The Summer 2017 Mathematics A To Z: Young Tableau


I never heard of today’s entry topic three months ago. Indeed, three weeks ago I was still making guesses about just what Gaurish, author of For the love of Mathematics,, was asking about. It turns out to be maybe the grand union of everything that’s ever been in one of my A To Z sequences. I overstate, but barely.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Young Tableau.

The specific thing that a Young Tableau is is beautiful in its simplicity. It could almost be a recreational mathematics puzzle, except that it isn’t challenging enough.

Start with a couple of boxes laid in a row. As many or as few as you like.

Now set another row of boxes. You can have as many as the first row did, or fewer. You just can’t have more. Set the second row of boxes — well, your choice. Either below the first row, or else above. I’m going to assume you’re going below the first row, and will write my directions accordingly. If you do things the other way you’re following a common enough convention. I’m leaving it on you to figure out what the directions should be, though.

Now add in a third row of boxes, if you like. Again, as many or as few boxes as you like. There can’t be more than there are in the second row. Set it below the second row.

And a fourth row, if you want four rows. Again, no more boxes in it than the third row had. Keep this up until you’ve got tired of adding rows of boxes.

How many boxes do you have? I don’t know. But take the numbers 1, 2, 3, 4, 5, and so on, up to whatever the count of your boxes is. Can you fill in one number for each box? So that the numbers are always increasing as you go left to right in a single row? And as you go top to bottom in a single column? Yes, of course. Go in order: ‘1’ for the first box you laid down, then ‘2’, then ‘3’, and so on, increasing up to the last box in the last row.

Can you do it in another way? Any other order?

Except for the simplest of arrangements, like a single row of four boxes or three rows of one box atop another, the answer is yes. There can be many of them, turns out. Seven boxes, arranged three in the first row, two in the second, one in the third, and one in the fourth, have 35 possible arrangements. It doesn’t take a very big diagram to get an enormous number of possibilities. Could be fun drawing an arbitrary stack of boxes and working out how many arrangements there are, if you have some time in a dull meeting to pass.

Let me step away from filling boxes. In one of its later, disappointing, seasons Futurama finally did a body-swap episode. The gimmick: two bodies could only swap the brains within them one time. So would it be possible to put Bender’s brain back in his original body, if he and Amy (or whoever) had already swapped once? The episode drew minor amusement in mathematics circles, and a lot of amazement in pop-culture circles. The writer, a mathematics major, found a proof that showed it was indeed always possible, even after many pairs of people had swapped bodies. The idea that a theorem was created for a TV show impressed many people who think theorems are rarer and harder to create than they necessarily are.

It was a legitimate theorem, and in a well-developed field of mathematics. It’s about permutation groups. These are the study of the ways you can swap pairs of things. I grant this doesn’t sound like much of a field. There is a surprising lot of interesting things to learn just from studying how stuff can be swapped, though. It’s even of real-world relevance. Most subatomic particles of a kind — electrons, top quarks, gluons, whatever — are identical to every other particle of the same kind. Physics wouldn’t work if they weren’t. What would happen if we swap the electron on the left for the electron on the right, and vice-versa? How would that change our physics?

A chunk of quantum mechanics studies what kinds of swaps of particles would produce an observable change, and what kind of swaps wouldn’t. When the swap doesn’t make a change we can describe this as a symmetric operation. When the swap does make a change, that’s an antisymmetric operation. And — the Young Tableau that’s a single row of two boxes? That matches up well with this symmetric operation. The Young Tableau that’s two rows of a single box each? That matches up with the antisymmetric operation.

How many ways could you set up three boxes, according to the rules of the game? A single row of three boxes, sure. One row of two boxes and a row of one box. Three rows of one box each. How many ways are there to assign the numbers 1, 2, and 3 to those boxes, and satisfy the rules? One way to do the single row of three boxes. Also one way to do the three rows of a single box. There’s two ways to do the one-row-of-two-boxes, one-row-of-one-box case.

What if we have three particles? How could they interact? Well, all three could be symmetric with each other. This matches the first case, the single row of three boxes. All three could be antisymmetric with each other. This matches the three rows of one box. Or you could have two particles that are symmetric with each other and antisymmetric with the third particle. Or two particles that are antisymmetric with each other but symmetric with the third particle. Two ways to do that. Two ways to fill in the one-row-of-two-boxes, one-row-of-one-box case.

