean space, hyperbolic space, topological space, Minkowski space);
I can still explain:
(5) relation, in terms of structure, using axiomatic development of relations and function;
I can still explain:
(6) pattern, as a type of structure in space, giving examples of integer sequences, tessellations, symmetries, orbifolds, ...;
I can still explain:
(7) form, as I did pattern, within the more general "structure" -- given that the structure may be very esoteric or abstract; and
I can still explain:
(8) entity, if only by saying that Mathematics is not about any specific entities at all, with the evasive "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." -- Bertrand Russell
Steen, L.A. (April 29, 1988). The Science of Patterns. Science, 240: 611â616. and summarized at Association for Supervision and Curriculum Development.
Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 9780716750475.
Orthogonal to that, so far as I see, most teachers do not understand multiple levels of abstraction. Even in Math, which so much depends on this. Most teachers are clueless about how the human brain works. I keep saying to my students that science and math are about:
(2) structures of stuff, and
(3) properties of structures of stuff.
(that being a deep insight from Category Theory). And relentlessly connecting that to their own lives. And making it hands-on, collaborative, and fun.
Do you not make hypotheses, try them, and draw conclusions in solving Sudoku? That's Math. Benjamin Peirce defined Mathematics as: "the science that draws necessary conclusions."
Albert Einstein wrote: "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
Note that I can bind my original list of 3 to "spiraling" through the curriculum, coming back again and again to basics, but elaborating and showing more sophisticated alternatives.
(1) Quantity starts with "number sense" that children have even in the extremely rare cultures that don't count; "God created the integers," wrote mathematician Leopold Kronecker, "All the rest is the work of Man." And then spiral through (as Mark CC has done in this blog) rational numbers, real numbers, complex numbers, quaternions, octonions, transfinite numbers, , cardinal numbers, and so forth;
(2) Structure, starting with examples of mathematical objects with internal structure; and I have students build Platonic and Archimedean solids of paper and glue, or toothpicks; and use them to illustrate (after they learn by tactile-kinaesthetic means, and are led towards Euler's polyhedral formula) with groups, rings, fields and other abstract systems, which are themselves such objects, and so on in level after level, as the field of abstract algebra. (including vectors, the generalization to vector spaces, and more useful things in linear algebra. Work with vectors nicely combines quantity, structure, space, and further generalization to vector calculus gets us back to change (the next time around the spiral we hit tensor calculus to appreciate symmetry and changes of vectors under rotation.
(3) Change: without this branch, we could not do natural sciences, as Calculus was discovered by Newton, Leibnitz, et al for Science. We get a deeper understanding of functions here, to clarify changing quantities, even though it took a century or so to get rigorous with real analysis, and then with complex analysis, and next (typically infinite-dimensional) functional analysis for applications such as quantum mechanics. Connecting quantity and rate of change gives us differential equations, which leads to more generalized dynamical systems, culminating in the tremendous new field of chaos theory, as Mark CC has recently explained.
So I like my list of 3. And Sudoku is Math from so many different approaches in this classification.
While most computer-generated Sudoku's are indeed not very good, I don't believe that this is a consequence of no human interaction. I think the qualities of a good Sudoku could certainly be incorporated into a good Sudoku generator.
For solving Sudoku, I like the article by Helmut Simonis that discusses what techniques from constraint programming that are needed to solve Sudokus of different hardness level without search.
As a side note, while 9x9 Sudokus are fairly simple to solve for a computer (no search, less than 0.1 ms), a 25x25 Sudoku is quite hard to solve.
I was interested in your comments on Sudoku, Mark. Having only done newspaper Sudokus, I started to lose interest in them after a while, once I'd worked out a system for solving them. Perhaps I should get hold of one of Shortz's books, and see if that revives my interest.
I still enjoy Killer Sudokus, which I've only done a few times. But I generally prefer cryptic crosswords, my favourite setter being Araucaria (a noted British setter). I particularly like his "jigsaw" crosswords, where the clues are unnumbered and you have to work out for yourself how to fit the solutions into the grid.
