Fun math paradoxes, problems, and stuff

[HINDYhat]

For those of you who love math as much as I do, here are some fun math things. Try to contribute some good problems to the thread and I'll update the OP with them if I like them. Also, post what's wrong with the problems in the OP, if anything, and I'll update it with the correct solutions (mathematical rigour is preferred, but I doubt most of you are literate in the formalities of mathematics)!

---

It is known that:
ln 2 = 1 - 1/2 + 1/3 - 1/4 + ...

So, consider the following:
ln 2 = 1 - 1/2 + 1/3 - 1/4 + ...

Partition the terms into odd and even parts:
ln 2 = (1 + 1/3 + 1/5 + ...) - (1/2 + 1/4 + 1/6 + ...)
= (1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...) - 2*(1/2 + 1/4 + 1/6 + ...)

Combine the first two terms and distribute the 2 into the third term:
ln 2 = (1 + 1/2 + 1/3 + ...) - (1 + 1/2 + 1/3 + ...) = 0
e^{ln 2} = e⁰
2 = 1

---

Consider the expression: the smallest positive integer not representable in under twelve English words.

Does such an integer exist? (this problem is tricky as shit)

---

Contributed by Zarathustra
Which is greater, the number of natural numbers or the number of squares of natural numbers? Note that for every natural number, there is a corresponding square... but some natural numbers (like 2 or 3 or whatever) are not the squares of natural numbers.
For those of you who are total noobs, the natural numbers is the set of positive integers, sometimes including 0, i.e. {0, 1, 2, 3, 4, ...}

Solution

For each natural number n, associate with n the number n². And for each square n², associate the natural number n. Since you can associate exactly one square with one natural number, and exactly one natural number with one square, then the set of natural numbers and the set of squares of natural numbers must have the same size.

For all of you noobies, this is called a bijection. A bijection is defined as an injection and a surjection.
An injection is a one-to-one function, that is if f(x) = f(y), then x = y. This means that for every element in the domain, there is exactly one mapped element in the range.
A surjection is an onto function, that is, for any element y in f's range, there exists an element x in f's domain such that f(x) = y.
Here is a more formal solution:
Consider the bijection f: ℕ → {n² : n ∈ ℕ}, defined as f(n) = n²
f is injective since:
f(n) = f(m) ⇔ n² = m² ⇔ n = m, since the natural numbers are positive.
f is surjective since for any square n², f(n) = n²
So, f is indeed a bijection.
Since f is a bijection, then the size of f's domain and range is the same. (this is an easy result)

 

[Zarathustra]

Partition the terms into odd and even parts:
ln 2 = (1 + 1/3 + 1/5 + ...) - (1/2 + 1/4 + 1/6 + ...)
= (1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...) - 2*(1/2 + 1/4 + 1/6 + ...)

(1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...) - 2*(1/2 + 1/4 + 1/6 + ...) = (1 + 1/2 + 1/3 + ...) - (1 + 1/2 + 1/3 + ...)

The sum over n of (1/n), from n = 1 to infinity is a divergent series. You can get a paradox when you try to subtract an infinite quantity from an infinite quantity.

Consider the expression: the smallest positive integer not representable in under twelve English words.

How do you define "representable"?

Try to contribute some good problems to the thread and I'll update the OP with them if I like them.

Which is greater? The number of natural numbers, or the number of squares of natural numbers?

For every natural number, there is a corresponding square... but some natural numbers (like 2 or 3 or whatever) are not the squares of natural numbers.

 

[The Reborn Devil]

Edit: Lol, I should click Refresh when I've had a page open for a long time. Opened this in the beginning of class and responded now when it ended. Anyway, I won't delete my post so you can laugh at it :>
Btw, I was thinking it had something to do with infinite decimals and the fact that the series continues infinitely (the 1/2+1/4+1/6 ...), but I wasn't quite sure.


Not sure what you did wrong with the first one, but somehow you got from ln(2) to ln(1). My calculator doesn't agree on this part:

Partition the terms into odd and even parts:
ln 2 = (1 + 1/3 + 1/5 + ...) - (1/2 + 1/4 + 1/6 + ...)
= (1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...) - 2*(1/2 + 1/4 + 1/6 + ...)

For me it makes sense that "-something = something - 2*something", but my calculator says that this: "- (1/2 + 1/4 + 1/6 + ...) = (1/2 + 1/4 + 1/6 + ...) - 2*(1/2 + 1/4 + 1/6 + ...)" is false.

 

[Adiktuz]

Consider the expression: the smallest positive integer not representable in under twelve English words.