This isn’t merely a neat, aesthetically interesting coincidence. I wouldn’t spend so much time on it if it were. There’s a matching here that’s built on something meaningful. The different ways to arrange numbers in a set of boxes like this pair up with a select, interesting set of matrices whose elements are complex-valued numbers. You might wonder who introduced complex-valued numbers, let alone matrices of them, into evidence. Well, who cares? We’ve got them. They do a lot of work for us. So much work they have a common name, the “symmetric group over the complex numbers”. As my leading example suggests, they’re all over the place in quantum mechanics. They’re good to have around in regular physics too, at least in the right neighborhoods.

These Young Tableaus turn up over and over in group theory. They match up with polynomials, because yeah, everything is polynomials. But they turn out to describe polynomial representations of some of the superstar groups out there. Groups with names like the General Linear Group (square matrices), or the Special Linear Group (square matrices with determinant equal to 1), or the Special Unitary Group (that thing where quantum mechanics says there have to be particles whose names are obscure Greek letters with superscripts of up to five + marks). If you’d care for more, here’s a chapter by Dr Frank Porter describing, in part, how you get from Young Tableaus to the obscure baryons.

Porter’s chapter also lets me tie this back to tensors. Tensors have varied ranks, the number of different indicies you can have on the things. What happens when you swap pairs of indices in a tensor? How many ways can you swap them, and what does that do to what the tensor describes? Please tell me you already suspect this is going to match something in Young Tableaus. They do this by way of the symmetries and permutations mentioned above. But they are there.

As I say, three months ago I had no idea these things existed. If I ever ran across them it was from seeing the name at MathWorld’s list of terms that start with ‘Y’. The article shows some nice examples (with each rows a atop the previous one) but doesn’t make clear how much stuff this subject runs through. I can’t fit everything in to a reasonable essay. (For example: the number of ways to arrange, say, 20 boxes into rows meeting these rules is itself a partition problem. Partition problems are probability and statistical mechanics. Statistical mechanics is the flow of heat, and the movement of the stars in a galaxy, and the chemistry of life.) I am delighted by what does fit.

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Reading the Comics, May 20, 2017: Major Computer Malfunction Week Edition


I was hit by a massive computer malfunction this week, the kind that forced me to buy a new computer and spend half a week copying stuff over from a limping hard drive and hoping it would maybe work if I held things just right. Mercifully, Comic Strip Master Command gave me a relatively easy week. No huge rush of mathematically-themed comic strips and none that are going to take a thousand words of writing to describe. Let’s go.

Sam Hepburn’s Questionable Quotebook for the 14th includes this week’s anthropomorphic geometry sketch underneath its big text block.

Eric the Circle for the 15th, this one by “Claire the Square”, is the rare Eric the Circle to show off properties of circles. So maybe that’s the second anthropomorphic geometry sketch for the week. If the week hadn’t been dominated by my computer woes that might have formed the title for this edition.

Werner Wejp-Olsen’s Inspector Danger’s Crime Quiz for the 15th puts a mathematician in mortal peril and leaves him there to die. As is traditional for this sort of puzzle the mathematician left a dying clue. (Mathematicians were similarly kind to their investigators on the 4th of July, 2016 and the 9th of July, 2012. I was expecting the answer to be someone with a four-letter and an eight-letter name, none of which anybody here had. Doesn’t matter. It’ll never stand up in court.

John Graziano’s Ripley’s Believe It Or Not for the 17th features one of those astounding claims that grows out of number theory. Graziano asserts that there are an astounding 50,613,244,155,051,856 ways to score exactly 100 points in (ten-pin) bowling. I won’t deny that this seems high to me. But partitioning a number — that is, taking a (positive) whole number and writing down the different ways one can add up (positive) whole numbers to get that sum — often turns up a lot of possibilities. That there should be many ways to get a score of 100 by adding between ten and twenty numbers that could be between zero and ten each, plus the possibility of adding pairs of the numbers (for spares) or trios of numbers (for strikes) makes this less astonishing.

