I have concluded there are
Let's take an example.
"How many rectangles are on this chessboard?"
People typically answer in one of the following (general) ways:
Type #1: “Huh? I see 64 squares. That’s way too hard for me.”
Type #2: “Uh…well, let me guess? 100? No? 200? 500? Okay, I give up. That’s a stupid question. When would you use it in the real world?”
Type #3: [Counts to a number, n, such that 64 < n < 1296. In other words, comes up with an incomplete (and incorrect) answer.]
--This type was added in the second edition. I’m having trouble getting inside the head of someone who could see some, but not all, of the rectangles on the board, so I can’t model their verbal explanation. –Ed.
Type #4: “Well, let me count. For the 1x2 rectangles, I see [mumbles to self] 1, 2, 3, 4, 5…56 vertical rectangles…and the same number of horizontal ones…then there are the 2x3 rectangles…um, do you have a sheet of scratch paper…?”
Type #5: “Oh, I read about this in my logic/puzzle book. The answer is 1296.”
Type #6: “This entire question is arbitrary. I mean, I COULD solve it, but why bother? Let me know if you have a specific problem you want me to work on. Preferably one that involves ballistics.”
--This type was added in the second addition. Thanks, Ronald. –Ed.
Type #7: “Squares are also rectangles, but rectangles are not also squares. So we need to count all the squares PLUS all the rectangles…let’s multiply. We have 8x8 = 64 1-by-1 squares, plus 7x7 = 49 2-by-2 squares, plus 6x6 = 36 3-by-3 squares…THEN we need to multiply all the 1-by-2 rectangles…7x8, plus 6x7 for the 2-by-3s…oh, drat! I was only counting the vertical rectangles. I need to multiply them by 2 for the horizontal ones…”
Type #8: “Great question! I will look up the formula so I can derive the answer myself! Drat, google gave me the answer first. Ah well...here's the formula! What does that C mean? ‘Choose?’” [Spends another twenty minutes clicking links on Wikipedia, chasing down new knowledge.]
Type #9: “Well, for the squares, you would sum the squares of all numbers from 1 to 9. That number is 285 -- I happen to remember that because it shows up a lot in math competitions… Then for the rectangles it wouldn’t be squares, it would be a n(n-1) but multiply that times 2 since we’re doing horizontal and vertical...so 0 plus 2 plus 6 plus 12 plus 20 plus…where was I? 6x5 = 30, I think…”
Type #10: “Let’s see…there would be nine dividing lines including the ones on each end, and you have to select two, one to be the left boundary of the rectangle, and one to be the right boundary. So you take 9 choose 2 squared, which I believe would be 9!/2!7! [Consults calculator] Which comes out to 1,296.” –(That was a direct Eric quote, by the way.)
Type #11: “Well, it would be nine choose two squared. Which should come out to…[ostentatiously does NOT use a calculator]…1296. [Everybody glares at #11]
Type #12: “The answer is 1,296 but I can’t explain to you how I know it. It’s kind of an intuitive autistic thing.”
Type #13: “There’s a formula for this…I should remember it from college. It’s going to drive me crazy…no, don’t give me any hints. It will be good for me to re-derive it.” [Spends twenty minutes with paper and pencil muttering things like “for n, [n(n+1)/2]…but then for (n-1)…” Everybody else wanders off, bored. Person 11 finally surfaces] “Okay, the universal formula should be [n(n+1)/2]^2. You could use that for any n x n board. In this case, for an 8x8 board, it should come to 1,296.
Type #14: [Yawning] It’s the sum of all cubes from 1 through 8, whatever that comes to. 1300 ish? [Everybody else strangles #14.]
Type #15: “It would take too long to count. I’ll write a nifty little PERL script to do it for me.”
--This type was also added in the second addition, thanks to Jon.
Type #16: [Pedantically] Well, there are several ways to approach this. The first, obviously, is to count, but that brute force method is inefficient. The next is to multiply, which is an improvement but still takes too long. Can anyone suggest a better way? [Expectant pause met by silence] Well, I see that Eric here did 9 choose 2, which is a good way to approach it. Let’s break that math down step by step….now, also, a few of you 11s out there might have used this universal formula, which is quite useful. But the fascinating thing is that the formula [n(n+1)/2]^2 is ALSO equivalent to the sum of the cubes of 1 through n. Does anyone have any guesses as to why that happens? [Expectant pause met by crickets chirping. The Aspie math professor shrugs and continues to lecture, unconcerned, and possibly unaware, that he is speaking into an empty lecture hall.] Here is where we get into the really gritty math, which I think is the most fun. You see… [He spends the next 30 minutes scribbling mathematical notations nobody would understand even if they were present.]
I’m a 7 or 8. Maybe 9 on a good day. Eric is a 10 leaning toward 11. ( He says he only used his calculator a little bit.) I suspected, in my first edition, that Jon would get an answer using the "9 choose 2" method or maybe multiplying, then work on deriving a universal formula for fun. His comment below disputed his tentative typing as an (adjusted) 13 and invented #15. My Grandpa Haupt, in good company with Sir Isaac Newton, was a 16.
I have accepted that Eric and Jon are just smarter at math than am I. It rankles, but it’s true.
Where all do you fall?
--My thanks to http://mathforum.org/library/drmath/view/56760.html for my research on #11 and #12.
9 comments:
Well...before my head injury i was probably only a 3. N since then, a 1!!! (Do i get points for honesty??)
Absolutely! Good job being honest. Like I said, one type of brain isn't necessarily better than another. It just means you shouldn't pursue a career in engineering.
How can you seriously say that there are only 12 options here? Also we can only choose one? I refuse to be held back by your ridiculous restrictions. I'm done with this blog.
Okay, fine. I guess I'll stick around just a little longer. Anyway my first approach would have been a #3 for about a minute or two before realizing that the problem would be faster solved through math than through brute force.
Then I would have been a #13 and written a quick Perl script to figure it out for me while I got REAL work done.
I'm open to revision. I've been thinking of others, like all the people who would count some but not all. And then can't see the extras. I'm having trouble figuring out the most common types of mistakes they'd make, though, so I don't know how to write their commentary. (Basically, I can't "get inside" their heads.)
I think if I had to do the work myself, I would have done something like #8 (of course) except I may not have realized that it could be simplified to factorials.
Jon--I am amazed that you would start with #3. I would also start, briefly, with #3 and then try to find a better pattern. I love your #13. Way to go, gamma guy! This goes to show that my own brain is severely limited about mathematical possibilities. But then, I already confessed you were smarter than I am at STEM.
I'm more of a literalist. It's an 8x8 grid of squares, contained in a giant square. the ability to draw a rectangle around some of those squares but not others is just arbitrary sillyness. you can use the board as a coordinate system, or define a system for dictating zones of controls, but there's no special need to define every possible rectangle on the board. Pick the rectangles you need and be done with it.
I'm certainly not going to create a program to list every possible grid combination without a clear reason why it matters and why i should invest the time on it. Which is not to say that I couldn't do it, mind you, i just don't want to.
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