Does such an integer exist? (this problem is tricky as shit)

-->yeah, its a paradox since that expression is 11 words only...

 

[Skycraft]

Here is a very difficult problem:

2 + 2 = x - 1.
Find what 'x' is, and you are a maths genius.

 

[Nightblade]

The simple version of 1=2 is far more appealing.

Heres an easy one.

1$=100 cents
=10 cents x 10 cents
10 cents are 0.1 dollars
therefore
1$ = 0.1$ x 0.1$
1$ =0.01$

 

[Adiktuz]

2 + 2 = x -1
2 + 2 + 1 = x
x = 5

 

[Skycraft]

Very well, you have deciphered my code. You will now be given an even more excruciating task of mathematics.

IF:

5 ^ 2 - x = 13

AND:

7 - (y ^ 3) = -2

THEN WHAT IS:

x + y ^ 4 = z.

Solve for 'z'.

 

[The Reborn Devil]

x = 12
y = 2.0800838230519
z = 12+2.0800838230519^4 = 30.720754407467

You don't have anything harder, Skycraft?

 

[Dr Super Good]

Generate the fourier series with n = 5 of the wave...
5 between 0 and pi/2
3 between pi/2 and pi
1 between pi and 3pi/2
-1 between 3pi/2 and 2pi

The wave is even

Hint
As the wave is even this means there will be no sin terms (as those provide odd components).
The period of the wave is 4pi so omega is 1/2 or 0.5.

 

[HINDYhat]

(1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...) - 2*(1/2 + 1/4 + 1/6 + ...) = (1 + 1/2 + 1/3 + ...) - (1 + 1/2 + 1/3 + ...)

The sum over n of (1/n), from n = 1 to infinity is a divergent series. You can get a paradox when you try to subtract an infinite quantity from an infinite quantity.

There is another mistake in the problem! However, this is also true, but I feel like it's a less fundamental mistake (or at least a less awesome mistake).

How do you define "representable"?

Well I guess I'd have to define a phrase. I guess a finite ordered set of words. Word being defined as a finite ordered set of alphabetic characters. I think that's a reasonable definition of a phrase. So a phrase representation would be a function which maps an integer to a phrase. An integer would be representable if the function is defined at this integer under the function's range constraint (under twelve English words).

Which is greater? The number of natural numbers, or the number of squares of natural numbers?

This is a good one for someone who has only taken high school math. I'll add it to the OP.

---

-->yeah, its a paradox since that expression is 11 words only...

The solution has greater implications than this. Here is a better formulation for what you've said:
Suppose such an integer exists. Then, 'the smallest positive integer not representable in under twelve English words' is a phrase of eleven words, which is a contradiction.
From this, you know that the integer does not exist. This is not a paradox, and it is not the complete solution!

---

The simple version of 1=2 is far more appealing.

But far more mainstream and obvious.

Heres an easy one.

1$=100 cents
=10 cents x 10 cents
10 cents are 0.1 dollars
therefore
1$ = 0.1$ x 0.1$
1$ =0.01$

Simply a matter of properly converting between units.
100 ¢ = (10 * 10) ¢
100 ¢ * 1 $/100 ¢ = (10 * 10) ¢ * 1 $/100 ¢
1 $ = 100 ¢ $/100 ¢ = 1 $

I don't think this is tricky or mathy enough to include in the OP.

---

Generate the fourier series with n = 5 of the wave...
5 between 0 and pi/2
3 between pi/2 and pi
1 between pi and 3pi/2
-1 between 3pi/2 and 2pi

This feels like a purely computational task. Let me know if otherwise, but it certainly doesn't feel mathy.

 

[Dr Super Good]

This feels like a purely computational task. Let me know if otherwise, but it certainly doesn't feel mathy.

Its a combination. You need to know how to generate the answer but I agree that it is largly brute force (aka just lots of computation at the end).

It is cool though making a sum that as N increases looks more and more like the wave you were asked for. Ofcourse it can never be a perfect representation of the wave due to the undifined differentials of the wave at certain points (something that sin and cos functions can not produce). However when you go to 100 terms it will certainly look like the wave on the display.

Heres an easy one.

1$=100 cents
=10 cents x 10 cents
10 cents are 0.1 dollars
therefore
1$ = 0.1$ x 0.1$
1$ =0.01$

Wrong...
10c * 10c = 100c^2 != 100c
You messed up with the units as c and c^2 are completly different units its like comparing 100m (distance) with 100m^2 (area).
likewise 0.1$ * 0.1$ = 0.01$^2

what you should know though is that 0.01$^2 is the same as 100c^2.

