Tuesday, December 31, 2013

Article: The Sums of Consecutive Squares

Fun fact for the 365th day of the year:

102 + 112 + 122 = 365 and 132 + 142 = 365

So 365 is the sum of two sets of consecutive squares, but, more importantly, those two sets are themselves consecutive: {10, 11, 12} and {13, 14}.

Now that's interesting! Okay, so it's also a co-incidence, really. Sums of consecutive squares have to add up to something, and, occasionally, those "somethings" will be the same number. But can we write a general rule for this?

Of course, we can. If we can write it, we can (hopefully) solve it. (Again, of course I can solve it, or I wouldn't be asking the question, but I'm sure that there are many, many rules which we could pose which I, personally, couldn't solve. However, this is a simple one to work with.)

At the very simplest level, we have the sum of the squares of two consecutive positive integers. It's obvious that these two numbers can't equal the sum of two higher integers, so the sum has to be of only one number. Not much of a sum, I grant you, but we're starting with a trivial case.

We want to find consecutive positive integers, a, b, and c such that

a2 + b2 = c2

so we'll use the variable n to stand in for the lowest integer, (n+1) for the next consecutive integer, and (n+2) for the third consecutive integer. Now we can rewrite the equation as

n2 + (n+1)2 = (n+2)2

Squaring the binomials, we get:

n2 + n2 + 2n + 1 = n2 + 4n + 4

Combing like terms gives us:

2n2 + 2n + 1 = n2 + 4n + 4

Rewrite this as a quadratic equation by subtracting the right side of the equation from both sides:

n2 - 2n - 3 = 0

Which factors into: (n - 3)(n + 1) = 0. Therefore, n = 3 or n = -1, but because we want a positive whole number, we'll discard the -1 and accept the 3. That makes the three consecutive integers 3, 4, 5 and, therefore, 32 + 42 = 52 .

But, of course, you already knew that. So why do all that work? Because now we can move up to four or five consecutive numbers. We can use the same procedure to find solutions to

a2 + b2 + c2 = d2 or a2 + b2 + c2 = d2 + e2

To save space, and to be as annoying as those textbook writers of my youth, I'll leave a, b, c, d to you to try. I'll give you a hint: there aren't any positive integer solutions, but you can prove that for yourself instead of taking my word for it. Go ahead -- challenge authority!

For the sum of the squares of three consecutive positive integers equal to the sum of the squares of the next two integers, this is the equation we write:

n2 + (n+1)2 + (n+2)2 = (n+3)2 + (n+4)2

When the dust settles, what will be the value of n? If you didn't get it, you weren't paying attention. It's in the first paragraph of this article. The solutions are n = 10 and n = -2. Once again, we toss the negative and we're left with: {10, 11, 12} and {13, 14}.

This brings two closing questions: The obvious question is what seven consecutive positive integers a, b, c, d, e, f, g fall into this pattern? (I didn't say that the answer was obvious, but it's easy to figure out.)

And another question about another pattern: In the first case, we threw out the solution n = -1. In the second case, we discarded n = -2. I'll go ahead and tell you that in finding the answer to the next sequence of numbers, you'll have to get rid of the solution n = -3. My question: will the negative solution we discard always have the same absolute value as the number of terms on the right side of the equation?

I'll leave that as an exercise to the reader. You have a whole, exciting, brand New Year to work it out!

Saturday, December 28, 2013

Yesterday and Tomorrow

If from the title, Yesterday and Tomorrow, you were expecting something profound, philosophical or metaphorical about the Past and the Future, then I'm sorry to disappoint. No, quite literally, I meant yesterday, December 27, and tomorrow, December 29. And a little bit of today being book-ended in-between.

Yesterday would have been my mother's birthday. As regular readers of the blog know, we lost her this past August so this was our first birthday without her (and first Christmas and Thanksgiving and ...). Tomorrow will be the second anniversary of my father's passing. He hung on with his very being to have one more Christmas with the family. He even told me that I shouldn't have bought him a Nook because (he said), "I won't be around to use it." I told him not to talk like that, but a few days later he was back in the hospital. Part of me "knows" that he held on until after my mother's birthday.

