# My Favorite Math Comic Strip

This began life as an answer to “What is your favorite math comic strip?” on Quora. Within hours it became my most viewed, most commented and most liked answer on that site. I share it here for your enjoyment.

What is your favorite math comic strip?

There’s a lot to choose from. You really can’t go wrong with Calvin and Hobbes, Foxtrot, XKCD and Math With Bad Drawings. I’m hoping to find some new favorites when I read through all the other answers here.

But, three comics immediately come to mind. Here’s two runners-up and my favorite.

The second runner up:

As a mathematician, I can’t help but appreciate this one.

The first runner-up:

Based on the artwork, his one has to be pretty early in the strip’s history. The look on Calvin’s face as he exclaims, “Imaginary Numbers?!” makes me chuckle to this day. Hobbes’ definition, “Eleventeen, thirty-twelve and all those” is priceless. What is your favorite math comic strip Lovely.

And the winner is…

I knew immediately this one was the answer because I remember reading it in the Palm Beach Post and laughing really hard. It’s interesting to me that the real punchline is in the third panel. In retrospect, the first panel may be even funnier once you’ve read the rest of the comic.

Somewhere there’s more. If I can find it, there’s a file of the comics I used to have taped to my office door at the University of Miami. It’s not directly Mathematics-related, but it contains a nice comic about the “Academic Beer Head Theory.” The basic idea is that you shouldn’t cram for exams because if you pour the knowledge into your brain too quickly it gets all foamy and spills out your ears. I’ve been quoting that to students for years. When it surfaces, if it ever surfaces, I’ll add a couple more here.

# When Jonah Learned Peter is Spider-Man

This was initially published in a slightly altered form on Quora.

I hope everyone who’s planning to see Spider-Man: Far From Home has seen it by now. (If not, Spoiler Alert! Stop reading!) If you have, you know that the mid-credits scene involves J. Jonah Jameson broadcasting Peter’s identity to the world. We won’t know how that will play out until Spider-Man: The Cows Come Home or… Spider-Man: Phone Home or… Spider-Man: Something Else Home (I don’t know the title. I’m just guessing.) but maybe the comics can give us a hint. Maybe not. The MCU is a very different place from Marvel’s mainstream continuity; in particular, it seems far more hostile to secret identities. Let’s check it out anyway, just for fun.

As far as I know, Jonah learned Peter’s secret twice. The first time was in Civil War (2006) #2 and simultaneously in Amazing Spider-Man (1963) #533. To set the stage, Peter had been working with Tony at Stark Enterprises. The universal brouhaha over the Super-Hero Registration Act began with Tony leading the Pro registration forces. He convinces Peter to reveal his identity in a televised press conference as part of the Act’s media strategy.

Jonah watched this on television, it goes more or less the way you’d expect.

So, Jonah’s more hurt than angry, but he’s still angry. He sues Peter for fraud asking for the money he’d paid for photographs of Spider-Man over the years. When Robbie Robertson stands up to Jonah, arguing that the vendetta has gone too far, Jonah fires him.

Of course, Jonah forgets about Peter’s secret identity after it’s magically made secret again in the One More Day storyline.

I don’t believe that Peter would actually reveal his identity on television; he was famously careful about his secret identity and always refused to put his loved ones at risk. Still, the writers laid some groundwork for the decision and the reveal was one of the few compelling things about Civil War.

Back to the topic at hand. Time passes. Jonah has a heart attack, and the Bugle is sold out from under him. He becomes Mayor of New York City; his wife, Marla dies; he is forced out of office and becomes a commentator on The Fact Channel. Meanwhile, Aunt May meets, falls in love with and marries John Jonah Jameson Sr. This makes Jonah and Peter family in a very real sense. Functionally, they’re step-brothers. Not long before Jonah learns Peter’s secret a second time, his father dies, Marla is resurrected and dies again and Jonah is fired from the Fact Channel, after which he begins writing a blog. Also, at some point, his adopted-daughter died. Whew. Comics… am I right?