This is why maths at a higher level should be compulsory.

 

[HINDYhat]

It is cool though making a sum that as N increases looks more and more like the wave you were asked for.

It's cool, but it's not fun or interesting for the layman.

This is why maths at a higher level should be compulsory.

Unit conversion has nothing to do with higher level mathematics. However, I do believe that lower level math should be more concerned with problem solving (which is actually a useful skill to develop, and happens to be of more importance in higher level math), rather than boring mindless computation.

 

[Zarathustra]

There is another mistake in the problem! However, this is also true, but I feel like it's a less fundamental mistake (or at least a less awesome mistake).

I guess more generally, addition isn't associative in infinite series where the sum of f(x) doesn't equal the sum of |f(x)|.

You grouped the terms in a way that created infinities. If you group them like (1 - 1/2 - 1/4 + 1/3 - 1/6 ... ) you can also get weird results: (1/2 - 1/4 + 1/6 ...) which converges to (ln2)/2. You could prove 1=2 that way, too.

Well I guess I'd have to define a phrase. I guess a finite ordered set of words. Word being defined as a finite ordered set of alphabetic characters. I think that's a reasonable definition of a phrase. So a phrase representation would be a function which maps an integer to a phrase. An integer would be representable if the function is defined at this integer under the function's range constraint (under twelve English words).

What a train wreck. I think this is a mess as long as you use a "function" that relies on the subjective meanings of English phrases and then try to analyze the function in that same vague context.

Let's say I have two "functions," A and B. Function A is just your interpretation of an English phrase, and B is the number that results when you type the phrase, convert the ASCII to binary, and then convert the binary to base 10. Both map an integer to a phrase, but there isn't as much confusion with function B. I doubt there is a problem with finding "the smallest possible integer not representable in under N English words input into function B".

Weird stuff with phrases

"Two times six times seven times five"
"Eight hundred forty divided by two"
- Function A gives you 420 for both, but function B gives you two different (incredibly huge) numbers.

"The limit of 1/n as n goes to zero from the right"
"This phrase refers to x unless the number represented by this phrase is x."
"I'm not bipolar, I'm bi-winning."
"This phrase represents an integer."
- Function A doesn't give you a specific integers (or necessarily anything at all), but function B again returns unique numbers.

If I use function B on "the smallest positive integer not representable in under twelve English words", I get a huge number that uniquely describes it.

Reveal

0111010001101000011001010010000001110011011011010110000101101100011011000110010101110011011101000010000001110000011011110111001101101001011101000110100101110110011001010010000001101001011011100111010001100101011001110110010101110010001000000110111001101111011101000010000001110010011001010111000001110010011001010111001101100101011011100111010001100001011000100110110001100101001000000110100101101110001000000111010101101110011001000110010101110010001000000111010001110111011001010110110001110110011001010010000001000101011011100110011101101100011010010111001101101000001000000111011101101111011100100110010001110011

or about 1.23657203120897 x 10¹⁸⁵

 

[HINDYhat]

I guess more generally, addition isn't associative in infinite series where the sum of f(x) doesn't equal the sum of |f(x)|.

You grouped the terms in a way that created infinities. If you group them like (1 - 1/2 - 1/4 + 1/3 - 1/6 ... ) you can also get weird results: (1/2 - 1/4 + 1/6 ...) which converges to (ln2)/2. You could prove 1=2 that way, too.

I'm not sure if associativity is the word you're looking for (rather a transposition of terms which includes associativity and commutativity), but essentially yes this is the problem. Since the series is not absolutely convergent, then for ANY real number α, there exists a rearrangement of terms in the series such that the series converges to α.

What a train wreck.

lol, I know right. D:

If I use function B on "the smallest positive integer not representable in under twelve English words", I get a huge number that uniquely describes it.

Great, but how is this integer the _smallest_ such integer?

Maybe I misunderstood something.

 

[Zarathustra]

Great, but how is this integer the _smallest_ such integer?

Maybe I misunderstood something.

I was just pointing out the fact that there are a lot of ways an English phrase can represent an integer, and some are more ambiguous than others. I asked you what you meant, and you told me "a phrase representation would be a function which maps an integer to a phrase". Precisely how that mapping is done was never specified, and until it is the statement is too vague to be meaningful.

So let's say it's done by function B from my earlier post: phrase -> string -> binary -> base-10 integer

Following that definition, you could probably write a program to find "the smallest positive integer not representable in under twelve English words".