The strange thing is that this weird alignment of these two biological anniversaries -- a birth and a death -- gets even stranger because it'll happen again next August. My mother passed away around my father's birthday. I guess she didn't want to be outdone. Maybe it was "payback".

Whatever it was, all I can do is think about the good times that I had with and enjoy good times with my family

Hmmm, I guess I did get a little metaphorical at the end.

Friday, December 27, 2013

Holiday Gathering Attendees

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(C)Copyright 2013, C. Burke.

They can also be organized by ''Wanted to Invite'', ''Had to Invite'', and ''Didn't Invite, But Showed Up Anyway''.

What's the one thing that we can deduce from this Venn diagram? If you're hosting the party then you'll be related to all the obnoxious people because a smart host/ess wouldn't invite obnoxious people they weren't related to. Those that are invited fall into the ''Had to Invite'' category on the other diagram, which has obnoxiously been left as an exercise to the reader.

Wednesday, December 25, 2013

Christmas 2013

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(C)Copyright 2013, C. Burke.

A last-minute comic on the night before Christmas with the candles 'blaising'...

Monday, December 23, 2013

Twas the Night Before Christmath

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(C)Copyright 2013, C. Burke.

Twas the Night Before Christmath and all cross the 'net,
Readers wanted their updates, or they might get upset . . .

And I sat and struggled to come up with rhymes,
Just lookin' for a chance to work in some primes:

''Now Thirteen! Now Nineteen! Twenty-Nine and Seven!
On Twenty-Three, Five, and Two, and Eleven!''

But I needed a hook; lest it all fall apart,
So I rushed to the end, and went straight for the heart!

Whether Heart of a Poem, or Heart of the Season,
The Heart is the Hope, the Heart is the Reason

That I can say to my readers, voice cheery and bright
Happy Christmas to All and to All a Good Night.

Friday, December 20, 2013

Let It Snow!

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(C)Copyright 2013, C. Burke.

I don't know if it's true that no two snowflakes are the same. I do know that they're lousy for tessellating!

I'm thinking that I should've included a snowflake-making activity. Or maybe if I'd just thought of this sooner and had more time to make it look nicer.
Deadlines ... Christmas doesn't wait.

Tuesday, December 17, 2013

Christmas Bells

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(C)Copyright 2013, C. Burke.

Play all morning and your ears will ring all day!

Sunday, December 15, 2013

Christmas Comics? I hope so!

With less than a week and a half until Christmas, it occurs to me that I need to switch gears and start doing some Christmas and holiday comics. My mind hasn't been on it because I've been thinking about regular strips, which I just haven't had time to do. Those can be put on hold, but I can't move December 25 on the calendar.

My only problem at this point? I used up a lot of my arsenal (as it were) last year. In the past, I've done takes on y=mx(as)+b and y=2^xmas, so I'm running out of things to do there (unless I'm seriously inspired to work backward from somewhere). And the one idea that I do have probably can't be done (by me, decently) in Windows Paint, so I may have to draw and scan it.

But as we're closing in on the Christmas break and things are getting hectic, remember to take time, even if just a few moments, to just enjoy the Spirit of the Season and reflect on another year gone by.

Thursday, December 12, 2013


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(C)Copyright 2013, C. Burke.

Happy First Anniversary to all the 12/12/12 couples!
(Make sure to buy her something nice!)

Thursday, December 05, 2013

Cyber Monday

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(C)Copyright 2013, C. Burke.

Yeah, it was supposed to be Monday's comic, but Real Life and All That . . .
The sentiment is still true, though.

Thursday, November 28, 2013

Happy Thanksgiving 2013: Food Pyramid

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(C)Copyright 2013, C. Burke.

I'll be up for seconds in a minute, then thirds will go forth.

Tuesday, November 26, 2013

Exponential Decay!

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(C)Copyright 2013, C. Burke.

A little late for Talk-Like-a-Pirate Day, but I wasn't up to this topic yet.

Friday, November 22, 2013

The Torch Has Been Passed

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(C)Copyright 2013, C. Burke.

Fifty years since his death. I thought it was worth noting.

