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This little booger took me the better part of a year to figure out, and then only with the help of some friends.

I hope y'all have as much fun unraveling this as I did.  :up:

I'm an old puzzler myself - that's a classic!

I ain't givin' it away!!!
JoeActor

The answer is...

the BUTLER did it!

I like the smiley!  (https://www.happyatheistforum.com/forum/proxy.php?request=http%3A%2F%2Fi89.photobucket.com%2Falbums%2Fk227%2FMommaSquid%2Fsmileys%2Fthumbsup.gif&hash=0ce3c02a9002cdabb779afffcf9880e713b89826)

erm...if the pieces really are all the same size as before; it doesn't make any sense.  How can something take up less space just because it is arranged differently?

Or, is that the solution...they arrangement made it take up less space just like organizing things in a drawer can make them take up less space.  

Hmm...It's still odd though.

edit:  I know the answer now because it annoyed me so I cheated and found the answer online.

First you move the blue triangle to the red triangle

Second you rotate the yellow bit

THEN A MIRACLE HAPPENS

THE END

That's easy. Look at the individual pieces. They never change in proportion or size, it's just the way the green and orange come together. The top one is slightly slimmer than the bottom one.

Sorry but I don't see the mystery. Thanks for sharing though.

Quote from: "Sophus"That's easy. Look at the individual pieces. They never change in proportion or size, it's just the way the green and orange come together. The top one is slightly slimmer than the bottom one.

Sorry but I don't see the mystery. Thanks for sharing though.

Sorry, nope. The grid makes squares, so the non-triangular pieces are the same over-all height and width, as are both resulting shapes.

Click on the image to see it undistorted. Try printing it and cutting out the prices.

Quote from: "JillSwift"

Quote from: "Sophus"That's easy. Look at the individual pieces. They never change in proportion or size, it's just the way the green and orange come together. The top one is slightly slimmer than the bottom one.

Sorry but I don't see the mystery. Thanks for sharing though.

Sorry, nope. The grid makes squares, so the non-triangular pieces are the same over-all height and width, as are both resulting shapes.

Click on the image to see it undistorted. Try printing it and cutting out the prices. ;)  But I don't understand the question. I see how they can be arranged to form a different shape but what's so fascinating about it? Ha ha... I could rearrange it to make a totally different looking shape and nobody would be in awe.

Quote from: "Sophus"Ah. Well at any rate whether they intended it to be or not the top one is slimmer. Take a close look.    But I don't understand the question. I see how they can be arranged to form a different shape but what's so fascinating about it? Ha ha... I could rearrange it to make a totally different looking shape and nobody would be in awe.

The area of the shape changed. That's supposedly impossible, if you recall your Euclidean geometry.

Christ on a Bike. It's Maths in 3rd year all over again. I swear I've seen this in my maths class at some stage in my short life so far. All I can think of is that the shapes are all the same as the top one, yeah? Well, when you out those two ( making the 'hole' ) beside each other in the way that they are, they'll always make a little 'hole' because that green one isn't long enough to fill the space.

Pfff that's all I got. I'm wrong too :B

I wish i didn't already know the answer to this. I love puzzles which truly squeeze my brain.  

OHHH I figured it out I just used "Christin Science" and the answer is the same for everything we don't know sure.... "God did it" woo hoo now I understand whenever there an answer I can't figure out "God did it". I think you're puzzle inadvertantly converted me. Now I just have to never use logic or reason again or ever ever ever think for myself and I'll always have the right answer.

Hmmmmmmm. Still thinking.

Quote from: "Sophus"I see how they can be arranged to form a different shape but what's so fascinating about it? Ha ha... I could rearrange it to make a totally different looking shape and nobody would be in awe.

Keep thinking about that...it will help you solve the puzzle.

I find this answer to be unsatisfying:
[spoiler:85jx4qg8]http://puzzles.nigelcoldwell.co.uk/twenty.htm
Is this the "real" answer? When I printed it out there was no noticeable difference in angle --- the two triangles still lined up on each other.[/spoiler:85jx4qg8]
Is there a better solution than this?

Quote from: "karadan"I wish i didn't already know the answer to this. I love puzzles which truly squeeze my brain.  :bananacolor:

Quote from: "BadPoison"Hmmmmmmm. Still thinking

Take a look at the hypotenuse (the longest side).

