Interlude: Not withering on the vine, no sir

So, after an epic writing marathon under extremely challenging circumstances yesterday, I’m no longer too worried about my writey muscles atrophying. I still feel like a big hand-flapping litwanky crybaby about it all, but everyone was very kind to me and all in all it was a good day. Lots of running around outside, followed by an easy stint at child-minding in the evening. Mrs. Hatboy was at a concert, Wump was absolutely exhausted and Toop was, while slightly fractious, asleep sweetly by eleven.

Amazing, what a good day’s writing and a good night’s sleep will do for the mood. My dad woke up on the wrong side of the bed this morning, but he’s just finished cutting a whole trailer-load of wood for us so the poor bloke’s probably wrecked. Plus, he’s got an iPad and is having trouble getting his e-mails read. Shocker.

Then today we got ourselves up to the “shitpile” (Högberget, the highest “mountain” in Vantaa) and enjoyed an Easter barbecue. That was nice, and a good workout carrying a backpack full of Tomps[1] and sausages and also Wump on my back, all the way up there. Then home to prepare for a grand old lamb roast dinner with the combined families. Busy day.

[1] If you don’t know what these are, you need to read this blog more carefully – or get out here to Bar Äijä’s more often.

What kept me distracted this morning, though, was a far wackier – and yet simpler – problem.

Now, I’m not a big maths-type person, but this was weird.

Take a square, right, divided up into a million little squares (I went with a million because the example I was actually thinking about, with ten million, wouldn’t square-root into a nice round number). This makes a square, yeah, with a thousand little squares on each side.

Right? 1,000 x 1,000 = 1,000,000. Easy.

And the circumference of that square would be four thousand, because there’s four sides and you count the outside of each side and each side is a thousand square long. So in terms of length, the circumference is easy too.

Of course, in terms of how many actual squares make up the circumference, it’s only 3,996. Because you only count each corner once, so it’s 1,000 + 999 + 999 + 998. Yeah?

So it’s 4,000 sides of squares, but only 3,996 actual squares.

And then, if you think about it, 3,996 is 4,000 minus 4. And those 4 are the four corners.

So you’re only counting each corner once, and at the same time you’re not actually counting any of the corners at all.

And then, if the ten-million-area was a circle instead of a square, the circumference would be nowhere near four thousand.

Just … what sort of a silly universe do we live in, anyway?

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10 Responses to Interlude: Not withering on the vine, no sir

  1. stchucky says:

    Need to keep an eye on the autocorrect, and make sure to fix my paragraph spacings. MC3’s Word-to-blog system doesn’t seem to double space. Maybe the WordPress app would work better.

  2. dreameling says:

    So it’s 4,000 sides of squares, but only 3,996 actual squares.

    And then, if you think about it, 3,996 is 4,000 minus 4. And those 4 are the four corners.

    I dunno, it kinda makes sense, since 4 of the 3996 squares are corners, meaning they count twice for sides. So, you have 3992 squares contributing only 1 side to the perimeter and 4 corner squares contributing 2 sides, which gives you a total of 4000 sides (3992 x 1 + 4 x 2). It all fits, right?

    Plus, a perimeter or circumference is basically a line, so it’s one-dimensional, while a square is two-dimensional, so they’re somewhat different mathematical beasts — you can’t directly compare them (4000 sides is a different thing than 3996 squares). Right?

    I’m not a math wiz either, but I think math is friggin beautiful. It’s like ultimate conceptual purity and consistency. I wish I understood it better.

    • stchucky says:

      Yeah, the “how the Hell can it be 3,996 when I know the circumference is 4,000 ohhh wait, of course, the line around the outside is 4,000 and has nothing to do with the number of squares” breakthrough felt like a world-class mathematical proof when I was out walking my second-born at six in the morning.

      I’m still no closer to understanding how 3,996 squares can count each corner of the circumference-squares once, but yet not count any of them.

  3. dreameling says:

    And then, if the ten-million-area was a circle instead of a square, the circumference would be nowhere near four thousand.

    But that one is weirdly hard to get intuitively. I guess a circle is just a more economical shape that gives you constant shortcuts around its perimeter / circumference?

    • stchucky says:

      Exactly, and then you end up with the conceptual nightmare of the single-sided shape of the circle, with a fractally infinite number of such cut corners, still somehow having a measurable circumference instead of a circumference of zero. Don’t even get me started. No wonder pi is so messed up.

  4. aaronthepatriot says:

    Behold why writers should not do math! Even when “math” is simply counting boxes and wondering why you “don’t have to count” some of them. *eyeroul*

    You count the corners, they’re just SHARED with other sides. When you count 999 on one side, you’re counting a corner. It’s 1000×1000 but that doesn’t mean 1000 UNIQUE squares….Aw forget it!

    This does remind me of a neat puzzle about persons paying for a hotel room, then receiving change, and somehow $1 of the money goes missing when you look at the math involved. Can’t remember exactly how it goes, let me try.

    Ok so 3 guys go to this hotel and rent a room for $10 each. So they pay $30. But then the hotel manager realizes the group rate should be $25 for all 3, so gives the bellhop $5 in one-dollar bills to return to the guests.

    But, the bellhop, crafty bloke that he is, sees an opportunity here. See, the 5 one-dollar bills can’t divide evenly amongst the 3 guys. So, he decides to tell them the rate is $9, give them each $1 refund, and pocket the remaining $2 for himself.

    So he does. But then as he’s walking back to the lobby, he thinks… “OK so they each spent $10 – $1 = $9 on the rooms. $9 x 3 = $27. And I pocketed $2. $27 + 2 = $29! Damnit where’s the last dollar I could have wrung out of this deal? They gave us $30 to begin with, not $29!”

    Can you find his missing dollar?

    ;D

    • dreameling says:

      Fuck me. I couldn’t figure this one out when reading your text, but when I paraphrased the puzzle to my wife, she somehow caught it immediately: There’s no missing dollar, since the hotel didn’t get $30. They got $25, because they gave back $5. Of that $5, $3 found its way back to the guys while the bellhop kept $2. The guys paid $27, yes, but only because the bellhop screwed them over, meaning they paid too much, above what the hotel got, so you subtract rather than add the $2. $27 – $2 = $25. It all makes sense! The original sum ($27 + $2 = $29) is a dirty trick.

      Right?

      • Quite right! I hope she does your finances! And it may be a dirty trick, but as they say “if you’re not cheating, you’re not trying!”

      • dreameling says:

        In retrospect, real world objects do not inexplicably vanish and valid math is fool-proof, so of course it had to be a trick in the setup of the puzzle. Bastards.

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