7 Drinks - 7 Flavors - Infinite variety?

Me and three friends are trying to find the answer to a question I posed about a self-service drinks machine in our local Burger King:

There is a drinks machine that has 7 varieties of drinks (coke, lemonade, orange-aid etc) and each drink has seven varieties of flavor (vanilla coke, lime coke, vanilla lemonade, lime lemonde etc). If you can choose the ratios of the different drinks in your cup, could you make an infinite number of different flavored drinks?

My friends feel it is important to point out that the cup is 500ml ( I don't think this makes any difference).

So far no-one believes me in saying it is infinite, even after we sent the same questions as stated above to a renowned text answering service, whose answer was:

There is an infinite number of drinks combinations with 7 drinks varieties with 7 flavors if you can choose the ratios, as there are infinite numbers.

My friends response to this was that they (the text answering service) didn't understand the question and got it wrong.

I hope you can answer my question and, if I am right, could you please explain your answer so my friends won't think you have misunderstood the question as well.

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I think your friend might be thinking about combinations of flavors instead of ratios. – krikara Dec 9 '12 at 21:28
I miss coke vanilla in the UK :( – rickyduck Dec 10 '12 at 9:19

This may be one of those times where it's important to distinguish between the mathematical and the physical.

If you are allowed to choose arbitrary ratios, then mathematically, the number of flavors are infinite. This is true even with two flavors, $a$ and $b$. If we let $0$ denote the flavor $a$ and $1$ denote the flavor $b$, then the line joining $a$ and $b$ is just the unit interval, which has uncountably many points. If each point represents a unique flavor, it follows that there are an infinite number of flavors.

In reality though, the number of flavors is finite for two important reasons. First, the human tongue cannot distinguish flavors to arbitrary accuracies. If nobody on Earth can tell the flavors $0.499$ and $0.5$ apart, do you still call them different flavors? Secondly, there are only a finite number of atoms which fit into the cup so the flavors are trivially bounded by the combinations of atoms you can choose to make the drink, which is finite.

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"We can't distinguish $0$ from $0.001$ which we can't distinguish from $0.002$ ... which we can't distinguish from $1$ so all flavors are the same." – xavierm02 Dec 9 '12 at 20:12
@xavierm02 That just shows that the physical question is not well-defined. What exactly do you define as a "flavor"? How exactly do you distinguish one flavor from another? – EuYu Dec 9 '12 at 20:16
It's just my way of telling you I don't like that part of your answer. – xavierm02 Dec 9 '12 at 20:20
"we can't distinguish from" is not the same as "is the same as"; "we can't distinguish x from y" is not a transitive relation! – Jason Polak Dec 9 '12 at 23:06
Regardless, simply writing 'there is a finite number of atoms which fit into the cup' is sufficient to show that you cannot make an infinite number of flavored drinks. – LJ2 Dec 10 '12 at 13:21

The answer is no, because of limitations in the physical world. In a container of given size, a limited number of soda molecules will fit.

For example, assume you have a very small cup only capable of containing $100$ molecules of soda. (To simplify the problem, assume all soda molecules are the same size.)

So, you have $100$ "slots" for soda molecules. Each slot can have one of up to $7\cdot7=49$ possible flavors. Thus, you only have $49^{100}$ possible combinations of flavors--a large number, for sure, but still a long ways off from infinite.

EDIT:

The limited capacity of the cup is important. If we had no limit of capacity, we could form any ratio we so wished between any mixture of drinks. However, this is not possible with limited capacity. For example, let's say we wanted a ratio of $100:101$ molecules of soda (in the previous example). We'd need a total capacity of $201$ molecules to have this precision. Unfortunately, we only can fit $100$ molecules in a cup.