Wikipedia led me to this page, from Balmoral Software, about all the different ways there are to score different numbers. The most surprising thing it reveals to me is that 100 isn’t even the score with the greatest number of possible scores. 77 is. There are 172,542,309,343,731,946 ways to score exactly 77 points. I agree this ought to make me feel better about my game. It doesn’t. It turns out there are, altogether, something like 5,726,805,883,325,784,576 possible different outcomes for a bowling game. And how we can tell that, given there’s no practical way to go and list all of them, is described at the end of the page.

The technique is called “divide and conquer”. There’s no way to list all the outcomes of ten frames of bowling, but there’s certainly a way to list all the outcomes of one. Or two. Or three. So, work out how many possible scores there would be in few enough frames you can handle that. Then combine these shortened games into one that’s the full ten frames. There’s some trouble in matching up the ends of the short games. A spare or a strike in the last frame of a shortened game means one has to account for the first or first two frames of the next one. But this is still an easier problem than the one we started with.

Bill Amend’s FoxTrot Classics for the 18th (rerun from the 25th of May, 2006) is your standard percentages and infinities joke. Really would have expected Paige’s mother to be wise to this game by now, but this sort of thing happens.

The End 2016 Mathematics A To Z: Voronoi Diagram


This is one I never heard of before grad school. And not my first year in grad school either; I was pretty well past the point I should’ve been out of grad school before I remember hearing of it, somehow. I can’t explain that.

Voronoi Diagram.

Take a sheet of paper. Draw two dots on it. Anywhere you like. It’s your paper. But here’s the obvious thing: you can divide the paper into the parts of it that are nearer to the first, or that are nearer to the second. Yes, yes, I see you saying there’s also a line that’s exactly the same distance between the two and shouldn’t that be a third part? Fine, go ahead. We’ll be drawing that in anyway. But here we’ve got a piece of paper and two dots and this line dividing it into two chunks.

Now drop in a third point. Now every point on your paper might be closer to the first, or closer to the second, or closer to the third. Or, yeah, it might be on an edge equidistant between two of those points. Maybe even equidistant to all three points. It’s not guaranteed there is such a “triple point”, but if you weren’t picking points to cause trouble there probably is. You get the page divided up into three regions that you say are coming together in a triangle before realizing that no, it’s a Y intersection. Or else the regions are three strips and they don’t come together at all.

What if you have four points … You should get four regions. They might all come together in one grand intersection. Or they might come together at weird angles, two and three regions touching each other. You might get a weird one where there’s a triangle in the center and three regions that go off to the edge of the paper. Or all sorts of fun little abstract flag icons, maybe. It’s hard to say. If we had, say, 26 points all sorts of weird things could happen.

These weird things are Voronoi Diagrams. They’re a partition of some surface. Usually it’s a plane or some well-behaved subset of the plane like a sheet of paper. The partitioning is into polygons. Exactly one of the points you start with is inside each of the polygons. And everything else inside that polygon is nearer to its one contained starting point than it is any other point. All you need for the diagram are your original points and the edges dividing spots between them. But the thing begs to be colored. Give in to it and you have your own, abstract, stained-glass window pattern. So I’m glad to give you some useful mathematics to play with.

Voronoi diagrams turn up naturally whenever you want to divide up space by the shortest route to get something. Sometimes this is literally so. For example, a radio picking up two FM signals will switch to the stronger of the two. That’s what the superheterodyne does. If the two signals are transmitted with equal strength, then the receiver will pick up on whichever the nearer signal is. And unless the other mathematicians who’ve talked about this were just as misinformed, cell phones pick which signal tower to communicate with by which one has the stronger signal. If you could look at what tower your cell phone communicates with as you move around, you would produce a Voronoi diagram of cell phone towers in your area.

Mathematicians hoping to get credit for a good thing may also bring up Dr John Snow’s famous halting of an 1854 cholera epidemic in London. He did this by tracking cholera outbreaks and measuring their proximity to public water pumps. He shut down the water pump at the center of the severest outbreak and the epidemic soon stopped. One could claim this as a triumph for Voronoi diagrams, although Snow can not have had this tool in mind. Georgy Voronoy (yes, the spelling isn’t consistent. Fashions in transliterating Eastern European names — Voronoy was Ukrainian and worked in Warsaw when Poland was part of the Russian Empire — have changed over the years) wasn’t even born until 1868. And it doesn’t require great mathematical insight to look for the things an infected population has in common. But mathematicians need some tales of heroism too. And it isn’t as though we’ve run out of epidemics with sources that need tracking down.