You seem to have hit a nerve Mark! It certainly bugs me, the number of people that seem to think that because I'm good at Maths (as we call it on this side of the Atlantic) I must also be good at arithmetic.
Also, I've seen Sudoku described as a denial-of-service attack on the human brain. YMMV.
@Richard Wein: Yes, newspaper Sudokus are pretty boring. I've done them for some time, and they are solvable in relatively short time, lets say 15 minutes or so. But they have to. Newspaper is read and then dumped, no need to introduce Sudokus that may take hours, they may leave the reader with a bad feeling ;)
BTW, I really like those crosswords without any hints, where all the letters are coded with numbers. That way, you have to work out the statistics of that crossword and also watch for sensible revealing combinations of letters (for example in german "sch" or doubled letters like "nn" or "tt". Great fun!)
For my studies I had the task to solve sudokus using a) a Hopfield network and b) a Markov Random Field with an incremental approximate algorithm. Also great fun!
Slightly offtopic, but did you read "Where Mathematics Comes From" by Lakoff and Nunez? I found this book very interesting, and wanted to hear your opinion :)
Is there any research on numeracy amongst mathematicians? I would bet good money that their much vaunted claim to be "not good with numbers, I'm a mathematician" would turn out to be hot air.
One reason I like Sudoku better than crosswords is that success depends on your reasoning ability, not your recall of trivia.
If a crossword has a clue like "Dodgers Pitcher in 1986", either you know or it your don't know it. If you don't, you're never going to solve the puzzle.
I was in high school after Newton derived fluxions, but still a long time ago. My math learning included Boolean algebra (actually, a project I did) and logic puzzles (Wiff N Proof - not sure of the spelling). My senior year math teacher loved logic puzzles. (We had all had calculus as juniors, so senior year was a free-for-all.) Somehow I learned intuitively that math was more than just arithmetic.
I was a physics teacher for a long time, and got to teach math by the way to physics students who never quite got algebra, trig and calculus. But I am horrible about calculating in my head. Maybe it's the pernicious influence of calculators, but I am never sure whether my "mental math" is correct. I can estimate like no tomorrow, but doing sums in my head is an iffy thing for me.
And for some reason, I never quite nailed the times tables for 7, 8 and 9, so even now I have to stop and think if I'm right.
Reminds me of the story Raymond Smullyan once told.
A woman told him that her son loved Smullyan's books, but that Smullyan shouldn't tell him he was doing math, because the kid hated math and would stop reading it immediately if he learned he was doing math.
I'm proud of my oldest boy - he's a junior school maths-medal winner. But I often worry that his ability with numbers will give him too much of an ego to reach further in maths.
I keep telling him that numbers is really only a tiny part of what maths is about.
I take heart that he still would like to be a "Mathematician" as a career choice.
It is good to hear confirmation from someone who is a real mathematician. I'm going to see if he is up to reading your blog post.
Thanks for the kind words, but I'm not a real mathematician. I'm not even a fake mathematician. I'm a software engineer who loves math, but I'm nowhere near good enough at it to be a mathematician.
What I used to tell the kids I was tutoring in Math was that Mathematics, in its essence, is about proof. Quantity, structure, change, and the like are all examples of different types of domains in which we apply mathematical reasoning, but the essence of all of them is the application of logical reasoning to prove theorems about the domain in question. Of course, this tends to get underplayed the way that math is typically taught in the schools, where it is typically presented as a bunch of facts and techniques to memorize, rather than a body of knowledge that is built up through reasoning.
I'm ... not so sure about that.
A given digit in a given square is log10(number of possible squares * number of posible digits) = log10(729) = 2.82 units of information. So you should only need about 8 placed digits. Of course, a placed digit doesnt quite have that amount of information, as each subsequent one is constained. So it'll be a bit more than 8.
I have found the puzzles generated by 5ud0ku to be very reasonable.
You should send your comment in to NPR and hopefully they will read it when they do their weekly comment reading.
