## I need a little help...

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svk1325
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### I need a little help...

This is not a spambot post even though the title may make it seem like it!

Anyways, I think that this is an appropriate forum to post this in, so here goes...

(If you don't like long background stories, just skip past this section.)

-----

My friend seems to think that Pi will eventually repeat itself an infinite number of times like a fraction in decimal form. He says that "there are only so many ways 10 numbers can be arranged before it must repeat themselves". Then he says that then it will continue to repeat itself indefinitely after that.

So I tried to convince him that intuition means nothing in mathematics and how it has to be irrational since his logic seems to imply that irrational numbers don't exist and that every number is rational. Unfortunately he doesn't exactly understand what I mean by "irrational number" and how Pi is one of them.

So today he kept bothering me about this and he wanted me to give him a probability of whether if Pi will repeat or not. Since he wouldn't stop bothering me, I just gave him "1/(10!^^10)" so he would stop. Now he is trying to find a way to calculate Pi to 10!^^10 digits thinking that it will repeat at that digit. The only thing is, he hasn't even found a way to compute that number.

So anyways, he now thinks that every mathematician in the world has this wrong and that Pi will eventually repeat itself. He thinks he's going to get rich from this "proof" that "no one has ever found before".

I don't want him to waste his whole life trying to do this. It's a stupid idea anyways. And the scary thing is that he's older than me and therefore thinks that his logic is right and mine is wrong. I might never get my point across because of this. And since he calls me a nerd almost every day, I want to prove that he is wrong, along with two of my other friends who think Pi repeats. I just feel that my logic can't get the point across since they can't understand it. I am wondering if any of you have anything useful, since I can't simplify my logic that well.

-----

Can any of you give me a simple, short, and understandable proof that Pi will never repeat itself? I couldn't find any that get the point across to my friend, considering he doesn't exactly understand the concept of "irrational numbers".

(We're both in high school, so we don't know that much compared to all of you. He doesn't know anything past Geometry and I don't know much past Calculus. Not to mention our high scool is pretty strict. Otherwise, I can't physically harm him or anything.)
Last edited by svk1325 on Wed Jan 24, 2007 1:29 am UTC, edited 1 time in total.
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jestingrabbit
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You've got to start by telling him about rational and irrational numbers, and in particular you've got to prove that the former have decimal representations that repeat or terminate and the later have representations that don't. If you can't get him to see that you're screwed.

Then, you could walk him through a proof like the one here. Its not easy, but I don't really see you finding a proof that anyone can get. Good luck, I think you're going to need it.

svk1325
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Ok, so I will try to explain the rational and irrational numbers to him better. The only thing is, he wouldn't understand the proof since he doesn't know Calculus... at all. (We're only in High School, after all.)

But just one more thing to ask, does anybody have any simpler proofs, or am I on my own from here?
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phlip
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Just show him: 0.101001000100001000001000000100...

Simple number to write down, but clearly never repeats, and is thus irrational.

Wikipedia has a reasonably simple proof that all rational numbers have recurring decimal expansions (and therefore that all non-recurring decimal expansions are irrational).

svk1325 wrote:"there are only so many ways 10 numbers can be arranged before it must repeat themselves"

This would be true if we were talking about strings of digits of a certain length, but remind him that the expansion for π is (or, at least as far as he's concerned, could be) infinitely long, and can therefore can have infinitely many arrangements before running out. For instance my example above never runs out of arrangements, even though it only uses 1 and 0, because the pattern gets longer each time.

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`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
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Framling
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The proof for sqrt(2) being irrational is a little easier to follow. That could at least get through to him that they exist. Wikipedia's writeup is pretty easy to follow.
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hyperion
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surely, if pi is truly infinite, it must contain a very large portion of itself at least twice?
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But I digress.

Framling
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HYPERiON wrote:surely, if pi is truly infinite, it must contain a very large portion of itself at least twice?

1) Your logic is flawed. Infinity does not imply "anything you can think of." There are and infinite number of even integers; none of them are seventeen.

