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

(Click on the cartoon to see the full image.)

(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

(Click on the cartoon to see the full image.)

(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

(Click on the cartoon to see the full image.)

(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!

(Click on the cartoon to see the full image.)

(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

(Click on the cartoon to see the full image.)

(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


(Click on the cartoon to see the full image.)

(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

(Click on the cartoon to see the full image.)

(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.