Spoiler Alert here, by the way. If you haven’t read Peter Parker, the Spectacular Spider-Man (2017) by Chip Zdarsky, go get the trade paperbacks or track down the back issues or something. It’s great. Especially issue #310 which just won the Eisner Award for Best Single Issue. I’ll wait.

So then, how does the second time that JJJ discovers Peter’s secret come about?

It starts in issue #5 of the aforementioned series; Jonah is on the verge of breaking a big story and needs to talk to Spider-Man. It’s about someone Peter is trying to help and Jonah agrees to share what he knows.

The interview takes up most of issue #6, (which is the other highlight of the series btw). It’s interesting and turns into quite a heart to heart. There are some expected dimensions and some that are less expected, like Peter admitting to Jonah why he became a crimefighter.

And of course, it gets heated.

Eventually, Peter realizes just how miserable Jonah is, “M-my father is dead! My daughter is dead! The Bugle, the only thing worth a damn in the world, has rejected me! My wife is dead! My wife is dead… Interview’s over! I’ve got — got nothing in my life now! You win —”

“You’re not!” I find this far more believable than the incident in Civil War. This action is born out of compassion and maybe a little responsibility. That’s exactly who Peter is.

And it pays off in an unexpected way. Without going into much detail (seriously get it, read it! This is not a paid endorsement!), Peter’s immediately put in danger because of the interview. In the next issue, (number 279 #MarvelMath) he temporarily gets clear and then this happens.

Jonah helps. He knows Peter and very clearly trusts him and on some level, he probably wants to make amends for what he’s done to Peter over the years.

This starts a new dynamic and Jonah becomes a sort of a side-kick, around frequently and determined to help Peter be a better hero. It’s delightful. This is from # 309.

It’s still noticeably Jonah. He’s still headstrong. He’s still sure he’s smarter than Peter. He still fails to think things through carefully. But after almost 60 years, it’s really nice to see a new dynamic between these two characters. I’m a bit disappointed that we haven’t seen more of this in Amazing Spider-Man and Friendly Neighborhood Spider-Man, but I hope this is the new status quo for years to come.

References:

# Dr. K’s Guide to Comics

This was originally written to answer the following question on Quora.

#### What are some things someone just getting into comic books should know?

If you’re just getting into comics, welcome to the club! You have years of entertainment to look forward to!

If you’re thinking about getting into comics, the first thing you should probably know is whether you actually enjoy comics; reading them, looking at the artwork, learning the contours of each new universe and so forth. If so, you’re a reader. You might also ask yourself if you enjoy experiencing comics: the feel of the paper, the smell of the ink, the satisfaction of completing a run and the look of a stack of books, all nicely bagged. In that case, you may just be a collector. Some people want to try and make some money on their collection; that’s fine. If you enjoy the other aspects of the hobby, you could become an investor. If you don’t enjoy the comics themselves, however, I would recommend against that. A portfolio of mutual funds is probably far easier to manage and more profitable than speculating on comics.

This is a good time to start.

You’ve decided to join the hobby at a good time; it’s easier than ever to find things to read whether you’re interested in new comics or back issues. Absolutely, your first step should be to find a Local Comic Shop (LCS) if there’s one nearby and get to know the folks there. Most of them love the hobby and enjoy talking about it. They can tell you which current titles have the best buzz and once they get to know you, they can point you towards things you might like, new and old. Many LCSs will also let you start a pull list and will put books aside for you. This helps you to avoid missing an issue and it helps them to know what to order. In any event, your LCS can be an essential resource.

If you’re interested in older stories, many comic stores have a selection of back issues that you can purchase.

I took this picture in Graham Cracker Comics last time I was in Chicago and the set up is pretty typical. High demand and interesting books are displayed on the wall while the more common back issues are alphabetized and in boxes. There are also significant on-line opportunities. I’ve bought many comics on e-Bay and most of my experiences have been positive ones. A lot of bigger stores also have their own websites where they sell comics. New Kadia and Mile High Comics are good examples.