Ironically, though I've wanted to do a comic for a while, I wasn't sure what or how to do it. Then I settled on a quote that I would use along with a simple illustration. And then when researching -- basically, double-checking the exact wording of the quote -- I'd discovered that I had my brain wires crossed and had picked a quote by the wrong Kennedy. Thank Heaven for the new age of information. Once again, the torch has been passed to a new generation.

Thursday, November 21, 2013

Blue Box and Whisker

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(C)Copyright 2013, C. Burke.

Eight minutes were added to Paul McGann's time.

For the non-geeks and geeks of a different calling, the numbers above represent the about of hours each actor starred as The Doctor in BBC's Doctor Who, which premeired 50 years ago. It had a 16-year hiatus, but came back in 2005, not as a reboot (I hate those) but as a continuation of the original. The numbers have been rounded to the nearest tenth of an hour, and do not, as far as I know, include later appearances in the series with other Doctors.

Paul McGann appeared on screen for only a 2-hour TV movie, intended as a pilot for a new series. (Eric Roberts played the Master.) It didn't get picked up. However, I included his recent Internet mini-episode, approximately eight minutes in length, because it's just that cool. Go watch it.

Finally: You don't have to write. I did the math. Despite his short TV tenure as The Doctor, 1.2 hours is NOT really an outlier. But it's not as funny the other way, and the back-up joke was too complicated. So you can save your Anonymous corrections. In this case, Math bows down to Humor. And I think it's "Fantastic" or even "Brilliant".

Monday, November 18, 2013


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(C)Copyright 2013, C. Burke.

Thanks to Gregory Taylor of Taylor's Polynomials for allowing me use of his Tangent character. It saved me from making another graph or creating a one-shot character.

That said, any occupational or relational issues discussed here do not extend back to Mr. Taylor's serial ... unless he chooses to incorporate it. Not my call. (but what a good storyline it could be ... dit-dooda-dit-dooda-dit ...)

EDIT: As Mr. Taylor pointed out to me in a tweet, the relationships mentioned couldn't carry over as-is into his strip, because I implied (stated, actually) Cotangent was male. However, in his serial, the trig functions (regular and "co-") are all female. It's the arc- functions which are male. It was right there in the Cast of Characters (from where I took the image) and I should've noticed it. Not that it would have changed anything one bit. Just a little trivia on the side. Now was that the opposite side or the adjacent side?

Wednesday, November 13, 2013


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(C)Copyright 2013, C. Burke.

If a line had any thickness, that could actually be a problem. I wonder what its origin could be?

EDIT: Fixed. An unsung anonymous poster pointed out I'd left out the "un" in "unusual".

Tuesday, November 12, 2013

Happy 11-12-13!

I didn't forget about it! I just didn't have the time to blog today. (I did tweet a message this morning. Do you follow me on twitter?)

HAPPY 11-12-13!
Those of you outside the U.S., we can revisit this next month. ;)

There's only one more Sequential Day left this century! After that, you'll have to wait until January 2, 2103.

Sadly, one of the students in my class said that next year will be the last one in "our lifetimes". I wanted to say (and I might've without thinking) "Speak for yourself!" I thought it sad because she's only, say, 15, which would make her 102 the next time it happens. We're living longer and they're are medical breakthroughs -- miracles, even -- happening all the time. No reason to believe that she couldn't reach that age.

As for me, okay, I'll probably only get that far with a robot body, and, hopefully, some cybernetic helper monkeys.

Monday, November 11, 2013

Vets 2013

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(C)Copyright 2013, C. Burke.

And I didn't even work in the vet's vest. And that's his best vest, which means that if he works the hardest, it'll the best vet's best vest. You bet!

Don't be surprised if I work in "Let's Go Vets!" next year. Just lobbing that out there.

Tuesday, November 05, 2013

Compound Inequality: From Take-off to Landing

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(C)Copyright 2013, C. Burke.

Once you're a Jet, you're a Jet all the way. Unless you get traded to another team.

Or if they tear down your section of the Upper West Side right after your movie is filmed to make way for a cultural institution.

Actually, this could be a great place to go off on tangents about the graph, the movie, the city, . . .