 

[Adiktuz]

it might not be small, but its still the smallest integer that would fit there... though, the statement itself , which is only 11 words long, makes that integer non-existent...

 

[puggsoy]

Here's a "paradox", if you can call it that. When I was reading some of Zero: The Biography of a Dangerous Idea, it apparently said something along the lines of "any number divided by zero equals any number divided by zero".

So a statement like this would be correct:

45/0 == 978/0

Therefore, since they are both equal, you can simplify it by crossing out the zeros, since they cancel each other out, and you will get this:

45 == 978

In this way he demonstrated that, because of zero, all numbers equal all other numbers. He also explained that this also means that, mathematically, anything in the universe is possible. If there are 0 elephants on my head, there are also 200 elephants on my head. If I have 2 arms, I also have 4 arms.

I think you could call that a paradox.

 

[Nightblade]

DSG,you know that the mistake was intentional...

 

[Dr Super Good]

DSG,you know that the mistake was intentional...

Errors should never be made intentionally.

Anyway, a common one of these "paradoxes" is integer float mathematics on computers. If you trace the modulus opperation that WC3 uses (the GUI one) without taking into account that integers are restricted to a certain domain you end up with a result of the number you put in. Although mathematically correct, one has to remember that integers have a restricted domain.

 

[HINDYhat]

I was just pointing out the fact that there are a lot of ways an English phrase can represent an integer, and some are more ambiguous than others. I asked you what you meant, and you told me "a phrase representation would be a function which maps an integer to a phrase". Precisely how that mapping is done was never specified, and until it is the statement is too vague to be meaningful.

I understand. The fact that this expression is not well-defined probably invalidates the whole problem. I'm not well-versed in this kind of math.

In this way he demonstrated that, because of zero, all numbers equal all other numbers. He also explained that this also means that, mathematically, anything in the universe is possible. If there are 0 elephants on my head, there are also 200 elephants on my head. If I have 2 arms, I also have 4 arms.

Oh god...

 

[En_Fuego]

Here's a "paradox", if you can call it that. When I was reading some of Zero: The Biography of a Dangerous Idea, it apparently said something along the lines of "any number divided by zero equals any number divided by zero".

So a statement like this would be correct:

45/0 == 978/0

Therefore, since they are both equal, you can simplify it by crossing out the zeros, since they cancel each other out, and you will get this:

45 == 978

In this way he demonstrated that, because of zero, all numbers equal all other numbers. He also explained that this also means that, mathematically, anything in the universe is possible. If there are 0 elephants on my head, there are also 200 elephants on my head. If I have 2 arms, I also have 4 arms.

I think you could call that a paradox.

In proportions, you can't just "cross out" your denominators...you would need to "cross-multiply" so the answer becomes 45*0 = 978*0

ie 0 = 0

 

[Adiktuz]

actually, I think it should be 0/0 == 0/0 because cross multiplying is actually just multiplying both sides by the number... http://en.wikipedia.org/wiki/Cross-multiplication

so that would be (45/0)*0*0 == (978/0)*0*0 ==> 0/0 == 0/0...

but I think the equation is still invalid as x/0 is indeterminate or sometimes we call it infinity...

 

[En_Fuego]

actually, I think it should be 0/0 == 0/0 because cross multiplying is actually just multiplying both sides by the number... http://en.wikipedia.org/wiki/Cross-multiplication

so that would be (45/0)*0*0 == (978/0)*0*0 ==> 0/0 == 0/0...

but I think the equation is still invalid as x/0 is indeterminate or sometimes we call it infinity...

While you're right in the fact that x/0 is undefined (how could I miss that...) cross multiplication still winds up with 45(0) == (978)(0). Or did you not read the first like 10 lines of the wikipedia article? By multiplying by the denominator you effectively "clear" the denominator. What you're actually doing is multiplying both sides to make up for the initial division.

 

[Adiktuz]

but you cannot do that because the denominator is 0...

try to do that on school, your prof would surely say its wrong...

but there are ways to solve that, I think you can perform some limit functions to effectively eliminate the zero denominator... but never by direct "cross" multiplication...

 

[En_Fuego]

but you cannot do that because the denominator is 0...

I already agreed with you. However, if the denominator wasn't 0, it would be the same thing.

 

[puggsoy]

Well, the theory may be wrong, but at least I got some conversation going

 

[The Reborn Devil]

Yeah, you trolled everyone :>

 

[Bloodprice]

Maths???... no thanks

 

[puggsoy]

Come on. Anyone that has anything to do with Warcraft 3/Starcraft 2 modding must like maths to some extent.