It wasn't "All Things Considered"; it was a local midday talk show (Leonard Lopate, I think). You need to comment while the interview is going on to get it on the air.
I can't count how many times I had this exchange back when I was in school:
Them: so, what's your major?
Them: oh, so you're going to be an accountant?
Semantics. He is using the common meaning of "math", which may have started as an abbreviation of "mathematics" but has grown to mean "common arithmetic of the sort taught in grammar school." You are using the academic meaning, which still is synonymous with "the domain of the field of Mathematics as taught in college."
It's annoying, but it's tantamount to me, a CS nerd, complaining when I hear a politics wonk describe distorting the interpretation of a statement as "parsing" or a businessperson describe tuning out of a meeting to check his voicemail as "multitasking."
Language evolves, and meaning is subjective. Sorry, but that's just how it goes.
Alex Chaffee is right. And this goes deeper.
When I was taking graduate courses in Computational Linguistics, I heard Lingustics grad students and post-docs complain that when they say "Linguistics" in a party, the usual response is "Oh, my son is taking French."
I recall a Cosmology professor gritting his teeth at a party when someone said: "Oh, my daughter is also studying Cosmetology. She wants her own salon some day.""
Re: comment #38, and apologies for off topic,
My husband is an accountant, and we have a friend who majored in math and is now a math teacher. Of my husband's profession, he said "Accounting is the stamp collecting of the math world."
In retrospect, I'm not sure if that's insulting to accountants or to stamp collectors, but it struck me as terribly funny at the time.
Hi Mark, why don't you do a post about why you love Sudoku and what makes an interesting Sudoku? I'd love to hear more about that!
Alex Chaffee hit the nail on the head. Thus one might reason as follows:
Sudoku is a logic puzzle.
Logic is part of philosophy. After all, some univerisities list both informal and symbolic logic as philosophy courses.
Therefore, when one does Sudoku, one is actually doing philosophy.
Of course, that's just one way of classifying Sudoku puzzles as already mentioned. Classifications like logic and maths this don't exist via nature. We create such classifications. A consequence of this comes as that we won't have universal consistent classifcations, as we've seen above.
Still, the cosmetology/cosmology example works out as something different, because we have different terms and ideas there. It sounds like someone misheard.
I am addicted to Sudoku. I love crossword puzzles too. They intrigue me in the same way, although I tend to swear at the stupid crossword puzzle author more than the Sudoku puzzle writer -- though a computer generated Sudoku puzzle may be boring, it's never actually wrong, and I have often found a perfectly good Sudoku right next to a perfectly terrible crossword in the newspaper, but not yet encountered the inverse.
It's true that you don't need numbers to do a Sudoku puzzle, which is why the whole "but it has to add up to something!" argument is silly. Yes, all the numbers in a line will add up to something specific (45 on a typical 9x9 puzzle), but this is not important; what's important is that no number gets duplicated in a column, row, or block. The same puzzle can be done with pictures.
My introduction to sudoku came from the BBC's Doctor Who website. Among the games is SuDocWho, where pictures of the first nine doctors (including Paul McGann as the eighth) must be placed onto the squares. Anything can be used -- letters, shapes, species of bird, anything.
One might argue that this is not mathematical if it doesn't have to involve numbers, but the strategies and logic involved can all be described mathematically. It's as much a math problem as trying to work out how to compress a digital image into a smaller file size.
"Will Shortz's Sudoku books have great ones; most computerized Sudoku games generate rather boring ones; the ones in most newspapers are obviously computer generated.)"
I think you are a sudoku-aesthetic person, seeking beauty in puzzles. Computer generates puzzles based on difficulty, not based on clue distribution, but I hope some minor tweaks in a program may be able to fix that.
I have a puzzle copied from another source, is it boring?
Please contact me. I am eager for your reply.
From an article in the June 19, 2006 "New York" magazine:
"Shortz was drawn into the craze when St. Martinâs called him in a panic last June. It demanded he produce three books of 100 Sudoku puzzles each in ten days, which he did with the help of a computer programmer in the Netherlands."