For example, I debated whether the endpoints of the compound inequality should be open or closed. Are you still a Jet if you're dead? Not according to the song, which is, in point of fact, a little too sad to believe that you're forgotten that easily. I mean, (SPOILER ALERT) Tony's dead and they carried him off together, but did they forget all about him a day later? I don't know the answer to this. However, it occurred to me that you'd still be a Jet on that day that you died, up to the point of death. Ergo, closed circle.

So "1st cigarette < Jet < last dying day".

Translating this into Set-Builder Notation is left as an exercise to the reader.
You may comment on this post.

Thursday, October 31, 2013

Check Out Who's Number 2 in this List

Moebius Noodles compiled a list of "5 Halloween Cartoon Treats for Little Geeks". Check out number 2.

Two of my recent Math Horror Movies made the cut. (You'll have to click the link to see which ones.)

Also on the list were a Spiked Math and an xkcd Halloween strip, so I'm in good company. (Theirs were from previous years -- I seem to remember seeing them before.)

Thanks for including me. Always fun to be included.

Wednesday, October 30, 2013

Math Horror Movies: Die! Agonal, Die!

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(C)Copyright 2013, C. Burke.

The Agony is only beginning!

(And then sneak out the Alley ... diagonally!)

EDIT: Reprinted here.

Monday, October 28, 2013

Math Horror Movies: Count Calcula

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(C)Copyright 2013, C. Burke.

He'll Count You Now, or He'll Calculator!

Friday, October 25, 2013

Math Horror Movies: The Bridge of Parallelostein

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(C)Copyright 2013, C. Burke.

I've seen flimsier premises for sequels!

Thursday, October 24, 2013

Happy Si(x)th Birthday to (x, why?)

I didn't forget! I just haven't had time. I was out early getting to the job, and I just got home from Parent-Teacher conferences. In between, there was limited computer availability, which had to be shared with the other teachers. I suppose I could've found my way to a fast-food place with free wifi and tapped away on my iPad, but that wouldn't have gone over very well.

But it was Six Years Ago (which is A Long Time Ago...) that the first Co-medians comic first appeared on this blog. So it's a Si(x)th birthday (and you can go find the x).

The new job is a little hectic, so I haven't been able to keep my usual pace. So no comic today.

I did want to get geeky and have a "Sith" birthday, with Star Wars references like this one, but I'd be lucky to have it finished before midnight Eastern time. Having a little fun on the dark side.

Alas, not to be. But at least I had a chance to mention it.

And a chance to say Thank You to those of you who have been here from the beginning, who came in somewhere in the middle, and who recently found us. I'm doing it for you as much as for me. Okay, I'm still doing it for me -- my sanity still needs saving!

Tuesday, October 22, 2013

Math Horror Movies: Parallelostein

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(C)Copyright 2013, C. Burke.

Halloween is coming again! (And with that, so is our sixth birthday!)

EDIT: Reprinted here.

Saturday, October 19, 2013

Does Anyone Know Anything About the Recent Comments Widget?

Does anyone know anything about the recent comments widget? (And if you do, can you tell me about it?)

If you scroll down the page, you'll notice that the sidebar has both a Recent Comments and a Top Comments widget. The reasons for this are that a) I didn't just want to see the "top" comments, and b)the Top feature stopped working for a while. In fact, it still doesn't work correctly -- when I see it now, it's usually just the recent comments. That is, unless all the comments are equally valued.

For the past few days, I haven't seen either the Top or the Recent appear in the sidebar (other than the headings). I don't know if they aren't working, or if it's a byproduct of the school networks I've been using.

So if anyone knows a widget I can use -- or how to program a new one -- I would be interested. I do get email when new comments are made, but I'd like other people to know about the comments as well because my hope is that one day there might actually be a discussion.

Right now, the only discussion that has gone more than, say, 4 responses in the Coffee Logic comic and that traffic mostly came from one source.

Actually, that reminds me: if there's a widget that lists which posts have the most comments, I'd be interested in that as well.


Friday, October 18, 2013

Climbing a Mountain

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(C)Copyright 2013, C. Burke.

And maybe they're part repitilian, too.