Voronoi diagrams turned out to be useful in my own meager research. I needed to model the flow of a fluid over a whole planet, but could only do so with a modest number of points to represent the whole thing. Scattering points over the planet was easy enough. To represent the fluid over the whole planet as a collection of single values at a couple hundred points required this Voronoi-diagram type division. … Well, it used them anyway. I suppose there might have been other ways. But I’d just learned about them and was happy to find a reason to use them. Anyway, this is the sort of technique often used to turn information about a single point into approximate information about a region.

(And I discover some amusing connections here. Voronoy’s thesis advisor was Andrey Markov, who’s the person being named by “Markov Chains”. You know those as those predictive-word things that are kind of amusing for a while. Markov Chains were part of the tool I used to scatter points over the whole planet. Also, Voronoy’s thesis was On A Generalization Of A Continuous Fraction, so, hi, Gaurish! … And one of Voronoy’s doctoral students was Wacław Sierpiński, famous for fractals and normal numbers.)

Voronoi diagrams have a lot of beauty to them. Some of it is subtle. Take a point inside its polygon and look to a neighboring polygon. Where is the representative point inside that neighbor polygon? … There’s only one place it can be. It’s got to be exactly as far as the original point is from the edge between them, and it’s got to be on the direction perpendicular to the edge between them. It’s where you’d see the reflection of the original point if the border between them were a mirror. And that has to apply to all the polygons and their neighbors.

From there it’s a short step to wondering: imagine you knew the edges. The mirrors. But you don’t know the original points. Could you figure out where the representative points must be to fit that diagram? … Or at least some points where they may be? This is the inverse problem, and it’s how I first encountered them. This inverse problem allows nice stuff like algorithm compression. Remember my description of the result of a Voronoi diagram being a stained glass window image? There’s no reason a stained glass image can’t be quite good, if we have enough points and enough gradations of color. And storing a bunch of points and the color for the region is probably less demanding than storing the color information for every point in the original image.

If we want images. Many kinds of data turn out to work pretty much like pictures, set up right.

Reading the Comics, July 2, 2016: Ripley’s Edition


As I said Sunday, there were more mathematics-mentioning comic strips than I expected last week. So do please read this little one and consider it an extra. The best stuff to talk about is from Ripley’s Believe It Or Not, which may or may not count as a comic strip. Depends how you view these things.

Randy Glasbergen’s Glasbergen Cartoons for the 29th just uses arithmetic as the sort of problem it’s easiest to hide in bed from. We’ve all been there. And the problem doesn’t really enter into the joke at all. It’s just easy to draw.

John Graziano’s Ripley’s Believe It Or Not on the 29th shows off a bit of real trivia: that 599 is the smallest number whose digits add up to 23. And yet it doesn’t say what the largest number is. That’s actually fair enough. There isn’t one. If you had a largest number whose digits add up to 23, you could get a bigger one by multiplying it by ten: 5990, for example. Or otherwise add a zero somewhere in the digits: 5099; or 50,909; or 50,909,000. If we ignore zeroes, though, there are finitely many different ways to write a number with digits that add up to 23. This is almost an example of a partition problem. Partitions are about how to break up a set of things into groups of one or more. But in a partition proper we don’t really care about the order: 5-9-9 is as good as 9-9-5. But we can see some minor differences between 599 and 995 as numbers. I imagine there must be a name for the sort of partition problem in which order matters, but I don’t know what it is. I’ll take nominations if someone’s heard of one.

Graziano’s Ripley’s sneaks back in here the next day, too, with a trivia almost as baffling as the proper credit for the strip. I don’t know what Graziano is getting at with the claim that Ancient Greeks didn’t consider “one” to be a number. None of the commenters have an idea either and my exhaustive minutes of researching haven’t worked it out.

But I wouldn’t blame the Ancient Greeks for finding something strange about 1. We find something strange about it too. Most notably, of all the counting numbers 1 falls outside the classifications of “prime” and “composite”. It fits into its own special category, “unity”. It divides into every whole number evenly; only it and zero do that, if you don’t consider zero to be a whole number. It’s the multiplicative identity, and it’s the numerator in the set of unit fractions — one-half and one-third and one-tenth and all that — the first fractions that people understand. There’s good reasons to find something exceptional about 1.

dro-mo for the 30th somehow missed both Pi Day and Tau Day. I imagine it’s a rerun that the artist wasn’t watching too closely.

