Comments on: 64=65?

64=65?

pyramid termite — Wed, 18 Aug 2004 05:42:23 -0800

64=65? there must be some kind of trick to this, right?

By: PenDevil

PenDevil — Wed, 18 Aug 2004 05:47:50 -0800

Yup, the line seperating the blue/red triangle from the orange/green one is not perfectly straight. There is actually a gap between the two.

By: pyramid termite

pyramid termite — Wed, 18 Aug 2004 05:49:34 -0800

oh ... duh ... ok, matt you can kill me now

By: gimonca

gimonca — Wed, 18 Aug 2004 05:54:15 -0800

It's early, but here's my best attempt on short notice.... Let's call the smallest angle in the red/green triangles A. Before rearranging, tan A = 3/8. After rearranging, tan A = 5/13. 3/8 != 5/13. Shenanigans, QED.

By: Johnny Assay

Johnny Assay — Wed, 18 Aug 2004 06:29:46 -0800

Obviously you didn't grow up as a math geek — I remember this trick from half a dozen math books for young folk. Still amusing, though.

By: Outlawyr

Outlawyr — Wed, 18 Aug 2004 07:48:33 -0800

There's no trick. 64 actually does equal 65.

By: Civil_Disobedient

Civil_Disobedient — Wed, 18 Aug 2004 08:00:41 -0800

I saw this on Mr. Wizard's World back in the 80's.

By: chicobangs

chicobangs — Wed, 18 Aug 2004 08:06:09 -0800

Go ask Alice when she's ten feet tall.

By: brownpau

brownpau — Wed, 18 Aug 2004 08:08:46 -0800

This is why I went to art college instead, where the only mathematics lesson we learned was "Bush = Hitler."

By: sad_otter

sad_otter — Wed, 18 Aug 2004 08:09:20 -0800

Qix rules.

By: kurumi

kurumi — Wed, 18 Aug 2004 08:15:39 -0800

Non-geometric proof: Let a = b = 1 a^2 = ab a^2 - b^2 = ab - b^2 (a + b)(a - b) = a(a - b) (a + b) = a 2 = 1 (ok, this is also an old math chestnut) 2 + 63 = 1 + 63 65 = 64

By: cyberbry

cyberbry — Wed, 18 Aug 2004 08:26:20 -0800

(a + b)(a - b) = a(a - b) (a + b) = a Riiiiight, because (a-b) = 0 and we all know that dividing by zero works so well in algebra.

By: mfbridges

mfbridges — Wed, 18 Aug 2004 08:40:42 -0800

shouldn't that be: (a + b)(a - b) = b(a - b) (a + b) = b because: ab - b^2 = b(a - b), not a(a - b) Doesn't change anything though.

By: psmealey

psmealey — Wed, 18 Aug 2004 08:45:07 -0800

It's been too long since Trig for me, so gimonca's post flew over my head, but just from a casual look, the slope of one of the diagonal lines is 8/3 and the other is 5/2. If I had a protractor I guess I could figure this out, but if you were to arrange these shapes and keep them constant sizes, there's no way you could make them come together as cleanly as is showing in the Flash animation. Or is there something else goiong on here? I have always been horrible with this kind of thing (it still takes me 20 minutes to figure out the 5 gallon jug and 3 gallon jug to get 4 gallons of water thing).

By: codger

codger — Wed, 18 Aug 2004 09:27:21 -0800

(a + b) = a works because b = 0, since a - a = b

By: leapfrog

leapfrog — Wed, 18 Aug 2004 09:28:53 -0800

Fill the 5 gallon jug. Pour from the 5 gallon to the 3 gallon, leaving 2 gallons in the 5 gallon jug. Then, empty the 3 and transfer the contents of the 5 into the 3. Fill the 5 again, then pour off the top 1 gallon into the 3 (since there are already 2 gallons in the 3 jug, stop when it's full). You will now have 4 gallons of water in the 5 gallon jug. Yay! spoiled!

By: spazzm

spazzm — Wed, 18 Aug 2004 09:40:53 -0800

"Fill the 5 gallon jug..." I knew there was a good reason for using metric.

By: joaquim

joaquim — Wed, 18 Aug 2004 09:51:05 -0800

64 actually does equal 65. Yes, but only for very large values of 64.

By: abez

abez — Wed, 18 Aug 2004 09:52:50 -0800

If you calculate the area of each of the parts the area of the whole never changes. Also if you look at the angles the angle of the triangles is different from the angle of the trapezoids. The image makes an assumption about how well these shapes fit together.