2) Even if some huge chunk repeated itself (which, in this case, is most certainly the case), it would have no more significance than '1' showing up twice in the first three decimal places. Of course parts will repeat. We know parts repeat. But that doesn't make it rational.
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Framling wrote:
HYPERiON wrote:surely, if pi is truly infinite, it must contain a very large portion of itself at least twice?

1) Your logic is flawed. Infinity does not imply "anything you can think of." There are and infinite number of even integers; none of them are seventeen.

I'm not sure what his logic was, and whether or not it is flawed, but he is right. It has an infinite decimal expansion, so it has infinitely many seperate substrings of length a=Ack(g64,g64). Since there are only 10Âª decimal strings of length a, one must occur at least twice (in fact, infinitely many times). So there are arbitrarily large substrings that occur infinitely often. Of course, they may also occur extremely sparsely throughout the sequence.
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phlip
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To clarify: to be rational, the decimal must not just have a section which repeats... it must, after some point, be composed entirely of infinitely many repetitions of the same sequence.

Take, say, 3/28: 0.10714285714285...
It starts off 0.10, but after that, the digits 714285 repeat endlessly.

But with, for instance, 0.10100100010000100000100...
There is no point where the same pattern repeats endlessly... there's more 0's each time. Now, a large portion of it does turn up several times (after a certain point, "10000000000" will turn up as part of every step of the pattern, for instance) but that doesn't imply rationality... to be rational the whole pattern would have to be just one repeating sequence, and it's not.

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`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
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phlip wrote:To clarify: to be rational, the decimal must not just have a section which repeats... it must, after some point, be composed entirely of infinitely many repetitions of the same sequence.

I think everyone here is aware of this, but perhaps this is where svk's friend's misunderstanding comes from. It might be worth making sure he's clear about that.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

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German Sausage
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### Re: I need a little help...

svk1325 wrote:... He thinks he's going to get rich from this "proof" that "no one has ever found before"....

is there a reward for finding a proof that pi is rational?
who would pay him?
would this continue to provide income?
no, no and no. if anything, there would be a mob of angry scientists chasing him because they are now unemployed, and need a new test for their supercomputers
really, get these point across as well as that he;s wrong, and even if you dont convince him, there will be no incentive left.

problem solved
[dusts hands and walks away]
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Framling
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skeptical scientist wrote:
Framling wrote:
HYPERiON wrote:surely, if pi is truly infinite, it must contain a very large portion of itself at least twice?

1) Your logic is flawed. Infinity does not imply "anything you can think of." There are and infinite number of even integers; none of them are seventeen.

I'm not sure what his logic was, and whether or not it is flawed, but he is right. It has an infinite decimal expansion, so it has infinitely many seperate substrings of length a=Ack(g64,g64). Since there are only 10Âª decimal strings of length a, one must occur at least twice (in fact, infinitely many times). So there are arbitrarily large substrings that occur infinitely often. Of course, they may also occur extremely sparsely throughout the sequence.

His logic was apparently

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`if     infinitethen     big ol' repeatsendif`

But you just have to look at phlip's example of 0.10100100010000100000100... to see that that logic doesn't hold.

It's one of those misconceptions about infinity that bug me. "If the universe is infinite, then there must be a planet somewhere where everyone looks exactly like Gary Busey!"

And besides, pi isn't infinite. Hell, it isn't even 3.15. I think we can all agree that 3.15 is less than infinity. But pi's decimal expansion extends infinitely.
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ekim
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In the 0.010010001... example you can find arbitrarily long strings of zeroes that repeat infinitely many times. So yes, the statement is true, as Skeptical points out.

Similarly, if the universe is infinite (unlikely, depending on what we mean), and if space exists in some kind of discrete quanta (plausible), and those quanta can be filled by only finitely many different types of things (extremely plausible), then by the exact same (correct) logic, you can find arbitrarily large chunks of space that repeat infinitely many times (these chunks are not likely to hold interesting things, like a bunch of Gary Buseys)

Framling
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In any case, regardless of whether it's true (which I agreed all along was the case), it's no more significant than Skeptical's revelation in the xkcd solution thread that the tangent to the lowest point on a French curve, no matter how it is oriented, is horizontal; or mine that a sixteenth-degree polynomial fits the seventeen data points of prime knot crossing numbers.

