# 64=65?

August 18, 2004 5:42 AM Subscribe

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

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

posted by pyramid termite at 5:49 AM on August 18, 2004

posted by pyramid termite at 5:49 AM on August 18, 2004

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.

posted by gimonca at 5:54 AM on August 18, 2004

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.

posted by gimonca at 5:54 AM on August 18, 2004

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.

posted by Johnny Assay at 6:29 AM on August 18, 2004

posted by Johnny Assay at 6:29 AM on August 18, 2004

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

posted by Civil_Disobedient at 8:00 AM on August 18, 2004

posted by Civil_Disobedient at 8:00 AM on August 18, 2004

Go ask Alice when she's ten feet tall.

posted by chicobangs at 8:06 AM on August 18, 2004

posted by chicobangs at 8:06 AM on August 18, 2004

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

posted by brownpau at 8:08 AM on August 18, 2004

posted by brownpau at 8:08 AM on August 18, 2004

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

posted by kurumi at 8:15 AM on August 18, 2004

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

posted by kurumi at 8:15 AM on August 18, 2004

(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.

posted by cyberbry at 8:26 AM on August 18, 2004

(a + b) = a

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

posted by cyberbry at 8:26 AM on August 18, 2004

shouldn't that be:

(a + b)(a - b) =

(a + b) =

because:

ab - b^2 = b(a - b), not a(a - b)

Doesn't change anything though.

posted by mfbridges at 8:40 AM on August 18, 2004

(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.

posted by mfbridges at 8:40 AM on August 18, 2004

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).

posted by psmealey at 8:45 AM on August 18, 2004

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).

posted by psmealey at 8:45 AM on August 18, 2004

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!

posted by leapfrog at 9:28 AM on August 18, 2004

Yay! spoiled!

posted by leapfrog at 9:28 AM on August 18, 2004

*"Fill the 5 gallon jug..."*

I knew there was a good reason for using metric.

posted by spazzm at 9:40 AM on August 18, 2004

*64 actually does equal 65.*

Yes, but only for very large values of 64.

posted by joaquim at 9:51 AM on August 18, 2004

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.

posted by abez at 9:52 AM on August 18, 2004

posted by abez at 9:52 AM on August 18, 2004

*Yes, but only for very large values of 64.*

Okay, now I want to have joaquim's children or something.

posted by Ethereal Bligh at 10:00 AM on August 18, 2004

(mfbridges):

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.

posted by kurumi at 10:09 AM on August 18, 2004

*shouldn't that be:*

(a + b)(a - b) =(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.

posted by kurumi at 10:09 AM on August 18, 2004

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.

posted by gimonca at 11:15 AM on August 18, 2004

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.

posted by gimonca at 11:15 AM on August 18, 2004

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

posted by betaray at 11:57 AM on August 18, 2004

posted by betaray at 11:57 AM on August 18, 2004

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.

posted by Veritron at 3:09 PM on August 18, 2004

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.

posted by Veritron at 3:09 PM on August 18, 2004

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

funniest ever.

posted by Satapher at 3:17 PM on August 18, 2004

*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.

posted by konolia at 6:49 PM on August 18, 2004

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.

posted by daHIFI at 6:57 AM on August 19, 2004

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?*posted by daHIFI at 6:57 AM on August 19, 2004

They paid $8.33333333333333333333333333333333333...each. Guffaw.

posted by Saddo at 10:26 AM on August 19, 2004

posted by Saddo at 10:26 AM on August 19, 2004

*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.

posted by kindall at 11:42 AM on August 19, 2004

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

posted by soyjoy at 1:05 PM on August 19, 2004

posted by soyjoy at 1:05 PM on August 19, 2004

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.

posted by betaray at 8:53 PM on August 19, 2004

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.

posted by betaray at 8:53 PM on August 19, 2004

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posted by PenDevil at 5:47 AM on August 18, 2004