How is this an optical illusion? It's a well known 8th grade math puzzle.

If you don't remember Algebra I, then none of what they're doing makes any sense and it's all math gibberish. If you do remember basic math, then the problem is easy to spot:

In the step where you cancel out (a-b), you are dividing by zero, which is a mathematical no-no. (Since a=b, a-b is always zero)

I think it has something to do with the fourth line. (a - b) = 0. I don't think you can cancel zero from both sides. I guess the problem is actually the third line, then. a^2 - b^2 = 0 and so does ab-b^2. Propagating zero down the line leads to 2*0 = 0, which is true.

... but really, I can't see the problem in the algebra. Are you not allowed to subtract and arbitrary number from both sides? I think you are...

grrrrrrrr (I love it...)
(;

. . . . . . .

Now we know the answer to the age old question of what came first: the chicken or the egg.

Clearly it was the entrepreneur (the ultimate professional) that came first. He/she introduced the question of the chicken or the egg to academia, so that academics could waste there lives in a circular reference, enabling the entreprenuer to get on with real life...

From 4th to 5th line, you divide by 0 (a=b so a-b=0), which is not possible.

Step 4 to 5 :
You can't divide by (a-b) because a=b. This is equivalent ti dividing by 0...

2x0=3x0
but u cant remove "0" so u can say 2=3
as in :

(a+b)(a-b)=b(a-b) ===> a-b=0

Yeah, I remember this from school. In the first line you state that a equals b. You go from line 4 to 5 by dividing both sides by (a-b). If a = b, then you are in fact dividing by zero. The answer to that is undefined and causes the funny result.

Wtf?.. (a-b)=0, so (a+b)(a-b)=b(a-b)
(a+b)*0 =b*0
0=0
because you can't divide by zero. Hehehe!

The mistake is here:

(a+b)=b --> -b
a=0

not a+a = a

If 2a = a then a can only be 0 and you cant divide by zero.

This makes no sense what so ever!
Its as if each new line is a different problem with some info taken from the last line kept to the next. This IS NOT an equation being worked out, and after line 4 all the computations are totally off

a+a != a
a+B != b

even if a=b, ESPECIALLY if A=B

Lets assing arbitrary values, 5 in this case. So if A=B (or == b in my notation :P) then:

5+5 =10 or 2a, 2b not 5

I'm not going to tackle line 4 because I can't remember the equasion you run it through when you have to multiply two parenthasies but I don't think its equal.

Yet another example of this website going way way way down hill.

i think your mother is going downhill, you should be grateful for this website like rest of us are, its easy to give bad comments...be contructive

I'm in 8th grade, and this is impossible! How in the world is A=B? and how did they get 2=1? But, A is the first letter in the alphabet, and B is the second, so maybe thats how they could have gotten it. I don't know.

division by zero is undefined

iiii.... am twelve, sixth grade... soooo... yea. It was waay over my head...

uhhhhhhhhh!

The funny thing is that even if you ignore the fact the we divide by (a-b) which is zero, still at the end if 2a=a it doesn't mean at all that 2=1.

We want to solve for either a or b.

How about we stick to the a+a=a ignoring the dividing by zero
giving us 2a=a

we don't divide both sides by a but rather subtract a from both sides and get

2a-a=a-a
a=0

The last line says 2a = a, and therefore, a = a.

To find out what "a" is, you'd divide the equation with "a" entirely - and when you do, you're left with a = 0.

Therefore, it means that the answer is a=0

I think what happened was the twin on the left wrote the left side, while the twin on the right wrote the right side, so they both got different answers. Course, I'm only in 6th Grade so, whatever!

Chuck Norris can divide by 0 (zero)

Simply put, the Exterminator got it right.
On line 5, you have a corret statement: (A + B) = B
Forget the fact that A will solve to zero and all that bull about dividing by zero. At this time, all you know is A + B = B, therefore you reduce this equation by subtracting B from both sides leaving A + B - B = B - B, reducing we get A = 0.

u guys r thinking this too hard.....the twin on the left first put (a+b) then put a+a thats not logical....i mean its common sense

Ladies and gentlemen:

If a = b as given,

then (a-b)= 0 , correct???


so, (fourth line becomes) (a + b) x 0 = b x 0
then (fifth line becomes) 0 = 0

I see many right answers and many wrong answers.