 

[HINDYhat]

Most people here think that math is arithmetic and formality is high school geometry.

 

[puggsoy]

Well this is how I see it.

Maths = mind sports

And geometry and trigonometry are awesome.

 

[Alain.Mark]

x=0.999...
10x=10(0.999)
10x-x=9.999...-0.999
9x=9
9x/9=9/9
x=1

how ever this is wrong in my opinion...

 

[HINDYhat]

I don't like that explanation. I prefer writing 0.999... as an infinite series, and then the result is immediate:
0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...
= 0.9*(0.1 + 0.01 + 0.001 + ...)
= 0.9*1/(1 - 0.1) = 0.9/0.9 = 1

But yeah, the statement 0.999... = 1 is correct.

EDIT:
You can probably do it this way too.
Consider the sequence x_n = 1 - 0.99...9, where n is the number of 9's in 0.99...9.
Then x_n = 0.00...1, where 1 is at the n^th position.
x_n = 10^-n
So x_n → 0, which means that 1 - 0.999... = 0, so 1 = 0.999...

It's pretty clear if you use any sequence or series representation of 0.999...

 

[fladdermasken]

Euler's Formula should be mentioned, not really a paradox but quite peculiar nonetheless.

e^(2πi)=1

Or if you will:

e^(πi)=-1

Complex=Real.

 

[HINDYhat]

For those who have been intoduced to Euler's formula but haven't studied it:

e^πi = -1
e^2πi = 1
ln e^2πi = ln 1
2πi = 0

Of course, this is silly, since the complex logarithm is multi-valued.

 

[puggsoy]

Wait... isn't Euler's Formula the thing you use to figure out how many sides/vectors a 3-dimensional polygon has? Or is that something else?

 

[fladdermasken]

Of course, this is silly, since the complex logarithm is multi-valued.

Infinitely so

Wait... isn't Euler's Formula the thing you use to figure out how many sides/vectors a 3-dimensional polygon has? Or is that something else?

You may be refering to his polygon division problem, which is to determine how many possible means there are to partition a plane convex polygon into triangles. Euler has quite the arsenal of formulas.

 

[HINDYhat]

Yeah, in fact I think I refer to the e pi i thing as Euler's identity. Whatever.

 

[Zarathustra]

On an infinite, two-dimensional, rectangular lattice of 1-ohm resistors, what is the resistance between two nodes that are a knight's move away?

- From the amazingly awesome Google Labs Aptitude Test / xkcd nerd sniping strip

 

[D4RK_G4ND4LF]

On an infinite, two-dimensional, rectangular lattice of 1-ohm resistors, what is the resistance between two nodes that are a knight's move away?

- From the amazingly awesome Google Labs Aptitude Test / xkcd nerd sniping strip

zero?

hmmmm no probably it's not :/

 

[Alain.Mark]

What if:
Pi=4.00?
Just came to me because if you try trapping a circle inside a square and continue dividing its corners until you get a circular like polygon then Pi would be fixed. In connection to wc3 since we are not realy making circular circles but rather polygons that were arranged well to be more circular, i think this is applicable. I also tried it on a knockback spell and it makes sense though the units push at a side a little but it really makes sense... (btw, just a theory)

 

[D4RK_G4ND4LF]

what if:
Pi=4.00?

NEVER!

 

[puggsoy]

Posting mathematical humour images are we?

 

[Alain.Mark]

I believe the internet more, than you all!!!

Just kidding ^^

 

[puggsoy]

That's a good one Alain.

 

[D4RK_G4ND4LF]

....

that's because you still have the jitter on the circle
imagine a wave
if you squeeze it together (like that: >>>>~~~~~~<<<<) you still have the ups and downs
make sure your lines are one-dimensional

 

[Alain.Mark]

*offtopic

What you said reminds me of the above. Imagine if you have a powerful magnification device (too bad i have non but my imagination) and you magnify a thread you will see > /\/\/\/\/\/\/\/\/\/\/\/\ and for every "/" you see you can zoom in again and see smaller > /\/\/\/\/\/\/\/\/\/\/\/\/\/\/\ . But is that even possible?

 

[Magtheridon96]

But is that even possible?

It depends on your definition of the word "possible"

EDIT:
LOL It reminds me of string theory
According to string theory, that's impossible. At one point, you'd have reached the "string" level where you'll
find 1 Dimensional strings forming a 2D and 3D universe =p You can't go further than that .. or can you

 

[Alain.Mark]

Oh yes Magtheridon96 that was the name I forgot : D
I call them "spring wavies" when i talk to ppl.