If you’re interested in older stories but you don’t want to collect back issues, your LCS probably also stocks “collected editions” which come in a variety of shapes and sizes. Here’s a random selection from my shelves.

The Marvel Masterworks in the center is a prestige format hardcover collecting Fantastic Four (1961) #1-10. The other two are “trade paperbacks,” which is the most common format of collected edition. Some of those, like the book on the left, try to reprint a lot of books inexpensively. That one is printed in black and white on pulp paper. Others collect a smaller number of issues of an ongoing series or a complete mini-series. There are also “graphic novels.” I personally use that term to refer to any complete comic story that’s more than a couple of issues in length. Occasionally complete stories will appear in trade paperback for the first time; those are called “original graphic novels.”

There are also digital options. Individual issues can be purchased at DCComics.com and ComiXology,com. You can buy individual comics at Marvel.com and, when you buy one of their physical comics, you get a code to download a digital copy of the same issue. ComiXology Unlimited and Marvel Unlimited are Netflix style unlimited services where a monthly or annual fee gives you full access to a large digital library of comics.

One cost-effective approach to reading a large number of titles is to buy the titles you really want to collect from your LCS and then wait for the remaining titles to hit an unlimited service.

There are two basic strategies when it comes to buying back issues. You can focus on buying issues to collect or buying issues mainly to read. Either way, you don’t want to overpay or pay a premium for the wrong book.

Condition Can Really Matter

Possibly the best illustration that condition matters is a story from about 6 years ago. Deanna and David Gonzales purchased a home in Elbow Lake, Minnesota. In the process of renovating, they ripped open the walls and amidst the old newspapers that were used for insulation, they found a copy of Action Comics #1.