I think there was an implied "for all a and b such that a = b". From that, we can go all the way to "for all a and b such that a = b, (a + b)(a - b) = b(a - b)". We CANNOT get to "for all a and b such that a = b, (a + b) = b" unless we know that (a - b) is never zero, when, in fact, it is always zero.

Anyway, this leads to "for all a and b such that a = b and (a - b) /= 0, 2 = 1". That statement in and of itself is perfectly true, but it's a vacuous truth.

1st, i thought it is in d 6th line,i.e.,(a+b)=a
how can this be written as (a+a)=a

ihope said it all... but for the ones that ask how a+b=b makes a+a=a appear is quite simple, since b=a...

The only problem with the equation is really the fact of a - b = 0...

This is really a good one that tricks many people... but 2 can be equal to 1 .... like 3 + 3 = 7, but that has another trick

The problem is obvious, you can't divide by zero to get from the fouth to the fifth line. It's funny how many people keep trying to prove that a = 0, that would mean that given a=b, a=0, which essentially implies that all numbers equal zero.

if a =b then a-b =0 but in step no 4 a-b has been canceled from both sides which is not possible as we cannot cancel out 0

4th line is 0=0.

Wow. i dont get it at all. Cause im not even in 8th grade yet..... i dont like it.

FIRST off
it's not a math problem
it a bunch of equalities
there are 4 that are correct(the first 4)
and 4 that are incorrect (the second 4)
if you know algebra even a little you would know that it isn't a problem at all
it is just confusing to the stupid
it's like me saying
1=1
2=2
2-2=2-2
0=0
2=1
2=1
2=1
2=1

i have 1 more
to prove door open = door closed
solution: 1/2 door open = 1/2 door closed
therefore 1/2 gets cancelled on both sides,
hence door open = door closed

the last line states 2=1

HOW CAN 2=1 ?!?!?

i am really really confused i cant do maths!

It looks very hard to figure out. Is it a2 = b2+b2 ???

So many answers to the same question
With reply to Rembrance's answer, try uneducated rather than stupid.

All you numbskulls no nothing the answer lies here.

a=2a-(2b or not 2b) that is the question
or is that the answer???

wheres your sense of fun!

forget the last line, A and B are both zero. The last line is where they went wrong.

Let's Begin with your Algebra I (Everything you do to one side, you must do to the other to keep the equation balanced):
a=b
If I were to multiply both sides with a, we get:
a*a=a*b or
a^2=ab
Now let's add -b^2 to both sides (or subtract b^2 from both sides if this makes you comfortable):
a^2-b^2=b-b^2
The next trick is knowing about "the difference of squares." This just states that if I have a^2-b^2 then you can factor the problem as (a+b)(a-b). Same thing with ab-b^2. I can factor out b and get b(a-b). So now we have:
(a+b)(a-b)=b(a-b)
But now we can divide the whole equation by (a-b). This is where the argument begins. Since a=b, you cannot divide the equation with (a-b) because a-b=0, and any number divided by 0 is undefined.
BUT, if you COULD, then you would be left with:
a+b=b
Now remembering that a=b then we can substitute b for a and we get:
a+a=a or
2a=a
Now we divide the whole equation by a and we are left with:
2 = 1.
A great teaser for any math teacher out there (Hint, Hint)
(^-^)

wat im only in 5th grade......ive no clue

the step 4 is wrong.
(a-b) can be cancelled both the sides unless a is not equal to b. since its given that a=b in the first step (a-b) cantt be cancelled both the sides and the result should be 0=0

=b

axa is a2 not a+a.

Yes, a-b=0, how can a number devided by zero, is infinity! The answer is not accurate! The working steps got problem. How ever, 0 divided by 0 that's the answer is accepted by any numbers.....

They tell you that a=b
a and b are variables an can be any number and they equal.
a-b is equal to zero and you can't divide by zero.

r u kidding me?! like im really gonna sit here all day trying to figer out this shit!!....but maybe i'll give it a try.....oh i get it! on the 5th equation, (a+b)=b is wrong. its (a+b)=(a+b) because at the top it says a=b, so if a were to be, let's say 2, so would b. so it would be incorrect to say (2+2)=2.

Dude, the only thing weird is how they go from 2a=a, to 2=1. Don't you subtract "a" from each side at that point?

some guys really rack and crack their heads dont they

robert, rememberance ihope are some.

simple answer is devision be zero is not defined.

