Film-Tech Forum ARCHIVE: 64 = 65?

This is topic 64 = 65? in forum Film-Yak at Film-Tech Forum ARCHIVE.

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Posted by Joe Redifer (Member # 3) on 03-12-2004, 01:36 AM:

[av=k]

Posted by Gilbert Travin (Member # 2138) on 03-12-2004, 01:53 AM:

Hi Joe !

The apparent contradiction is perhaps a problem of rounding when boxes are divided? [Confused]

Posted by Daryl C. W. O'Shea (Member # 1303) on 03-12-2004, 01:58 AM:

Uh, those triangles wouldn't fit together like that, their slopes aren't the same. Close enough to fudge though.

Posted by Dustin Mitchell (Member # 372) on 03-12-2004, 02:11 AM:

*waits for Evans*

Posted by Daryl C. W. O'Shea (Member # 1303) on 03-12-2004, 02:23 AM:

What for? We already know what the illusion is.

Posted by Wolff King Morrow (Member # 2226) on 03-12-2004, 02:29 AM:

Take particular note of the blue and green on the final joining. The 3rd square over from the left is extremely "estimated" and doesn't line up at all.

Posted by Brian Michael Weidemann (Member # 2243) on 03-12-2004, 05:05 AM:

Yeah, look at those center three boxes in the 5x13 one, the horizontal lines on the tops and bottoms shift. If you were actually to cut these shapes out from the first and arrange the "5x13", there would be precisely one square unit of gap over that whole diagonal. Narrow, but definitely there. This wouldn't be convincing at all if the demonstration were larger and the shapes didn't use such a thick border.

The slope shifts from 2/5 to 3/8 or vice versa at the joints. 0.4 versus 0.375 Close, but no cigar.

Posted by Manny Knowles (Member # 1171) on 03-12-2004, 09:42 AM:

Yeah.

I can't believe Joe actually fell for that.

What a gullible idiot he is!

Posted by Tim Reed (Member # 137) on 03-12-2004, 09:53 AM:

The angles/colors are there to throw you off. They cover the deception of your perception. Everyone knows you can't split your blocks.

The real trick would be to make it work and stay within the squares! [Razz]

Posted by Thomas Procyk (Member # 1170) on 03-12-2004, 12:53 PM:

WTF? This thread is dumb.

I don't see boxes or triangles or anything anyone is talking about. Just "64=65 ?"

Explain before I condemn this thread to total stupidity! [evil]

=TMP=

Posted by Mike Blakesley (Member # 26) on 03-12-2004, 01:27 PM:

Brings to mind an old Mad Magazine "puzzle:"

"Arrange these shapes into a square"

The solution was to just throw the shapes into the square any old way, with the solution: "You probably couldn't make it work becauase you wouldn't overlap the shapes...but nobody said you couldn't, you dolt!"

Posted by Dustin Mitchell (Member # 372) on 03-12-2004, 02:58 PM:

I'm waiting for Evans to provide us with a essay utilizing multi-variable calculas to explain the problem, as opposed to your one line explanation Daryl [Smile]

Posted by Steve Kraus (Member # 476) on 03-12-2004, 11:18 PM:

Thomas, it's an animation and you need to lower your shields to see it play.

Posted by Evans A Criswell (Member # 381) on 03-16-2004, 09:06 PM:

Joe emailed me and pointed out to me that I missed this thread. The pieces do not fit together perfectly when arranged, as several of you have pointed out.

I first saw this kind of thing back in 1987 in the book "Aha: Insight". It involved a similar thing: a square rug of a certain size that could be cut up, almost exactly like the one shown in this thread, so that its rearranges size had an area that differed from the original by 1. It was a 13 by 13 (area 169) that rearranges into a 21 by 8 (area 168). Unless the pieces are cut with extreme precision, you won't notice that the pieces don't fit together perfectly (the beauty of real-life situations).

This trick works with numbers from the Fibonacci sequence:

Note that the one I saw years ago used numbers 5, 8, 13, 21.

Any 4 consecutive Fibonacci numbers can be used if you go far enough into the sequence. The farther the better the actual fit (the more closely the slopes of the pieces of the triangles actually are to each other):

3 5 8 13 (8x8 rearranaged into a 5x13)
5 8 13 21 (13x13 rearranged into a 8x21)
8 13 21 34 (21x21 rearranged into a 13x34)

and so on.

Posted by Dustin Mitchell (Member # 372) on 03-16-2004, 10:26 PM:

You see Daryl, that was much more interesting than your answer [Smile]

.

Evans, thank you.

Posted by Brian Michael Weidemann (Member # 2243) on 03-17-2004, 02:34 AM:

Oh, yes, the beauty of the Fibonacci sequence! Math rules!!!

Posted by Michael Schaffer (Member # 1204) on 03-17-2004, 03:01 AM:

Fibonacci? Isn`t that a pasta sauce?

Posted by Evans A Criswell (Member # 381) on 03-17-2004, 10:49 AM:

Today, since I'm in my office, I can provide better references to this subject.

The title of the book where I first saw this phenomenon illustrated was aha! Gotcha: Paradoxes to puzzle and delight by Martin Gardner (1982). I got the book free one year (it came on March 18, 1986) when I subscribed to Scientific American. While at Jacksonville State University, I checked a book out of their library that had a chapter on this kind of thing (most likely, a Martin Gardner book).

On the Fibonacci sequence, to generate it, just start with 0 and 1 and make each next number be the sum of the two numbers before it, like this: 0 1 1 2 3 5 8 13 21 34 55 89 144 ...

The growth rate converges to the "golden ratio", which is equal to (sqrt(5) + 1) / 2 or 1.6180339887...

After lunch, I'll scan a couple of things. I have another one of these "rearrange and get a different area" puzzle that's a bit harder to see.

Posted by Evans A Criswell (Member # 381) on 03-17-2004, 01:12 PM:

OK. Here are the diagrams from the book I mentioned. The first one (the top two figures) are like the one Joe used, except it uses a 13 by 13 rectangle rearranged into an 8 by 21 rectangle.

The second one, however, uses a 12 by 12 rectangle with a 1 by 1 hole in it, which is rearranged into a 12 by 12 rectangle without a hole. I find the second one (bottom two figures) much more interesting. Feedback welcome!

Here's the image:

Posted by Thomas Procyk (Member # 1170) on 03-18-2004, 10:27 AM:

Fibonacci is the name of my friend's secksy cat.

=TMP=