Action Comics #1, for those of you who don’t know, is the first appearance of Superman and probably the quintessential collectors’ item. The advent of superheros, beginning with this particular issue remade the entire medium and there are only about 100 copies known to exist. Needless to say, every copy is highly sought after. The nicest existing copy sold on e-Bay for $3.2 million in 2014. The Gonzales’ copy could have sold for$250,000 but they tore the back cover when they were removing it from the wall. That tear cost them about $75,000. The moral? You should have at least a rough idea of how to grade comics, that is, to determine their condition. If you’re looking for something particular, you should have an idea of its value. The Overstreet Comic Book Price Guide is a good resource for both of these things. There are also online resources; I tend to use ComicsPriceGuide.com and they have their own explanation of the grading system. A quick internet search will reveal more options. If you’re looking for fair market value another place you can check is the sold listings on e-Bay. Lots of recent comics are actually available for less than cover price in Near Mint, but it seems to me that the price guides are hesitant to list books for less than that. Using the sold listing also gives you immediate information; prices can fluctuate after a guide is printed. Whether you’re buying a comic to read or to collect, you should decide the minimum condition you’re comfortable with and get a sense of what a fair price for that book would be in that condition. Why worry about that? The good comic stores will generally know how to grade books and price them accordingly. But there are less professional vendors or even vendors, like pawn shops and used book stores who think any comic that’s old is valuable who may price books without investigating the value. There are others who will price every book at near mint regardless of the condition either because they don’t know how to grade or because they don’t understand how it effects the price. If you know what to expect from the price, you’ll be able to avoid over paying. Sometimes, You Can Compromise on Condition The point where condition isn’t very important is when you’re buying back issues to read rather than collect. You, of course, need to decide for yourself the condition with which you’re most comfortable. But comics that are in very good (VG) condition or better should be complete and readable. VG books should sell for about 10% of the mint price. Below that there are still plenty of books that are nicely readable but you may run into issues like fragile paper or cutouts that can diminish the reading experience. When I was collecting in the 1970s and 80s I collected a lot of reprint titles like Marvel Tales, Marvel’s Greatest Comics and Marvel Super Action. These reprinted Amazing Spider-Man, Fantastic Four and Avengers respectively and gave me an inexpensive way to read a lot of classic stories from the Silver Age. But there were drawbacks. A Comic reprint can be significantly different from the original. The print quality can be noticeably worse as the company generally no longer had access to the original artwork. Further, the comic could be recolored and sections of the story might be deleted to fit the comic into a smaller page count. Whereas an original comic might be highly sought out and valuable, mostly these reprints are barely even considered collectable. Unless you’re collecting them for sentimental reasons, there are much better options for reading old stories such as trade paperbacks and Marvel’s and ComiXology’s unlimited services. You Can Be Really Intense About Condition If You Want. If you have a special comic that you want to carefully preserve you can send it out to a third party grading service. The most famous of these is the Certified Guarantee Company or CGC. For a fee, these companies will grade your comic and then seal it in a hard plastic shell that documents the condition and presumably protects it from further harm. This process is called “slabbing” and it’s controversial within the collecting community. The upside is that, if you want to sell your comic, its condition is well established so buyers can have confidence in its accuracy. Because of this, slabbed books tend to sell for more than their free range counterparts. The downside is that the slab prevents you from even touching the comic itself. In a medium that’s supposed to be read, experienced and enjoyed, some collectors find slabbing offensive. Caveat Emptor Another thing to keep in mind as you shop for back issues is that you want to be sure you’re buying the correct book. Consider the following listing that I encountered on a well known auction site a few weeks ago. If you’re not knowledgeable about comics, and you don’t look too closely, it might seem to you that this is a copy of Silver Surfer #4 from 1969. After all, that’s what the listing claims to be. The problem is, it isn’t. Here’s a picture of the genuine article. The comic in the listing is actually Fantasy Masterpieces #4 from 1979. It’s a reprint of the Silver Surfer comic. And the big issue here is the price. ComicsPriceGuide.com tells us that Silver Surfer (1968) #4 in near mint (NM) condition is worth$1000, while Fantasy Masterpieces (1979) #4 lists at $4. That’s a huge difference and I hope no one actually paid$290 for a $4 book. This issue isn’t just restricted to auction sites. At a store in St. Louis I saw the comic on the right, priced as though it were the comic on the left. This is an easier mistake to make than the Silver Surfer/Fantasy Masterpieces confusion, and this example, it turns out, is trickier than most. But there’s still a big difference in terms of price. ComicsPriceGuide.com tells us that the original book goes for$160 in NM while the reprint is worth \$4 in the same condition. This was in a large store that had many boxes of back issues but also sold used books, used DVDs and used video games. The staff may have been somewhat knowledgeable about comics, but comics certainly weren’t the focus of their business.

So, how do you avoid this? Well the first thing you need to do is identify the actual title of the series you’re dealing with. That’s not necessarily the title on the cover. To get the official title of the series, you need to look at the indicium , which is the comic’s publishing information. It’s usually found at the bottom of the first page or on the inside front cover of the comic. Here’s the indicium for the older comic.

The title of this series is clearly indicated: “Nick Fury, Agent of SHIELD.” Not “S.H.I.E.L.D.” like you might think from looking at the cover and with a comma that we don’t see on the cover. On the other hand, this is the indicium from the 1983 series.

So the title of this series is “Nick Fury Agent of SHIELD.” No comma. Ironically, the series with the comma on the cover has no comma in the official title and the series without the comma on the cover does has a comma in its title. This series is also a “volume 1” since, because of that comma, the titles are not exactly the same. It’s important to know the proper title because that’s how the book will be listed in the Overstreet Price guide.

Here’s the listing for these comics in the 2009 edition of Overstreet. It’s easy to see which listing goes to which book since we’ve figured out the real titles. If there’s ever a question, you can go by the dates of the issues, which are also in the indicia. Modern conventions delineate the different series by the start year so in comicbookdb.com, you see these titles as Nick Fury, Agent of SHIELD (1968) and Nick Fury, Agent of SHIELD (1983). You can see the values for each issue on the right for the conditions good, very good, fine, very fine, very fine/near mint and near mint.