All I can see is that if you multiply anything by 0 then the answer is O as seen in line 4. Besides, there should be a minus sign inbetween the two brakets on line 4.
Anyway its no big deal, we all know the answer is wrong so we don't need to debate.

who cares, its wrong, and the right answer is 2=2 anyway, no biggy

that isnt even an illusion!

on the 5th line.
if a=b,
then a=b cant =b.
thats like sayin,
1+1=1,
DOY

man im 12 [primary 6] yet i dunnoe!!! man i'm dumb )': WAAAA!!!!!!!!!!!!!

Guys, stop thinking so much, the fourth line isn't equal, distributive property.

(a-b)(a+b) = b(a-b) do the work out

a^2 + b^2 = ab - b^2

the second part equals zero, the first part is a positive number

huh???

this isn't even an optical illusion...

the mistake there is when 2a=a,
he used that property in equations wherein you transfer a variable to the other side of the equation,
so when you transfer a from 2a to the other side it would be a divide by a which in Algebra is equal to 1.
but the mistake is variables are only equavalent to numbers when they are predefined with a specific no.

This is NOT and OPTICAL ILLUSION. by this logic i could take any math problem....include it in some sort of picture and call it an optical illusion. First..Just because you threw in a picture of two guys does not cover the 'optical' requirement in optical illusion. second....This is not an 'Illusion' at all. its a math puzzle, a somwhat clever math puzzle yes, but in no way is it an illusion. Unfortunatly more and more of the "Illusions" that have been appearing on this site arent even close to the definition of optical illusion.

I never knew Chuck Norris had a twin, because Chuck Norris can divide by 0.

I really liked this one, it stumpted me for a while. I gotta show this to my Calc teacher.

1=2 sounds like Bush economics.

The only way for 2a to equal a is if they're both zero. The equation is totally ligitimate, but just cause 2a=a, doesn't mean 2=1. Also, (a+b)(a-b)=b(a-b), you're dividing by (a-b), not 0

the algebra in itself is perfect. there is only one problem: the first line:

a=b

which later leads to a division by zero, which is undefineable, and also against the eleventh commandment:

"Thou shalt not divide by zero."

the reason why this one got so many replies with the correct answers is people enjoy pointing out other people's errors... and frankly, so do I.


that's all, people

well, thas it's true... like the no gravity rule.

>.<

I'm stumped on the first line.
Shouldn't a = a? The supposed fact that "a = b" is just stupid.
End of story.
And nobody else post about "cant divide by Zero" I've read enough of those replies.

even I get this
(a-b)=0
x divided by 0=YOU FAILED MATH CLASS!!

When you divide members of the equation for a factor you have to remove the case when the factor goes zero. so when you divide for a-b you have to remove a-b = 0 from the solutions which is the first equations! so you can't divide for a-b, because you remove all the solutions!

OK! any given number divided by 0 is not defined therefore the algebraic sequence must come to a halt at that point, as anything after it will be incorrect, hence 2=1! capiche?

step 1 : a=b
step 2 : a2 = ab

but.. if both sides r being squared...
isnt a2 = b2 n not ab?

right??..
so basically .. der is only 1 mistake...
right..or not??

I figured it out. It's the sum that states "a+b=b". This is mathematically impossible and with these rules, I can see why the twins think that "2=1". So the mistake is in the 6th line.

As an engineer and former math teacher I'm saddened to see some of the ignorance displayed here. Taking the whole thing point-by-point (using standard Maple notation) let's talk about it...
The first line, Let a = b, is perfectly routine and shouldn't raise any issues.

The second line, multiplying both sides by a, we get a^2 = a*b which is also OK.

In the third line things get very slippery. BY subtracting b^2 from both sides to arrive at a^2–b^2= a*b–b^2 we must recall our premise. Since we are subtracting like terms because a^2=b^2=a*b, we are already in trouble since terms on both sides of our equation are zero! Everything that comes after this step is based on a false premise that IF a*0=b*0 THEN a=b. This is a somewhat subtle use of the zero property over multipication for equations. It is closely related to division by zero.

Factoring terms (we might think) we see that (a+b)*(a-b) = b*(a-b). We've now collected our zero into one of our terms, (a-b). This is crucial for the operation of the trick, but we actually left good math behind at step three.