Aaron McGruder’s The Boondocks rerun for the 2nd concludes that storyline I mentioned on Sunday about Riley not seeing the point of learning subtraction. It’s always the motivation problem.

Counting Things


I’ve been working on my little thread of posts about sports mathematics. But I’ve also had a rather busy week and I just didn’t have time to finish the next bit of pondering I had regarding baseball scores. Among other things I had the local pinball league’s post-season Split-Flipper Tournament to play in last night. I played lousy, too.

So I hope I may bring your attention to some interesting posts from Baking And Math. Yenergy started, last week, with a post about the Gauss Circle Problem. Carl Friedrich Gauss you may know as the mathematical genius who proved the Fundamental Theorem of Whatever Subfield Of Mathematics You’re Talking About. Circles are those same old things. The problem is quite old, and easy to understand, and not answered yet. Start with a grid of regularly spaced dots. Draw a circle centered on one of the dots. How many dots are inside the circle?

Obviously you can count. What we would like is a formula, though: if this is the radius then that function of the radius is the number of points. We don’t have that, remarkably. Yenergy describes some of that, and some ways to estimate the number of points. This is for the circle and for some other shapes.

Yesterday, Yenergy continued the discussion and got into partitions. Partitions sound boring; they’re about identifying ways to split something up into components. Yet they turn up everywhere. I’m most used to them in statistical mechanics, the study of physics problems where there’s too many things moving to keep track of them all. But it isn’t surprising they turn up in this sort of point-counting problem.

As a bonus Yenergy links to an article examining a famous story about Gauss. This is specifically the famous story about him, as a child, doing a quite long arithmetic problem at a glance. It’s a story that’s passed into legend and I had not known how much of it was legend.

Reading the Comics, February 6, 2016: Lottery Edition


As mentioned, the lottery was a big thing a couple of weeks ago. So there were a couple of lottery-themed comics recently. Let me group them together. Comic strips tend to be anti-lottery. It’s as though people trying to make a living drawing comics for newspapers are skeptical of wild long-shot dreams.

T Lewis and Michael Fry’s Over The Hedge started a lottery storyline the 1st of February. Verne, the turtle, repeats the tired joke that the lottery is a tax on people bad at mathematics. Enormous jackpots, like the $1,500,000,000 payout of a couple weeks back, break one leg of the anti-lottery argument. If the expected payout is large enough then the expectation value of playing can become positive. The expectation value is one of those statistics terms that almost tells you what it is just by the name. It’s what you would expect as the average result if you could repeat some experiment arbitrarily many times. If the payout is 1.5 billion, and the chance of winning one in 250 million, then the expected value of the payout is six dollars. If a ticket costs less than six dollars, then — if you could play over and over, hundreds of millions of times — you’d expect to come out ahead each time you play.

If you could. Of course, you can’t play the lottery hundreds of millions of times. You can play a couple of times at most. (Even if you join a pool at work and buy, oh, a thousand tickets. That’s still barely better than playing twice.) And the payout may be less than the full jackpot; multiple winners are common things in the most enormous jackpots. Still, if you’re pondering whether it’s sensible to spend two dollars on a billion-dollar lottery jackpot? You’re being fussy. You’ll spend at least that much on something more foolish and transitory — the lottery ticket can at least be used as a bookmark — I’ll bet.

Jef Mallett’s Frazz for the 4th of February picks up the anti-lottery crusade. Caulfield does pin down that lotteries work because people figure they have a better chance of winning than they truly do. Nobody buys a ticket because they figure it’s worth losing a dollar or two. It’s because they figure the chance is worth a little money.

Ken Cursoe’s Tiny Sepuku for the 4th of February consults the Chinese Zodiac Monkey for help on finding lucky numbers. There’s not really any finding them. Lotteries work hard to keep the winning numbers as unpredictable as possible. I have heard the lore that numbers up to 31 are picked by more people — they’re numbers that can be birthdays — so that multiple winners on the same drawing are more likely. I don’t know that this is true, though. I suspect that I could feel comfortable even with a four-way split of one and a half billions of dollars. Five-way would be out of the question, of course. Better to tear up the ticket than take that undignified split.