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Even with an unlimited capacity in the cup you'd eventually run out of resources to mix, unless you know a very special Burger King restaurant. – Joachim Sauer Dec 10 '12 at 12:16

Yes, if you can control the ratios to an arbitrarily (even if finite) precision then you can make infinitely many different drinks with only two flavours available. Simply take $p$ such that $0<p<1$ and take $q=1-p$. Now take $p\times 500\text{ml}$ of the first flavour and $q\times 500\text{ml}$ of the second flavour. There are infinitely many $p$'s between $0$ and $1$ so there are infinitely many combinations.

The surprising thing, if anything, is that with seven flavours or with two flavours -- the number of different drinks you can generate is the same. And the same means not just "infinite" but rather a very good mathematical notion of "same size" for infinities.

Do note, however, that this is all theoretical in an ideal universe of mathematics. Our [physical] mouths don't have arbitrary precision between flavours and we can't control the machine more than a finite and limited precision, moreover there is only a finite amount of drinks we can make during the lifetime of ours and of the machine. So there is only a finite number of combinations in reality from which we can choose after all.

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With a finite capacity, you cannot control ratios arbitrarily-- You have limited "resolution" so to speak... – apnorton Dec 9 '12 at 19:50
@anorton: So you're saying that there are only finitely many pairs of rationals $p,q$ such that $p+q=1$ and $p,q\geq 0$? – Asaf Karagila Dec 9 '12 at 19:54
No. The statement you just made in your comment is true. However, if you have a finite capacity for the drink (say 100 molecules), all ratios must be of the form $\frac{a}{b}, \frac{c}{d}$ where $\frac{a}{b}+\frac{c}{d}=1$ and $0 \lt b, c \le 100$, and $a, b \gt 0$. You cannot have a denominator greater than $100$, as you cannot deal in partial molecules. – apnorton Dec 9 '12 at 20:04
@anorton: Your argument is flawed for two reasons. The first is that indeed in the physical world this is true, but mathematics have little to do with the physical world. Therefore this question is best approached as a theoretical question in a mathematical universe, and then remarked (as myself and others have done) that in the physical world this logic would fail. The second reason is that if you wish to apply physical arguments finiteness of life is better, you can only make finitely many drinks during your life time, and the machine would break down long before that. – Asaf Karagila Dec 9 '12 at 20:13
I agree with all of your arguments if the question is taken in a purely theoretical context. I was approaching the question from the idea that we were manipulating physical (that is, discrete) quantities in a mathematical universe (no lifetime, wear on machine, etc.) In hindsight, I agree that assuming this is a theoretical question is probably a better approach. – apnorton Dec 9 '12 at 20:29

The text-answering service would be correct. For simplicity, we can assume there is only one choice of drink and one possible flavoring. Then we could have a $1:1$ ratio, where we have half of the drink and half flavoring, or we could have a $2:1$ ratio, where $\frac{2}{3}$ of the mixed drink is the drink itself while the remaining third is the flavor, and so on. Thus, we can have a ratio of $n:1$ for any possible $n$, and since there are infinitely many positive integers, we have infinitely many possibilities. Since this is a simplified version, certainly if we allow there to be 7 drink choices and 7 possible flavorings, there will be infinitely many choices then as well.

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Oh, observe that the argument doesn't rely on the cup size (because ratios don't care about capacity). – Clayton Dec 9 '12 at 19:33
Well, no. For the same reason that the Banach-Tarski paradox doesn't apply to reality (you can't have an infinitesimal part of a ball/amount of a liquid). – You Dec 9 '12 at 19:33

flavour n.
1. taste perceived in food or liquid in the mouth

Flavour is the perseption of chemicals that come in contact with our taste buds. As stated by EuYu, taste buds are capable of a finite resolution of perception, hence there is ultimately a finite number of flavours. You can have a much wider variety of chemical combinations, but again are limited by finite atomic granularity.

Another thing to consider, is how varied are the forty-nine sodas? I'm willing to bet that every available soda is at least 90% water. I was going to say "and 9% sugar", but "Diet" drinks provide alternatives. That leaves a whole 50ml of variety in the cup.

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