What I'm saying is it's not a property that's unique to pi, so I don't see how it says anything about it.
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svk1325
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skeptical scientist wrote:
phlip wrote:To clarify: to be rational, the decimal must not just have a section which repeats... it must, after some point, be composed entirely of infinitely many repetitions of the same sequence.

I think everyone here is aware of this, but perhaps this is where svk's friend's misunderstanding comes from. It might be worth making sure he's clear about that.

Yup, that's basically right. I tried explaining, but it didn't help much; he still doesn't get the point. Anyways, almost every time I try to explain, he cuts me off saying "What the heck does that have to do with Pi?"

German Sausage wrote:
svk1325 wrote:... He thinks he's going to get rich from this "proof" that "no one has ever found before"....

is there a reward for finding a proof that pi is rational?
who would pay him?
would this continue to provide income?
no, no and no. if anything, there would be a mob of angry scientists chasing him because they are now unemployed, and need a new test for their supercomputers
really, get these point across as well as that he;s wrong, and even if you dont convince him, there will be no incentive left.

problem solved
[dusts hands and walks away]

I told him and he says that he's still going to be famous for his "discovery". He's still going for it.

But anyways, I tried telling him today and it still didn't help. I guess I'll just have to wait until he realizes himself that it doesn't repeat forever. He says my logic is flawed, so I'm going to ask my math teacher to explain it to him on Thursday. If that doesn't end this, I don't know what will.
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Air Gear
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I'm gonna be a jerk and start by saying that if he calls you a nerd basically every day (hell, I'll go with a weaker condition and say "on a regular basis"), the first thing you have to do is punch him in the face. If he's good at dealing with that kind of thing, instead punch toward the face with your left fist so he's not paying attention to the fact that you're using the other arm to uppercut him in the stomach.

After that, the first thing I'd try would be...ok, how many ways are there to arrange the ten digits if you're going to 10 decimal places? Ok, so now what if you're going to 1000 decimal places? Huh, a lot. And you know what, if you go more and more at a time, there are more ways you can arrange it. So basically, you can make any size string you want at any time and it can NOT be a string you've already had.

So yeah, after (as they say in the vernacular) pwning him a little bit, try going to the "a lot of ways to arrange that" argument or possibly going with an EASIER irrational number as has been suggested earlier. If that doesn't work, just point him toward the Euler-Mascheroni constant and have him work on that one since at least it's not completely idiotic to try calling that rational.

svk1325
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Air Gear wrote:I'm gonna be a jerk and start by saying that if he calls you a nerd basically every day (hell, I'll go with a weaker condition and say "on a regular basis"), the first thing you have to do is punch him in the face. If he's good at dealing with that kind of thing, instead punch toward the face with your left fist so he's not paying attention to the fact that you're using the other arm to uppercut him in the stomach.

Can't. Teachers and "security" everywhere. I would get in huge trouble.

Air Gear wrote:After that, the first thing I'd try would be...ok, how many ways are there to arrange the ten digits if you're going to 10 decimal places? Ok, so now what if you're going to 1000 decimal places? Huh, a lot. And you know what, if you go more and more at a time, there are more ways you can arrange it. So basically, you can make any size string you want at any time and it can NOT be a string you've already had.

I actually tried to explain it that way today. He still doesn't get it.

Air Gear wrote:So yeah, after (as they say in the vernacular) pwning him a little bit, try going to the "a lot of ways to arrange that" argument or possibly going with an EASIER irrational number as has been suggested earlier. If that doesn't work, just point him toward the Euler-Mascheroni constant and have him work on that one since at least it's not completely idiotic to try calling that rational.