Finding the Right Comic

When I’m collecting, I like to find a complete run of a title before reading so that I can read a bunch of issues back to back. If you’re merely reading comics in trade paperback form or online this isn’t an issue, but if you’re collecting the individual issues, it can get tricky. Nothing here is a major problem as long as you know what to expect, but if you’re not careful, you can end up with a comic you don’t need.

This was not an issue when I started hunting for back issues around 1980. At that point, most titles had a single volume with a single consecutive numbering. There was one run of Amazing Spider-Man of about 200 issues. Flash was closer issue 300, but that included the entire run of Flash Comics, dating back to January 1940. This is no longer the case. Comics now restart their numbering frequently and it can be a nuisance to keep track. Sometimes the companies renumber a title because the creative team is changing, sometimes it’s because of some company wide event. Sometime there’s no apparent rhyme or reason except that maybe the numbers have gotten a bit too large. Depending on how you count, there have been 11 or 12 comics that are technically “Captain Marvel #1″ published by Marvel since 1968. There have been at least 8 volumes of The Avengers and that doesn’t count New Avengers or Mighty Avengers or All-New All-Different Avengers or West Coast Avengers or Avengers Forever or Avengers AI or Solo Avengers… you get the idea.

So, do you need a copy of Avengers #2? Which one? There are at least 8, not counting reprints that might look like Avengers #2 (see “Caveat Emptor” above). A simple answer might be to make sure you know the cover, but that too is not as simple as it once was. When I started collecting, every comic had but one cover, but that also is no longer the case.

This is because of another trend in comics. There is now a proliferation of variants. Variants are copies of the same issue of a series that are absolutely identical except for the cover. To me, this is getting excessive. There are variants to encourage dealers to order more; there are variants with some special gimmick on the cover; there are variants to celebrate a special event and variants to popularize a particular character. Many, like the “Special Inauguration Day Edition” of Amazing Spider-Man #583 are rare enough to be worth considerably more than the standard version but most probably have no additional value at all.

When the latest volume of Fantastic Four hit the stands last August, there were at least 42 different variants of the first issue, some of which are shown above (you can see three more here). This title has averaged roughly nine variants per issue since its inception less than a year ago.

So, if you can’t just go by number and you can’t just go by cover, what do you do? One answer is to make sure you know the publication date. If you’re looking at back issues in a comic store, you can always check the indicia. As long as your book has the correct publication date, it should be the correct book.

You can’t always see the indicia, such as when you’re shopping on line. Going by the cover can still be a good plan in a lot of cases. If there’s only one cover to the book you need, make sure you’re familiar with it. That’s the case with the vast majority of older books. Even if a comic has multiple variants, most copies will have the main cover so it pays to know it. But maybe you’d prefer a different cover or perhaps the main cover is sold out and isn’t an option. Maybe you just want to own the issue and you don’t care which variant you end up with. Many comic websites have a cover browser function. Here’s the cover browser for Hawkeye (2017) from comicbookdb.com.

This would be easy to print out or access on a tablet if your LCS has WiFi. Armed with this information, you have all the potential covers at your finger tips and it’s easy to tell if the comic you’re looking at is the correct one or not.

The final thing we’ll cover here is storage. Given the importance of condition, you’ll want to protect your books.

You might think that lying your comics flat and storing them in a stack would help them stay flat. In fact, since the spine is thicker than the rest of the book, the exact opposite is true; stacking comics can lead to “spine roll.” You’ll want to store your books vertically, as shown here. Luckily, there are supplies designed for just this kind of storage.

There are 2 sizes of comic boxes from which to choose called, creatively enough, long boxes and short boxes. A long box is about 28 inches wide and holds about 300-350 comics. A short box is half as wide and holds half as many. Long boxes cost about half again as much, so it’s more cost effective to use long boxes, but a full long box weighs about 50 pounds, making them more unwieldy and therefore more difficult to move and organize.