Wait a minute! You mean [spoiler:3g776jlv]the freaking hypotenuse isn't a straight line in either one?  :crazy:

Foul![/spoiler:3g776jlv]

Quote from: "McQ"Wait a minute! You mean [spoiler:16yvwlk0]the freaking hypotenuse isn't a straight line in either one?  :crazy:

Foul![/spoiler:16yvwlk0]

Exactly my thoughts. Lame.
[spoiler:16yvwlk0]Because I totally printed it out and cut it out to layer on top of each other. I really couldn't see it not being a straight line...[/spoiler:16yvwlk0]

[spoiler:pdzdpf8v]my comment is in the spoiler below[/spoiler:pdzdpf8v]
[spoiler:pdzdpf8v]I can't figure it out!  [/spoiler:pdzdpf8v]

major spoiler here
[spoiler:1gpwrhc3]Just check the slopes in the triangles.
We can do that seeing if the angles which are to the left (the most acute angles) are the same, just calculate the tangent. They are similar (thus and changing the color and size of each one gives the appearance to be equal), but they aren't equal. So if you just have 2 different angles and put them together as in the figure, the resulting line will not be a straight line, making the figure to be a quadrilateral, not a big triangle. The difference between the slighest convex polygon and the concave one sums that area which appears to be missing.

Being the difference so small between the two angles (~1,2º) this is not well perceived if you just measure them, unless maybe if you make it real big. But if you do the math, you can see it clearly.

Or so I think.
Sorry about the crappy English...

PS: I guess you aren't de coña,    but yeah it can be tought of as an optical illusion. But it's plain Euclidean geometry, squares and triangles you know[/spoiler:1gpwrhc3]

Don't spoil it if you're still working on it.  :D[/spoiler:28kl7e82]

I think that's why I don't understand these types of things..... they're all trick questions. The question almost always asserts a lie.  

Are there any more like these?

I just figured it out.

[spoiler:132ib05r]The red and turquoise triangles are not similar. Therefore the composite "triangle" is not a triangle. Abandon math. Lol.[/spoiler:132ib05r]

Quote from: "Sophus"I think that's why I don't understand these types of things..... they're all trick questions. The question almost always asserts a lie.  

Are there any more like these?

No, this is not tricky! It's tricky if you trust what apparently you see with your eyes, but doing some simple calculations lets you find the answer easily

I enjoyed that!  Thank you.  Got it on my own, but it took a while.  The hint about looking at the hypotenuse helped me.

[spoiler:1673hp46]If you look closely at the lower diagram, where the two triangles meet, you can't see the corner.  But if you follow that vertical line straight up to the top diagram, you CAN see the corner, proving the two smaller triangles are not similar triangles.[/spoiler:1673hp46]

Ok... How about a math based one?

This one's easy, but a little deceptive as well.

Three friends eat at a restaurant and agree to split the bill of $30 evenly.
Each pays $10.
The waiter comes back and tells them the manager is running a special.
Dinner will only be $25 tonight, so he give them $5 back
Since they wanted to split evenly, each takes a dollar, and they tip the waiter $2.

...So, each man paid $9 for his dinner.
$9 times 3 friends equals $27, plus the $2 from the waiter equals $29.

Where's the extra dollar?

JoeActor
(no, I didn't keep the dollar!)

Quote from: "joeactor"Ok... How about a math based one?

This one's easy, but a little deceptive as well.

Three friends eat at a restaurant and agree to split the bill of $30 evenly.
Each pays $10.
The waiter comes back and tells them the manager is running a special.
Dinner will only be $25 tonight, so he give them $5 back
Since they wanted to split evenly, each takes a dollar, and they tip the waiter $2.

...So, each man paid $9 for his dinner.
$9 times 3 friends equals $27, plus the $2 from the waiter equals $29.

Where's the extra dollar?

JoeActor
(no, I didn't keep the dollar!)

[spoiler:2d6ohme6]$9 times 3 friends equals $27. The $2 tip is included in the $27 so you would just add the $3 of change received to get the original $30.

Yippee![/spoiler:2d6ohme6]

Thanks Joe!

Got it in 6 minutes, BadPoison - kudos!

Quote from: "joeactor"and they tip the waiter $2.

Cheapskates.
15% tip on $25 is $3.75. ;)

Has anyone ever heard of the proof that one equals two?

X = Y

2X = YX  [Multiply both sides by X]

2X - Y^2 = YX - Y^2  [Subtract Y^2 from both sides]

(X-Y)(X+Y) = 2(X-Y)  [Factor; remember FOIL?]

X+Y = Y  [Divide both sides by (X-Y)]

1+1 = 1  [Assign value 1 to X and Y, since X = Y]

2 = 1

I promise you there's a flaw.  Can you spot it?  Took me a year.

Quote from: "HandsandDreams"X = Y
2X = YX  Multiply both sides by X

[spoiler:1ekse37i]Didn't you just multiply the left side by 2 and the right side by X? It should be

X = Y
X^2 = YX

Also, why does X = Y? I would agree with X = X or Y = Y.[/spoiler:1ekse37i]

I slightly modified this puzzle to make it harder to search for! Haha!

A woman will be elected as US President in the future.

This is a claim and has truth value. It is either true or false.

If the original claim is true:
Then it is true today that a woman will be elected as US President in the future.
Since it is true today, it is necessary that a woman will be elected in the future.