By: Ethereal Bligh

Ethereal Bligh — Wed, 18 Aug 2004 10:00:08 -0800

Yes, but only for very large values of 64. Okay, now I want to have joaquim's children or something.

By: kurumi

kurumi — Wed, 18 Aug 2004 10:09:28 -0800

(mfbridges): shouldn't that be: (a + b)(a - b) = b(a - b) You're right. As my 10th grade math teacher used to say in similar situations: "Stupid chalk!" Here's another fallacious proof, this time involving complex numbers.

By: gimonca

gimonca — Wed, 18 Aug 2004 11:15:23 -0800

Here's a more thoughtful description of what I was getting at previously: The assumption in the animation is that the red triangle, and the triangle made by combining red+blue, are congruent, that is, that their angles are all the same, which also means that their sides should form the same ratios to each other. (Exactly the same with green and green+orange.) Before separating the pieces, the ratio of the legs of the red triangle is 3 to 8 (or, .375). After rearranging the pieces, the ratio of the legs of the red+blue triangle is 5 to 13 (or, about .384). That's NOT the same triangle. Or, if you rearranged the pieces in real life, it would not be a triangle. Instead of a hypotenuse, it would be "bent" a little where the red and blue pieces join--it would actually be a four-sided shape. So red+blue and green+orange would not fit together exactly, there would always be a slight wedge separating them. This illustration "gets away with it" because the differences are small, and the illustrator can just overlap thick borders to "fudge" the differences and trick the eye. Also possible that the grid of squares is more rectangular than square, or changes size at different points in the animation--I haven't printed out any screen shots to measure.

By: betaray

betaray — Wed, 18 Aug 2004 11:57:09 -0800

I've had many a discussion trying to get people believe that .99... = 1

By: Veritron

Veritron — Wed, 18 Aug 2004 15:09:08 -0800

1/infinite->0? 1/10 = .1 1/100 = .01 1/1000 = .001 1/10000 = .0001 1/1...0 = .0...1 1/3 = .33... 3/3 = 1 (x/x = 1) 3/3 = .33... * 3 = .99... (.3*3 = .9, .33 * 3 = .99, .33 * 3 = .999 ...) .99... = 1 ? .99... = 1 - 1/infinite 1 - .0...1 = .99... .99... + 1/infinite = 1 assuming 1/infinite = 0. I honestly think it's an ambiguity.

By: Satapher

Satapher — Wed, 18 Aug 2004 15:17:39 -0800

There's no trick. 64 actually does equal 65. funniest ever.

By: konolia

konolia — Wed, 18 Aug 2004 18:49:54 -0800

This is why I went to art college instead, where the only mathematics lesson we learned was "Bush = Hitler." We actually studied Fibonacci numbers in art school along with the mathematics of the Golden Mean.

By: daHIFI

daHIFI — Thu, 19 Aug 2004 06:57:58 -0800

3 men travelling the country decide to stop at a hotel for the night. One of them goes to the clerk and asks how much the room is. "$30," replies the clerk. The man goes back to his friends and tells them. Each of them puts up $10 and the first one goes back and pays for the room, which they then go to retire in. The hotel manager, however had been watching from the lobby and had seen the clerk overcharge the man for the room. He told the clerk that the room was only $25, gave the clerk five $1 bills and told him to give it back to the men. The clerk, being the greedy bastard that he is, decided to pocket $2 before knocking on the door to the room and giving the man back $3, which the 3 men split between them. Now the question is: How much did each man pay for the room? $9. Ten dollars originally minus the dollar he got back from the clerk. If 9 times 3 is twenty seven, and you add the 2 dollars that the clerk kept you have twenty nine. Where did the other dollar go?

By: Saddo

Saddo — Thu, 19 Aug 2004 10:26:09 -0800

They paid $8.33333333333333333333333333333333333...each. Guffaw.

By: kindall

kindall — Thu, 19 Aug 2004 11:42:21 -0800

If 9 times 3 is twenty seven, and you add the 2 dollars that the clerk kept you have twenty nine. This is a classic of misdirection. You subtract the two dollars that the clerk kept and get $25, which is the amount the manager expected to see for the room.

By: soyjoy

soyjoy — Thu, 19 Aug 2004 13:05:49 -0800

Yeah, that's one of my favorites. Thanks for reminding me of it, daHIFI.

By: betaray

betaray — Thu, 19 Aug 2004 20:53:51 -0800

BTW, it's impossible to have 0.0...1 What place is that 1 at? Infinity + 1? And a understanding of calculus and limits makes the .99... thing pretty obvious, but like I said you're not the first person to try to argue this with me. It's just a syntax thing.