A problem here... He doesn't know what the "Euler-Mascheroni constant" is. Heck, I don't know either. We're only in High School. (And I better edit my first post to say that.)
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Air Gear
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svk1325 wrote:
Air Gear wrote:I'm gonna be a jerk and start by saying that if he calls you a nerd basically every day (hell, I'll go with a weaker condition and say "on a regular basis"), the first thing you have to do is punch him in the face. If he's good at dealing with that kind of thing, instead punch toward the face with your left fist so he's not paying attention to the fact that you're using the other arm to uppercut him in the stomach.

Can't. Teachers and "security" everywhere. I would get in huge trouble.

Seriously, though, if he goes with the "NERD" sort of thing, he needs to get smacked around a bit. Maybe you need to meet him outside of school for a little "special explanation".

Yes, I've learned to be extremely hostile toward the "HUH HUH NERD" attitude and other forms of hyperstupidity. At this rate, in a year or two I'll be taking my own damn advice in that sense.

So yeah, anyway, encourage him to do two things: 1) start doing some computation looking for those repeats and 2) shut the hell up. The second will have to be covert. In other words, let him waste as much time as he wants being stupid since, hey, the drugs we have for "stupid" are hit-and-miss at best.

tetoru
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Just to add a little something to the discussion...

It's an open question[1] whether pi is normal, i.e. whether each digit occurs equally often (asymptotically, of course.) It's not even known whether each digit occurs infinitely many times. (Although experimental data strongly suggests that both statements true.)

So, given a particular string of digits, it's not necessarily the case that you will be able to find it infinitely many times in the expansion.

[1] http://pi314.at/math/normal.html

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tetoru wrote:Just to add a little something to the discussion...

It's an open question[1] whether pi is normal, i.e. whether each digit occurs equally often (asymptotically, of course.) It's not even known whether each digit occurs infinitely many times. (Although experimental data strongly suggests that both statements true.)

So, given a particular string of digits, it's not necessarily the case that you will be able to find it infinitely many times in the expansion.

[1] http://pi314.at/math/normal.html

Yes, but it is certainly the case that there is some string of every length which occurs infinitely many times, even if it's not the case that every string occurs infinitely often. This is simply true by the pigeonhole principle - nothing deep is at work.

Additionally, that source is highly questionable, and says some things that are just wrong: "While an undergraduate at Cambridge University, D. Champernowne proved that 0.12345678910111213 ... is normal in base 10, [which is completely obvious] but an explicit example of a normal number is still lacking." There are certainly examples of normal numbers; Chaitin's constant is one.
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Framling wrote:
HYPERiON wrote:surely, if pi is truly infinite, it must contain a very large portion of itself at least twice?

2) Even if some huge chunk repeated itself (which, in this case, is most certainly the case), it would have no more significance than '1' showing up twice in the first three decimal places. Of course parts will repeat. We know parts repeat. But that doesn't make it rational.

I am not a math person at all (though, actually, I prefer the more theoretical, philosophical side of it, particularly those parts of math that refer to an infinity at some point), but I thought both of these things while reading the original part.
Yes, it is very likely true that part will repeat itself somewhere inside the infinity of pi.
Does it make a difference? Absolutely not.

tetoru
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Lethal Interjection wrote:
Framling wrote:
HYPERiON wrote:surely, if pi is truly infinite, it must contain a very large portion of itself at least twice?

2) Even if some huge chunk repeated itself (which, in this case, is most certainly the case), it would have no more significance than '1' showing up twice in the first three decimal places. Of course parts will repeat. We know parts repeat. But that doesn't make it rational.

I am not a math person at all (though, actually, I prefer the more theoretical, philosophical side of it, particularly those parts of math that refer to an infinity at some point), but I thought both of these things while reading the original part.
Yes, it is absolutely true that part will repeat itself somewhere inside the infinity of pi.
Does it make a difference? Absolutely not.
Is there anyway to prove either side conclusively? Nope, it is entirely theoretical. It can't be proven either way, though the further the number is figured, the more likely it is irrational.