I'm pretty sure all subsequent Shortz sudoku books were also computer-generated (though not all sudoku generators are created equal, I suppose).
I'm sure that most, if not all Sudoku's involve a computer at some stage of the process. But there are good Sudokus, and there are not-so-good ones. The role of the editor of a collection of puzzles is to recognize the good ones.
When you look at the puzzles in Shortz's books, they're not just typical computer generated puzzles. The really difficult ones are *really* difficult - they've got specific features in the puzzle that make it appear ambiguous. There's also frequently some kind of structure to the puzzle (particularly with the easy ones) that are very clever.
Like I said - I'm seriously hooked on Sudoku. Between my wife and me, we've probably got 30 or 40 books of Sudoku, and at least half a dozen different Sudoku games for laptops or cellphones. And there's a dramatic difference between puzzles from different places. The Mensa Sudoku books tend to be dull - they're on a par with one of the mediocre laptop games. The "black belt" sudo books are awful - on a par with the lousy computer games. The better computer games are pretty good, but not great. And Shortz's books are clearly on top. He's just *really* good at recognizing what makes an interesting, engaging puzzle. He's great.
For computer games, my personal favorite is Snoodoku. It generates respectable (if not great) puzzles, and the UI is very nice. I haven't found anything that I like as much. At one point, I found a mac-only Sudoku that generated better puzzles, but the UI was god-awful. It was the kind of UI that would work nicely on a cellphone with a touchscreen, but which was god-awful painful with a mouse and keyboard.
I love math, don't like Sudoku, but love KenKen (which involves math much more clearly than does Sudoku).
We may be dealing in semantics here, but I think it's an oversimplification to say "logic is math." Logic UNDERLIES math, but the two are certainly not exactly equivalent.
And a question: suppose someone solves Rubik's Cube through much visual trial-and-error effort -- has he solved it logically, mathematically, or neither of the above (or maybe it's a pointless question!)???
The key of Sudoku Mark hits on is the clever structure/features of good Sudoku. The same is true with good crosswords. Bad crosswords are just a bunch of trivia and/or go-to crossword words (epee, aria, etc). The best crosswords don't rely on those tricks, and often have a theme or structure. Once you see that structure, it opens the game wide open. Any good puzzle lends a little insight into what the puzzle writer was thinking when designing it, and seeing the puzzle from his or her perspective is the reward.
However, I have always preferred crosswords to Sudoku, because of the complexity, though maybe not in the formal sense. The question I ask myself is "How hard would it be to write a program that would solve a Sudoku?" It would be easy but slow to just crunch the numbers, but you could also apply some neat methods to improve upon that, because as pointed out, Sudoku is a constraint problem, and such problems are well characterized. But I can't even fathom how one could write a program to solve a cryptic crossword.
So how could you write a program to solve a crossword?
I particularly hate the crossword in the Big Issue with its definitional clues. It reads as though the compiler has taken a thesaurus, looked up a word that means something a bit like the answer, and then looked up another word that means something a bit like the intermediate word to use as the clue.
Give me a nice cryptic crossword anyday.
As for Sudoku ..... I thought I had written a computer program to solve them. It occasionally fails. The only way I have found to proceed with those puzzles is trial and error. I can't believe anyone would actually find that enjoyable, so I can only assume I'm missing something.
Interesting, though, that there is enough information in the puzzle as published to replicate the fully-solved grid. That's why I don't buy any cosmological fine-tuning arguments. We already proved that Îµ0 and Î¼0 are not independent variables, so why should the rest be?
Îµ0 and Î¼0 are not independent variables. If the measurements at NIST of Planck's constant is off, then our measurements of Îµ0 and Î¼0 are also off. And the two most precise ways of measuring Planck's constant do NOT agree. Sudoku is quantized, right? In the limit of the number of boxes going to zero, it deforms to the classical null puzzle.
This is a good sudoku variant puzzle. It took me several hours to do.
Also, the numbers to fill it in are also used as numbers, so the crossword-puzzle-guy on the radio would approve!
Nice article related to Maths puzzle.