If the original claim is false:
Then it is false today that a woman will be elected as US President in the future.
Since it is false today, it is necessary that a woman will not be elected.

Therefore, whether a woman will be elected as US President in the future is predetermined because it is already true or false.

Oh crap, you're right, sorry!  I made a mistake in step 4 also.

The real puzzle should read:

X = Y [Because we said so; X and Y have the same value]

X^2 = YX [Multiply both sides by X]

X^2 - Y^2 = YX - Y^2 [Subtract Y^2 from both sides]

(X+Y)(X-Y) = Y(X-Y) [Factor; remember FOIL?]

X+Y = Y [Divide both sides by (X-Y)]

1+1 = 1 [Assign value 1 to X and Y, since X = Y]

2 = 1

[spoiler:2n2sq3p9]Well then X-Y is potentially zero when you divide. You can't divide by zero. And as it turns that if X = 1 and Y = 1 then its a divide by zero. Is that it?[/spoiler:2n2sq3p9]

(https://www.happyatheistforum.com/forum/proxy.php?request=http%3A%2F%2Fweb.mit.edu%2Fpersci%2Fpeople%2Fadelson%2Fimages%2Fcheckershadow%2Fcheckershadow_illusion4med.jpg&hash=1b9011f68ef518b1951ca83316f811d71cbcec0f)

Squares A and B are exactly the same shade of grey.

This one does my head in. Not a puzzle, but certainly puzzling.

Quote from: "AlP"[spoiler:1khdf23l]Well then X-Y is potentially zero when you divide. You can't divide by zero. And as it turns that if X = 1 and Y = 1 then its a divide by zero. Is that it?[/spoiler:1khdf23l]

You got it!

Quote from: "HandsandDreams"Has anyone ever heard of the proof that one equals two?

X = Y
...
X+Y = Y  [Divide both sides by (X-Y)]

[spoiler:3dwo6y24]if X=Y then X-Y = 0, you can't divide by zero.[/spoiler:3dwo6y24]

Quote from: "karadan"This one does my head in. Not a puzzle, but certainly puzzling.

I hate those...I even checked to make sure they were the same...and they were.  It has to do with the way colors that are next to the color affect how it is perceived.

Quote from: "Whitney"

Quote from: "karadan"This one does my head in. Not a puzzle, but certainly puzzling.

I hate those...I even checked to make sure they were the same...and they were.  It has to do with the way colors that are next to the color affect how it is perceived.

Totally. I still can't look at it and see them as both the same shade of grey. I think it is impossible. Someone in my office wouldn't believe it, so i had to do a demo with MS paint.  

Spoiler alert!!!









How many triangles? How many line segments? How do you know?









I began by assuming three triangles in each figure, and calculated the areas using the formula 1/2*base*height. The small triangles don't add up to the same area as the large "triangles". Then I cheated and looked up why: basic trigonometry, i.e. compare the tangent ratios of the apparently equal angles. The larger figures are two different quadrilaterals whose area differs by exactly one square unit. The top figure's area is 1/2 square unit less that the imaginary larger triangle, and the bottom figure's area is 1/2 square unit larger than the imaginary larger triangle.

Quote from: "karadan"(https://www.happyatheistforum.com/forum/proxy.php?request=http%3A%2F%2Fweb.mit.edu%2Fpersci%2Fpeople%2Fadelson%2Fimages%2Fcheckershadow%2Fcheckershadow_illusion4med.jpg&hash=1b9011f68ef518b1951ca83316f811d71cbcec0f)

Squares A and B are exactly the same shade of grey.

This one does my head in. Not a puzzle, but certainly puzzling.

Huh? I don't really get this illusion. They are clearly different shades...how can they be the same?

Quote from: "NaturaLCalamity"Huh? I don't really get this illusion. They are clearly different shades...how can they be the same?

Use a paint tool to move one of the squares over the other. you'll see that they are indeed the same shade of grey!

Quote from: "NaturaLCalamity"Huh? I don't really get this illusion. They are clearly different shades...how can they be the same?

Perception. You have a brain that evolved to detect stuff in the shadows (among other detection tricks), so it's "adjusting" perceived luminescence of the squares that appear in a shadow.

What Jill said  

(https://www.happyatheistforum.com/forum/proxy.php?request=http%3A%2F%2Fhalshop.files.wordpress.com%2F2007%2F03%2Fphpw9jvl0pm.jpg&hash=742c90cc4cc3154598ec2220b48d3efb71fcd7aa)

As for the president one, yes, it is already predetermined, at least barring unknowable quantum effects. If you include the quantum effects, then the origianl staement has no truth value, but is in a superposition of truth states 0 and 1.

I don't see a problem with it being pre-determined, just like it is predetermined a ball thrown up will come down to earth, given the right info, you can make the right predictions.

I like the quantum bit though, my profs would be proud of me.