You should store your comics in acid-free bags. It can be bad for your comics to store them in a bag that’s too big or too small, but most comics published after 1974 fit nicely in a “current” comic bag. Comics have gotten narrower over time. Generally, comics made before the early 1950’s need “golden age” bags while comics published from the mid 50’s to 1974 require “silver age” bags. There are also different materials available.

Most comic bags are either polyethylene, which has more of a matte finish or polypropylene which is clearer and shinier. Opinions vary, but I prefer the polypropylene. I think bagged comics look better with the shiny bag and I find that polyethylene bags stick together after being stored for a long period of time. Many collectors also place a backing board in each bag with a comic. These are thin pieces of cardboard which prevent bending and protect the corners of the book from blunting. Boarding every comic in your collection is probably excessive, but it’s a good investment for your favorite or most valuable books.

That’s probably enough information to throw at you all at once; I hope it helped. If you think of additional questions, you can add them to the comments and I’ll try to answer. There will be more for beginning collector coming soon.

# How would presidential elections change if electoral votes were allocated by congressional district ?

First published on Quora.

This is the system currently used in Maine and Nebraska. In Maine and Nebraska the statewide winner gets the two electoral votes (EVs) that correspond to the senators and then the remaining votes are determined by the winner of each congressional district.

The Electoral College already has a “small state bias” that skews for the time being in favor of the Republicans, since the smaller states tend to be more Republican than the country as a whole. I haven’t checked the numbers, but California has the same population as something like the smallest 20 states combined. That’s two EVs for the statewide win in California compared to forty for the statewide wins in these other states. It’s this bias that is responsible for the two “electoral inversions” we had in 2000 and 2016. That is to say, the two elections where the winner of the Electoral College did not match the winner of the popular vote.

Choosing the remaining EVs by congressional district would further skew things in the Republican direction. This is due to the extreme partisan gerrymander that took place after the 2010 election. To put this into perspective, the Democrats won the “national congressional vote” (NCV) in 2018 by something around 7 percentage points. This will give them a majority of between 14 and 19 seats when the remaining races are determined. By contrast, the Republicans won the 2014 NCV by 5.4 percent in 2014 and that gave them a majority of 30 seats. Worse, in 2012 the Democrats won the NCV by 1.2% but the Republicans maintained a majority in the House of 16 seats.

So, at least until the the congressional districts are redrawn in the wake of the 2020 Census, the current small state bias that favors republicans would be exacerbated. I don’t know if it would be impossible for a Democrat to win the presidency under such a system, but it would certainly be more difficult and there would be many instances where this system would elect the Republican even if the American people preferred the Democrat.

Still, it’s easy to imagine a worse system. During the run-up to the 2012 election, I recall Nebraska debating a return to a winner-take-all system so that President Obama could not win an EV from Nebraska like he did in 2008. At at about the same time the republican-controlled Pennsylvania legislature debated switching to allocating EVs by congressional district to help Governor Romney. Imagine such a system implemented nationwide, with all the red states using winner-take-all and all the blue states allocating by congressional district or vice-versa. Such a system would virtually guarantee one-party control of the presidency.

References (all accessed 19 November 2018):

House Election Results: Democrats Take Control

RealClearPolitics – Election Other – 2014 Generic Congressional Vote

RealClearPolitics – Election Other – 2012 Generic Congressional Vote

114th United States Congress – Wikipedia

113th United States Congress – Ballotpedia

# What if Captain America became President?

First Published on Quora, 14 October 2019.

It’s worth pointing out here that this very idea has already been played around with in the comics. In Captain America 250 (October 1980) Steve is approached by the New Populist Party and asked to be their candidate for president. He gives it serious thought and spends most of the issue debating the pros and cons with his friends, Avengers and otherwise. The ending of the issue is bittersweet; Steve, of course, decides not to run for president and the enthusiasm that had been building within the NPP turns to a profound disappointment.

About six months later, Marvel itself answered this question in What If? #26. Of course we don’t really know whether WI#26 tells us what “really” would have happened, but it’s at least as valid as what anybody else would have said. And in some sense, that’s the ultimate answer to the question at hand. It really depends on who gets to write the story.

