Author |
Topic: Ping Pong Balls (Read 943 times) |
|
evergreena3
Full Member
  
Gender: 
Posts: 192
|
 |
Ping Pong Balls
« on: Jan 8th, 2007, 12:49pm » |
 Quote  Modify
|
(I know the gurus here will instantly know the answer to this, but what I am looking for is the explanation. I'm trying to wrap my mind around the infinity concept.)
In front of you is a line of ping pong balls, in single file, that goes on infinitely.
To the left of that is a square of ping pong balls, obviously as wide as it is long, that goes on infinitely.
Which construct contains more ping pong balls, and why?
|
|
 IP Logged |
You're welcome! (get it?)
|
|
|
Whiskey Tango Foxtrot
Uberpuzzler
Sorry Goose, it's time to buzz a tower.
Gender:
Posts: 1672
|
|
Re: Ping Pong Balls
« Reply #1 on: Jan 8th, 2007, 12:53pm » |
Quote Modify
|
They both contain an infinite amount.
|
|
IP Logged |
"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Ping Pong Balls
« Reply #2 on: Jan 8th, 2007, 12:58pm » |
Quote Modify
|
You can make a one to one mapping between the two, so there are equally many.
For example you can take ball 12345678 in the line, and map it to the ball at coordinate (1357, 2468) of the square. And other balls can be treated similarly; take every other digit fto form the first coordinate and the rest to form the second.
And the reverse just takes interleaving the coordinates (adding zeroes in front if necessary). So (12,345678) would map to 30405061728.
[edit]
I tacidly assumed the line and square start somewhere and move infinitely away from us. If they're infinite in both directions, the mapping needs to be altered slightly.
We can split the numbers into even and odd, and map those to positive and negative.
[/edit]
|
| « Last Edit: Jan 8th, 2007, 1:13pm by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Ping Pong Balls
« Reply #3 on: Jan 8th, 2007, 12:59pm » |
Quote Modify
|
on Jan 8th, 2007, 12:53pm, Whiskey Tango Foxtrot wrote:| They both contain an infinite amount. |
|
That's an incomplete answer. There are more real numbers than integers, even though both sets contain an infinite number.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
SMQ
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 2084
|
|
Re: Ping Pong Balls
« Reply #4 on: Jan 8th, 2007, 1:03pm » |
Quote Modify
|
Infinity is assuredly a difficult consept to wrap one's mind around. Obviously you already know they contain the same number of balls.
As for why, let's try numbering the balls. The line is easy: call the first one 1, the next one 2, etc. But what if you're looking at the middle of the line and not the end? No problem: pick any ball and call it 1; call one of the ones next to it 2; now take the ball on the other side of ball 1 and call it 3, then the one on the other side of 2 and call it 4, etc. It's obvious you can give every ball a number.
The square is a little trickier. First imagine you're looking at a corner of the square, so what you see looks like a triangle of balls stretching away to infinity. Call the corner ball 1, then the two balls in the next diagonal "row" 2 and 3, the next row 4, 5, 6, etc. And if you're starting in the midle of the square, just pick any ball and work in a spiral outward, giving each ball a number. Again, you should be able to see that you can give every ball a unique number.
Since for both the line and the square we can give every ball a unique number (in math terms they can be put in one-to-one correspondence with the natural numbers), they must have the same number of balls.
--SMQ
|
|
IP Logged |
--SMQ
|
|
|
evergreena3
Full Member
Gender:
Posts: 192
|
|
Re: Ping Pong Balls
« Reply #5 on: Jan 8th, 2007, 1:17pm » |
Quote Modify
|
I get what you each have said. The mapping, the one-to-one-ness, etc.
How do I break the seemingly-logical statement that instantly comes to my mind?
I would think that a line has X balls, and the square has X squared balls. And, any number greater than one, when squared, is greater than the original number. Therefore, if the line has N balls, the square has N squared, which must be larger than N.
Thanks.
|
|
IP Logged |
You're welcome! (get it?)
|
|
|
SMQ
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 2084
|
|
Re: Ping Pong Balls
« Reply #6 on: Jan 8th, 2007, 1:29pm » |
Quote Modify
|
You just have to realize that rules like that tend to break at infinity. 
If you were to change your statement to "any finite number greater than one, when squared, is greater than the original number" that would be accurate. However, a lot of rules which apply to "any finite number" stop working as expected with infinite (technically transfinite) numbers -- infinity is just weird like that.
Something that is bigger than the natural numbers (or any infinite structure made out of ping-pong balls) is the real numbers. If you take any two natural numbers (ping-pong balls), there are a countable number of balls inbetween them, right? But if you take any two real numbers -- no matter how close together -- there are still an "uncountable number" of real numbers in between! There are some famous proofs (most notably the Cantor Diagonal) that this makes the real numbers "more infinite" than the natural numbers.