Sorry if I misunderstood you, but pi is provenly irrational. In fact, it is transcendental, which is an even stronger statement: the irrationals can be partitioned into two collections, the first containing irrationals which are zeroes of polynomials with integer coefficients - numbers such as sqrt(2), because sqrt(2) satisfies x^2 - 2 = 0, and probably most irrational numbers you can think of off the top of your head - and second, numbers which are not roots of any polynomial (called transcendental.)

The first collection is actually really small. It is the same size as the natural numbers. The number of transcendentals, on the other hand, is huge, just as "big" as the set of all real numbers, in fact, but for some time their existence was known only as a theoretical fact without any specific examples of numbers proven to be transcendental. Pi, it turns out, is one of those.

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i don't know that much about maths, enough to basically understand what is going on here, but no more, but i would like to suggest a different tactic. if you've spent a fair amount of time trying to explain to him something which is generally understood to be true, and he doesn't believe it, then what does it matter? he's being ignorant and unreasonable, you've tried your best, there's no reason to keep pursuing the matter really...
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Axolotl
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svk, have you tried philip's '0.101001000100001000001000000100...' explanation? In my mind that's one of the more straightforward ways of disproving your friend's statement that "there are only so many ways 10 numbers can be arranged before it must repeat themselves".

svk1325
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I haven't tried that explanation yet, but I think that it's best for him to figure it out for himself. I tried to explain for this whole week already. If he brings it up, I'll explain it. Otherwise, I'll just ignore it. I'm basically just going to pretend it never happened because I'm tired of trying to explain basic concepts to him.

He seems to be sort of ignorant about what you can and can't do in mathematics. I tried explaining these things to him but I couldn't get the point across. The thing is, it seems like he needs me to help him with this "proof". He hasn't done any actual work on this yet. At least, I think. Heck, he's still thinking that he will be allowed to use supercomputers for this task. But mainly, if I don't help, he can't seem to do it. So I won't help. It's a bit mean, but that should finally get the point across.

Oh, and about the nerd-calling issue, I sorta learned to accept it. At least for now anyways. He says it's "just a joke" or it's a complement about my knowledge. I'm not mad or anything, it's just a bit annoying on my part, that's all. If it goes too far, then I'll do something. But for now, it's fine.
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taemyr
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hmm, 10!^10=3628800^10 Lets say (10^6.5)^10=10^65
http://pages.prodigy.net/jhonig/bignum/indx.html gives the number of atoms in our galaxy as 10^66.
So the number he is considering to calculate has one digit for each atom in the galaxy, no super-computer in foreseable future is going to be able to store this number.

Also calculating pi to this length is going to be meaningless. There are sequences in pi that occurs more than once, 3 digits is enough to spot the sequence "1". When we say that pi is non-repeating we mean that it does not reach a point where the rest of the decimal expansion is the same sequence repeated over and over. So even if he found that pi goes 3.141...123123123123 this would prove nothing, he would need to prove that it continues repeating 123.

But, I think that he could become rich by calculating pi to 10^65 decimal places, not so much from having this accuracy for pi as for devising the computer and algorithm for completing the task.[/url]

Torn Apart By Dingos
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taemyr wrote:hmm, 10!^10=3628800^10 Lets say (10^6.5)^10=10^65
http://pages.prodigy.net/jhonig/bignum/indx.html gives the number of atoms in our galaxy as 10^66.
So the number he is considering to calculate has one digit for each atom in the galaxy, no super-computer in foreseable future is going to be able to store this number.

No unary supercomputer. Storing it in binary only requires 28 bytes.

phlip
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10!^10 ≠ 10!^^10

10!^^10 = 10!^(10!^(10!^(10!^(10!^(10!^(10!^(10!^(10!^(10!))))))))) = Rather big number

Tetration, and its buddies, are responsible for all of the horrifying hugeness of Ack(g64,g64)...

Also, Dingos: A computer would be able to store the number 10!^10, but it wouldn't be able to store 10!^10 digits of pi... which you'd need to store if you wanted to show it repeats after that point.

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`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
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Pathway
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phlip wrote:Ack(g64,g64)

I propose that we define Ack(g64,g64) to be Axkcd

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