Cap #250 is a classic. What if #26 is pretty good. Both are worth checking out and are available on Marvel Unlimited.

To give my own opinion, there’s two things I think are worth addressing, how Steve would have governed and how the public would have been likely to respond.

Politically, I think Steve is likely to be a New Dealer. He was born in 1920 and came of age around 1940; FDR was popular and won a fairly lopsided electoral victory that year, although not nearly as lopsided as 1936. I think what we’ve seen from Steve over the years bares this out, from Englehart’s run in the 70’s to his reason for stepping out of the role in “Captain America No More” to his stance in Civil War and beyond (Hydra-Steve not withstanding). In foreign policy I think he would be an excellent diplomat, able to find common ground with other nations and move forward productively. He would certainly be more apt to use military force than Carter, but probably not nearly so apt as either of the Bushes. He would be relentlessly ethical.

But I think that the public’s response to Steve as president would be more indicative of his legacy as Commander-in-Chief than his political positions.

If he had been elected President in 1980 when “Cap for President” first hit the stands, conservative or liberal, I think President Rogers would have been a transformational  figure. Six years out from Watergate and a bit over a year after Carter’s malaise speech, the American Electorate was in flux. “Reagan Democrats” were becoming a thing while there was a candidate for the Republican nomination, John Anderson, who was arguably more liberal than the Democrats’ eventual nominee. If there’s one constant in all the portrayals of Steve Rogers, it’s in his ability to lead and inspire. Cap as president in the early 1980’s would have changed the political landscape for a generation or more.

On the other hand, had Steve been elected in the current political climate, I don’t think any of that would have mattered. Ed Brubaker (I’m pretty sure, I haven’t been able to locate the quote) made a relevant remark about the time Steve “died” in the aftermath of Civil War. He said that it was tricky to write Cap. One side of the political spectrum mainly wanted to see Cap beating up terrorists, while the other side mainly wants to see him giving speeches about rights and fairness. If anything, this aspect of has gotten more extreme over the past 11 years. Steve as president in the 21st Century probably presides over a lackluster presidency with one side of the aisle lauding his accomplishments and the other condemning his inadequacies, justly or unjustly.

And that, I think says much more about the state of politics in America today than it does about Captain America.

References:

Captain America #250 (October 1980)

What If? # 26

Captain America – Wikipedia

# Peter Parker’s Playlist

I just took a stab at answering the following question on Quora, and I thought I would share it here as well.

What would Spider-Man/Peter Parker’s music playlist be?

No definitive playlist, but some thoughts. The only mention I recall of Peter Parker’s music taste is from Marvel Team-Up Annual #4 from 1981.

Purple Man has Peter climb a lamppost to distract him and has him sing. His choice of music? Elvis Costello. Specifically “Oliver’s Army” from Armed Forces.

Extrapolating from this, here’s my guess. Peter probably listens to well regarded artists who are slightly out of the mainstream. Elvis Costello is established. Perhaps also artists like the Velvet Underground, Big Star, Nick Lowe or The Talking Heads.

Popular and mainstream artists might be less likely. Pulling from the same time period, maybe not Madonna, Michael Jackson or U2. These are mostly 1980’s examples since that’s when the comic came out, but you could extend the same thinking to other decades. It feels to me like it would hold true.

The one thing we can say for sure, is that Peter isn’t just listening to the top 40; he’s done some research and I suspect his tastes are fairly eclectic. It wouldn’t surprise me if he listened to some Big Band music if that was what he heard growing up with Aunt May and Uncle Ben.

It would be interesting

to see what other references to music we could find in the comics.

Update: Blaine Savini, a member of Old Guys who Love Old Comics on Facebook, pointed  out that we also learn in the Comics the Peter likes Ella Fitzgerald.  This is from Amazing Spider-Man #136, September 1974.

The Album in question is likely Ella in London (4 1/2 Stars, allmusic.com) which is the only 1974 issue listed in her discography on ellafitzgerald.com. It contains songs by George and Ira Gershwin, Duke Ellington and Cole Porter.