The question of whether or not there are any numbers bigger than the natural numbers but smaller than the real numbers is one of the most famous problems in modern mathematics, called the Continuum Hypothesis.
--SMQ
|
| « Last Edit: Jan 8th, 2007, 1:40pm by SMQ » |
IP Logged |
--SMQ
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Ping Pong Balls
« Reply #7 on: Jan 8th, 2007, 1:30pm » |
Quote Modify
|
on Jan 8th, 2007, 1:17pm, evergreena3 wrote:| I would think that a line has X balls, and the square has X squared balls. And, any number greater than one, when squared, is greater than the original number. Therefore, if the line has N balls, the square has N squared, which must be larger than N. |
|
That kind of thinking only applies when you're dealing with a finite number of balls. But there is always a next ball in the line (where intuitively you would think there are too few to cover the square).
And because there is always a next one, you can always map another ball from the square to the line (and vice versa)
Besides, from looking at the infinite line of balls, you wouldn't be able to tell whether they were X2 or X.
|
| « Last Edit: Jan 8th, 2007, 1:31pm by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: Ping Pong Balls
« Reply #8 on: Jan 8th, 2007, 6:21pm » |
Quote Modify
|
Here is a different approach that you may find thought-provoking. Ever since grade school you have been taught the number-line. But you can also form the number-circle. Put your number line in a plane and draw a circle of radius 1 centered at 0. Label the two points furthest from the line the North and South Poles (N and S). The tangent line to the circle at N is parallel to the number-line. Every other line passing through N must intersect the circle in one other point. Conversely, any non-tangent line passing through N is not parallel to the number-line, and so must intersect it in a unique point. This provides a one-to-one correspondence between the number-line and the points on the circle. For each real number x, draw the line passing through N and that number. It intersects the circle in another point, which we can label (x). Conversely, if we are given (x), then draw the line from N through (x), and it will intersect the number-line in x.
So now that we've turned our number line into a circle, what is the point? Well, there is one point on the circle that doesn't correspond to any number: N. This point we can identify with  . (Note that in the number circle, - = ).
0 is mapped by this process to the South pole, while 1 and -1 are the median points halfway inbetween. It is obvious as well that additive inversion, (x) --> (-x), reflects the circle through the line (0)- diameter. One of the great beauties of the number circle is that multiplicative inversion, (x) --> (1/x), reflects through the (-1)-(1) diameter.
What does squaring do on the number circle? It doesn't take too much to see that it maps the entire circle onto the (0)-(1)- side in a double covering. In particular, note that the behavior around is the mirror image of the behavior around 0, and just like 02 = 0, we have that 2 = .
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: Ping Pong Balls
« Reply #9 on: Jan 9th, 2007, 5:00am » |
Quote Modify
|
on Jan 8th, 2007, 12:49pm, evergreena3 wrote:To the left of that is a square of ping pong balls, obviously as wide as it is long, that goes on infinitely.
|
|
A square doesn't go to infinity. 
|
|
IP Logged |
|
|
|
evergreena3
Full Member
Gender:
Posts: 192
|
|
Re: Ping Pong Balls
« Reply #10 on: Jan 9th, 2007, 7:32am » |
Quote Modify
|
on Jan 9th, 2007, 5:00am, Grimbal wrote:
A square doesn't go to infinity. |
|
Assuming you are serious (the smiley indicates you may not be), why is this?
I can envision a single ping pong ball then 3 more appearing about it as a square. Then 5 more. Then 7 more, etc. As each side of the square advances one, at the same rate, the internal ping pong balls appear to make it a square.
Is there a mathematical issue with this going on infinitely?
|
|
IP Logged |
You're welcome! (get it?)
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Ping Pong Balls
« Reply #11 on: Jan 9th, 2007, 8:01am » |
Quote Modify
|
on Jan 9th, 2007, 7:32am, evergreena3 wrote:| Assuming you are serious (the smiley indicates you may not be), why is this? |
|
How would you distinguish an infinite square from an infinite rectangle, or even e.g. an infinite rightangled triangle?
Shape really only aplies if there are edges.
|
| « Last Edit: Jan 9th, 2007, 8:01am by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: Ping Pong Balls
« Reply #12 on: Jan 9th, 2007, 8:14am » |
Quote Modify
|
Well, I know what you mean (therefore the smiley) but at the limit, when the balls go all the way to infinity, it is not a square any more, so you cannot speak of a "square that goes to infinity". I would call it a quarter-plane (assuming you fix one corner when the squares tend to infinity).
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print