Now the Fait au complet comes in cancelling (dividing) the common (zero) terms on both sides giving us a+b = b (which can only be true for a=b=0, not all possible values of a!)

completing the illusion, the trickster resubstitutes a for b simplifying to 2*a = a (since a = b), and then cancels a to conclude 2 = 1!

Those who opined that the error came in step 4 or between 4 and 5 are close, but must remember that once you have a 0=0 situation (for BOTH RHS and LHS) in ANY equation it is useless. When that happens you must work the math differently. The best way to avoid this pitfall is to use AS FEW TERMS AS POSSIBLE!

Of my last group of AB Calculus students (who scored 3s and 4s on their AP exams) none solved this puzzle! I've had three students and 5 colleagues figure it out (correctly) in 23 years of engineering and math teaching. I think the students are doing quite well! They are always better at getting "out of the box." It's no wonder the Fields Medals only go to those 40 and younger!

Don't make it too complicated...

a=0
b=0

That's all there is to it, basically.

ok, this is where they really really messed up. there were only variables involved before and then suddenly- boom! numbers. real, rational, old-fashioned, numerical, numbers

the 2nd to last line says: 2a = a
they've divided by "a" to cancel where in fact they should minus the "a" to find the worth.

true story.

a=b
asq(a square)=ab
asq-bsq=ab-bsq
(a+b)(a-b)=b(a-b)
2a x 0 = a(a-a)
2a x 0 = 1
a x 0 = 1
1=1

I have not had math in 20 years but whatever I remember from grade 13 indicates that when you multiply a x 0 is always equals 1.

This all makes sense except for the very last line.

a=b or 2=2

a^2=ab or 4=4

a^2-b^2=ab-b^2 or 16-16=16-16 or 0=0

(a+b)(a-b)=b(a-b) or (0+0)*(0-0)=0*(0-0)

(a+b)=b or (0+0)=0

a+a=a or 0+0=0

2a=a or 2*0=0

all true to this point.

But 2*0 does not equal 1

Given that anything multiplied by zero is always equal to zero.
You could say for instance
5 x 0 = 2 x 0
doesn't mean that 5=2

MacGyver can divide by zero using a candle and a mirror..

This doesnt even make sense because if a=b then a+b cant equal b and a plus a cant equal a makin this whole thing false

I'm not so sure how this is an illusion...

if a=b, then its safe ot say that a and b hold the same value, which is pointless, but just for arguments sake, lets say it fits the scenario, if a is equal to b, then

(a+b) does not = b, it equals 2b
and a + a equal 2 a, so you have 2a = 2b, reduced means a=b, nothing proven.

Plus, in the 4th and 5th lines, im not sure what the author did, but they divided by -0 or something weird like that, because (a-b) becomes (a+b), in which case b on the outside would have become a negative b, which would then make the proof, 1=-1, and technically all numbers are reflexive to their intergers.

It took me a while though to figure this one out, props to the author.

it shouldnt be b(a-b) in the middle step. It should be (b+b)(a-b) which gives you (b+b)(a-b)

if someone already said this, fine, i didnt read past the first 10 really stupid answers.

both are equal to the product ab-b^2, but there is a difference between the two. that is why there is something wrong with it, it has nothing to do with dividing with zero until later on if you decide to continue with it, which you wouldnt because they would look exactly teh same and to simplify you would cancel it all out.

OK. THere are two mistakes, at least one of which has already been said.

#1. if a=b, then a-b=0, and you can't divide by zero.

#2. If 2a=a, a can only be zero, so you can't divide by a as the last step.


As a side-note, some people have the wierdest reasons for why it doesn't make sense. a and b are the first and second lettersof the alphabet? That's true, but what does it have to do with anything?

In the third line the right side equals 0 because b x b is the same as ab so ab - b squared is 0

What was Annie talking about earlier!?

'iiii.... am twelve, sixth grade... soooo... yea. It was waay over my head...'

Is she crazy? Oh yeah...I forgot I'm in higher math...lol

I'm eleven and seven months and in sixth grade, and this makes total sense to me!

I'm gonna show my teacher this! ^_^

-EXILE

Exile here! This is what I have to say!

Not sure, but this is what I think:

Problem: What went wrong?