Notice that MJ clearly implies that Peter doesn’t listen to much in the way of popular music.  He mentions that he’s a junior in college in this issue, if that means he’s 21, he would have been Spider-Man for about 6 years at this point.  In 1964 we would have been 11.

But,well regarded, check.  Out of the matinstream, check.  Ella Fitzgerald and Elvis Costello: Eclectic, check.

References:

# Writing a Combinatorial Proof

This started life as a post on Quora, answering the question:

For any positive integer $n$, how do you write a combinatorial proof of the identity

$\displaystyle {2n \choose n} = \sum_{i = 0}^{n} {n \choose i} {n \choose n-i}$?

To write a combinatorial proof, the idea is to describe how each side of the equation is actually counting the same set of objects.

Now, if you have a set of $2n$ objects, $\displaystyle {2n \choose n}$ (that is $2n$ choose $n$) is the number of subsets of that set which contain exactly $n$ of them.

Now pretend that you’ve taken your $2n$ objects and put them into two boxes with $n$ objects in each. Think about what each of your terms on the other side of the equation, $\displaystyle {n \choose i}{n \choose n-i}$ represent. Then consider what you get when you add all of these together. You should be able to explain that this really does count the same thing as $\displaystyle {2n \choose n}$.

Now when I wrote the above, I wondered if this was a homework question someone had posted to Quora.  I didn’t fill in all the details since, while I’m happy to help someone with their homework, I don’t want to do it for them.

But in case anyone wants to see a worked out example, here’s the standard initial example of a combinatorial proof.  This is the identity that makes Pascal’s Triangle work as nicely as it does.

Theorem: $\displaystyle {n \choose k} = {n-1 \choose k} + {n-1 \choose k-1}$.

Proof: Let’s start by thinking about the expression on the left.  If we have a set with $n$ objects in it, $\displaystyle {n \choose k}$ is the number of ways we can select a subset of $k$ objects.  To say that another way, it’s the number of ways we can pick $k$ objects out of our set without caring what order they’re in.

Now suppose one of our objects has decided to wear a hat.  If we’re looking to select a subset with $k$ objects, we can decide to include or exclude the one wearing the hat.

Say we don’t want to include the guy with the hat.  In that case, all $k$ objects, have to be selected from the $n-1$ objects that aren’t wearing hats.  We can do this in $\displaystyle {n-1 \choose k}$ different ways.

Now suppose we decide to include the one with the hat.  Well then, to get $k$ objects altogether, we need to select $k-1$ more from the set.  There’s $\displaystyle {n-1 \choose k-1}$ ways to get the rest of the objects that you need.

Putting these two together we see that $\displaystyle {n-1 \choose k} + {n-1 \choose k-1}$ is also the number of ways to select a subset of $k$ objects from a set containing $n$ things.

Therefore, our identity must be true.

If you’d like to try one on your own, $\displaystyle 2^n = \sum_{i = 0}^{n} {n \choose i}$ is another nice example.

# How do you prove a conjecture is false?

First posted to Quora on Friday, 7 September 2018

That depends on the nature of the statement.

If you have a universal statement, which is to say a statement that all of the things in some category share some property, you merely have to provide a counter-example.

So if you wanted to disprove the statement, “all prime numbers are odd” you’d merely have to point out that 2 is even and the statement cannot be true.

Disproving an existential statement is usually more work. These statements say that there is at least one thing that has a particular property. To disprove an existential statement, you need a general argument that that property can never happen.

So to prove that the statement “There is a pair of even integers whose sum is odd” is false, you must prove that the sum of any two even integers must be even.

Those are the cases “all” and “some.” The cases “none” and “some are not” are similar.

To disprove a statement like “None of the items in set A have property B” you simply have to find one that does. If you want to show a statement like “Some of the items in set A do not have property B” is false you need a general argument that everything in A has property B.

In any case, disproving a statement is equivalent to proving its negation.