Ahead of the exam, Ruthie asks, 'Instead of two number 2 pencils, can we bring one number 3 pencil and one number 1? Or one number 4 pencil or four number 1 pencils? And will there be any math on this test? I'm not good at math.'
In Rick Detorie’s One Big Happy for the 3rd of February, 2016. The link will probably expire in early March.

In Rick Detorie’s One Big Happy for the 3rd of February features Ruthie tossing off a confusing pile of numbers on the way to declaring herself bad at mathematics. It’s always the way.

Breaking up a whole number like 4 into different sums of whole numbers is a mathematics problem also. Splitting up 4 into, say, ‘2 plus 1 plus 1’, is a ‘partition’ of the number. I’m not sure of important results that follow this sort of integer partition directly. But splitting up sets of things different ways runs through a lot of mathematics. Integer partitions are the ones you can do in elementary school.

Percy Crosby’s Skippy for the 3rd of February — I believe it originally ran December 1928 — is a Roman numerals joke. The mathematical content may be low, but what the heck. It’s kind of timely. The Super Bowl, set for today, has been the most prominent use of Roman numerals we have anymore since the Star Trek movies stopped using them a quarter-century ago.

Bill Amend’s FoxTrot for the 7th of February seems to be in agreement. And yes, I’m disappointed the Super Bowl is giving up on Roman numerals, much the way I’m disappointed they’re using a standardized and quite boring logo for each year. Part of the glory of past Super Bowls is seeing old graphic design eras preserved like fossils.

Brian Gordon’s Fowl Language for the 5th of February shows a duck trying to explain incredibly huge numbers to his kid. It’s hard. You need to appreciate mathematics some to start appreciating real vastness. I’m not sure anyone can really have a feel for a number like 300 sextillion, the character’s estimate for the number of stars there are. You can make rationalizations for what numbers that big are like, but I suspect the mind shies back from staring directly at it.

Infinity, and the many different sizes of infinity, might be easier to work with. One doesn’t need to imagine infinitely many things to work out the properties of infinitely large sets. You could do as well with a neatly drawn rectangle and some other, bigger, rectangles. But if you want to talk about the number 300,000,000,000,000,000,000,000 then you do want to think of something true about that number which isn’t also true about eight or about nine hundred million. But geology teaches us to ponder Deep Time. Astronomy trains us to imagine incredibly vast distances. Why not spend some time pondering huge numbers?

And with all that said, I’d like to make one more call for any requests for my winter 2016 Mathematics A To Z glossary. There are quite a few attractive letters left unclaimed; a word or short term could be yours!

Reading the Comics, February 4, 2015: Neutral Edition


Several of the comic strips that’ve been sent my way the past couple days touch on cultural neutrality in mathematics problems. People like to think of mathematics as a universal language, which makes me think of, for example, the quipu — twisted woolen cords with smaller cords tied to the main one — that Incans used to represent numbers. Even knowing the number one is supposed to represent doesn’t help me work out how to read the thing, and that’s not even doing calculations, just representing a number.

Darby Conley’s Get Fuzzy (February 1) uses several mathematics questions as part of a “general knowledge” quiz. Mathematics questions, particularly reasoning questions, are held up a good bit as examples of general knowledge since we’ve always cherished reasoning as a particularly precious sort of thinking, and because it’s easy to convince oneself that arithmetic and logic problems are culturally neutral. They’re not, but I would agree that “one times four” or the candy-counting problem are more culture-neutral than naming places with “-ham” or (to invent something not in the strip) identifying prime ministers of Canada would be. Really intriguing to me, though, is that Conley has Bucky Katt mention the Times as a newspaper without comics and the Daily News as one with: I had believed the strip to be set in or around Boston in the past, while this is pretty soundly a New York reference. Perhaps Conley’s let his daily comics lapse into reruns because he’s been moving, very slowly, across Connecticut?

Mac and Bill King’s Magic in a Minute (February 1) isn’t really a mathematics puzzle, but it does employ mathematical symbols in a way that I remember fondly from a bunch of “stories with holes” — superficially nonsensical problems which have logical resolutions if you can avoid being hobbled by implicit assumptions — so it’s really well-fitted for kids of the right age.

Continue reading “Reading the Comics, February 4, 2015: Neutral Edition”