Given: a=b

a^2=ab
a^2-b^2=ab-b^2
(a+b)(a-b)=b(a-b)
(a+b)=b
a+a=a
2xa=a
2=1

a=b we will assume a and b are 2

a^2=ab would be 2x2=2x2 (4)
a^2-b^2=ab-b^2 is 2x2-2x2=2x2-2x2 (0)
(a+b)(a-b)=b(a-b) is (2+2)x(2-2)=2x(2-2) (0)
(a+b)=b is (2+2)=2 (4=2) incorrect
(a+a)=a is (2+2)=2 (4=2) incorrect again
2xa=a is 2x2=2 (4=2) incorrect a third time
2=1 is 2=1

that can't be right!

Any number but zero is wrong!

If we do try it with 0, however...

a^2=ab would be 0x0=0x0 (0)
a^2-b^2=ab-b^2 is 0x0-0x0=0x0-0x0 (0)
(a+b)(a-b)=b(a-b) is (0+0)x(0-0)=0x(0-0) (0)
(a+b)=b is (0+0)=0 (0) correct
(a+a)=a is (0+0)=0 (0) correct again
2xa=a is 2x0=0 (0) correct a third time
2=1 is wrong. this makes it 0=0

^_^

So technically, 2=1 is wrong, because the only thing possible is 0=0, not to mention variables aren't neede now that we know it's 0! Happy trails!

on the line after:
(a+b)=b
the next line written is:
a+a=a
this is impossible as the line should read:
a+b=b
which would make a=0 and b=0

lets start at the line that reads:
(a+b)=b
the next line after that reads:
a+a=a
this is impossible as the next line is supposed to read:
a+b=b
which would make a=0 and in turn making b=0...it is actually really simple cuz the one twim made the mistake...which is what your supposed to look for...duh

Huh?

THE MISTAKE THE TWINS MADE WAS WEARING THOSE CLOTHES.

seeing as you cannot solve an equasion with more then 1 unknown variable, the problem is unsolvable.. most of your explanations would be correct if A and B we're actually 0, but there's no way to tell that from the equation a==b. it simply states they're of equal value

it's realy simple, the mistake is made in the fifth line, (a+b)= a is wrong (a+b) = a+b so not b

easy problem..any tougher one?!

dude, i learned this crap in 6th grade, this is not ann illusion

i did this in A2A3 math. thats 8th grade stuff. its not that hard.

a squared is a times a not a times b

gothca

Given: a=b

a2 = ab
a2-b2 = ab-b
(a+b)(a-b)= b(a-b) remember given
therefore
(a+b)x 0 = b x 0
0 = 0
LHS = RHS


this guy sure failed math

If half door is open, it is equal to half door shut, therefore if full door is open, it means full door is shut ?!!!
(a-b)=0, therefore (a+b)(a-b)=b(a-b)=0, period.

I know something kind of like this. Three guys walk into an inn and pay ten dollars each for all three of them to stay the night. Later, the clerk realizes he overcharged them and gives five of the dollars to someone to give back to the three men. On the way, the person realizes there is no way to evenly split the five dollars, so he gives one to each of the men and keeps two for himself. So, each of the men only ends up paying nine dollars. Three times nine is twenty-seven. Add the two that the person kept and you get twenty-nine. What happened to the other dollar?

Scott, you have a very good point, but seriously... THIS IS NOTHING TO GET SO WORKED UP OVER!!! Anyway, even if they multiplied by (a-b), the math should still be constant, even if it equals zero. Don't go all psycho. This really messed up equation is just all in good fun.

i'm in 7th grade and i'm done with algebra 1. if a=b then onthe 4th line then if you use the distributive property then it looks like (a+b)x0=bx0 aka 0=0. case closed. By the way, why the heck did these twins waste their time with this anyway?

Hmm...

WOW....
barring the possibility that there are little kids posting...

if you are an adult and have gone through the education system and don't understand the simple math like the connection between a=b and a^2=a*b you shouldn't be allowed to have kids.

Sorry about being harsh, because its fine to be stumped by this, its not ok to not know basic elementary school math

so really the solution is 2/0 = 1/0 ?

And that is true...

So the real "mistake" is saying that 2/0 = 2 and that 1/0 = 1....

And if anyone reads this far down, get a f'in life

this is not an optical illusion,they just try to confuse you and check if you at lest remmember 6grade maths.everything that's below line 3 is not right:
(a+b)(a-b) will never equale the statement b(a-b)
always (a+b)(a-b)=a2+b2