But forget I said that if I want to use that at a later date!

No borders on the comic, because it would have interfered with the arrows on the vectors.

Wednesday, October 16, 2013


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(C)Copyright 2013, C. Burke.

Logic sometimes leads to unusual results. No need to harp on it!

Oh, c'mon! That was funny -- Look up, "Medusa" and then look up "harpies"!

Saturday, October 12, 2013

Welcome "Coffee Logic" and Spiked Math Readers

Once again, there has been a bounce in the stats for my site and the Postest with the Mostest, once again is Coffee Logic, which is usually the weeklong- and monthlog-leader about my comics.

However, at the moment, the numbers are so skewed that I can only assume that the Spiked Math post, Three Logicians Walk Into a Bar, has been reposted somewhere, and that the spillover traffic is heading over here.

So, please, if you're new here and looking around, I'm curious how you found my site. And if you came here through Spiked Math, I'd like to know how you found Mike's site. No, really, I would.

Mike has always enjoyed a larger audience that I have, and I'm fine with that, because he advertises and I rely on word of mouth. Seriously, there's no reason for me to advertise because I'm not selling anything. (Granted, more readers might give me a reason to eventually start selling things, but it's been six years now, and I haven't tried to sell anything yet.)

So, once again, "Welcome" to (x, why?). Read some of the over 800 comics, feel free to participate in, or start a discussion. You can post anonymously, but if you do, you won't know if someone responds to you or not.

And, please, be polite. And Thanks for stopping by.

Friday, October 11, 2013

Coplanar Points

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(C)Copyright 2013, C. Burke.

Oddly, they're all labeled Point m.

Thursday, October 10, 2013

Mentions in Wikipedia

A few years back, while I was still trying to promote this strip -- these days, I let others spread the word more than I do myself -- I looked for any list of webcomics I could find and then looked for a way onto that list. That's how I got onto The Webcomics List and some other comic rankings.

One list that I tried to get on was the list of comics in Wikipedia. (No link, they don't link here, so forget it. You know where it is.) I added myself, and the addition was deleted. The reason being -- as I was told -- that the list only included comics that had individual wiki pages.

Hey, no prob! I created one of those, using the wiki page I set up at Comic Genesis as a basis. And that page was fast-tracked for deletion because, I don't remember exactly, but something like, it wasn't important or significant enough to include.

Well, shoot, man, that hurt a little. (x, why?) had been around for almost two years at that point, and on two servers, with links coming in from all over. But okay, it was still a relative young comic.

I look at that wikipedia list now (I won't even capitalize them) and notice that there are plenty of webcomics listed that started after mine did in October of 2007. Sure, some are by more established artists, and one or two may have gotten their own TV series or something, but I'm still here, plugging away, too good for wikipedia.

And by total accident today, I discovered that I'm actually listed in wikipedia, not once, but twice. Just not for this comic.

I'm in the list of GURPS books for co-writing GURPS Autoduel. (This I actually knew about.)

But I also have an External Link for the Guardians CCG, which links to its current location, which was supposed to be a temporary placeholder, after the old ISP kicked me off a couple of years ago. (It was sold out from under me.) Here's the funny thing: I haven't updated the content of the Guardains page since BEFORE I started this comic. But that's okay for those folks. Go figure.

Wednesday, October 09, 2013

The Number of the Pins

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(C)Copyright 2013, C. Burke.

Imagine if the pins were numbered like Pascal's Triangle.

This should be the last of the "The Number of the" for a while.

And surprisingly, it seems to be only my second bowling comic, and the first in over 500 strips, since my 300th comic.

Monday, October 07, 2013

The Number of the Beast

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(C)Copyright 2013, C. Burke.

I'd like to try to increase the humor by an order of magnitude. Or at least the number of responses in the discussion area on the blog.

This follow-up was meant to be published last Wednesday. Or Thursday. Or Friday. Actually, I thought about doing it before the "Mole" comic.

EDIT: Typo fixed.

Tuesday, October 01, 2013

The Number of the Mole

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(C)Copyright 2013, C. Burke.

Guacamole: It's like whackamole, only with chips instead of hammer, and dip instead of an animal.