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I've tried various google and math.SE search strings but I'm having trouble formulating a query that gives me relevant information.

Questions

  • Does this table below accurately represent an acceptable answer to this question, (Drinking_Habits_Riddle), if the answer is stated with "100% of the people of said village drink some form of alcohol (assuming that the whiskey and gin are both alcoholic)"?
  • Or would it strictly require a numeric symbol translation to be acceptable?
    • If so how would it be derived directly from the graph alone ?
    • Or could it be that any correct answer be derivable from the table itself but it still needed to be in a numeric format ?

Perhaps I could have said something along the lines of "As is evident by this graph which holds strictly to the correct ratios of Tea:Coffee:Whiskey:Gin that there are no gaps where any percent of people drink no alcohol in this village and no percent of people drink all four beverages. Therefore 100% of the population drinks some form of alcohol." But even if the previous is True I guess that's more a logic answer.. but would that also be unacceptable/irrelevant?

In a small village 90% of the people drink Tea, 80% Coffee, 70% Whiskey and 60% Gin. Nobody drinks all four beverages. What percentage of people of this village drinks alcohol?

Table:

     T C W G
 1   + + + -
 2   + + + -
 3   + + + -
 4   + + + -
 5   + + - +
 6   + + - +
 7   + + - +
 8   + - + +
 9   + - + +
10   - + + +    


 where T == Tea
       C == Coffee
       W == Whiskey
       G == Gin

Extra if helpful

I was excited when I saw this question on stackoverflow as I'm better at understanding patterns and ratios then using symbolic notation manipulation. The table was part of my first answer on math.SE, but got several down-votes. I had added that "it seems any multiples of 10 are true as well" but didn't give proof of said statement. I admit perhaps my choice of words in several places seem to suggest I wasn't being purely mathematical, id est, that it could have seen to have been just a guess. I was told to delete it for it wasn't mathematics. I've since deleted the post after understanding that I didn't show proof of the 'multiples' but I never got an answer from anyone who was against my answer if the (table and accompanying statement) themselves, by themselves, was unacceptable. In either event if I happen to answer anymore questions I'll be sure to only state things I can prove to be true.

For a formal education setting I only have a GED so I apologize if this isn't my place. I have a rough time at judging delimitations of acceptance, and in choosing tone of formal:casual:readable.

Lastly, if this isn't the site for this sort of question please let me know. Maybe it could belong on meta.SE, Quora, or other? As this site is for mathematics I assumed asking here would be acceptable.


In advance thank you for any input

kit
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    Even though your solution is not complete, the aggressive language that a few users posted in the comments to your answer was not warranted. Let me apologize on behalf of the community, and I hope you stick around. – hmakholm left over Monica Aug 10 '18 at 11:36
  • This sort of question is suitable for this site. By the way, in my opinion, you are correct in deciding to post answers that you think you have completely proven, since it is better not to mislead others. That original question thread now has about 3800 views, and most of them are people from outside Math SE, who will not be able to tell what is correct mathematics and what is not. I personally would be very sad if any incorrect answer resulted in thousands of students learning wrong mathematics. I hope you understand this point that I'm making, and we can move on to actual mathematics. =) – user21820 Aug 10 '18 at 12:36
  • I have an unusual temperament. As I take truth and logic to be most important, although I originally misunderstood them, I couldn't find any flaw at an atomic level to their logic of why they said my answer wasn't complete. In that I have no quarrels over. In fact I thank them as I want nothing more then to be a little less ignorant then the day before. I do appreciate both of your inputs. – kit Aug 10 '18 at 14:32

2 Answers2

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Your table can be part of an answer -- namely, it shows that it is possible for everyone to be alcohol drinkers, given the provided information.

However, for a full answer you would also need some kind of argument that this is the only possible situation. (In some situations you might get away with claiming that whoever asked the question wouldn't have asked it that way unless they knew there was a unique result, so you don't need to be sure that is the case. But this is not usually considered satisfactory as a mathematical treatment of the question).

What you need for this other part is not -- as you seem to assume -- more numbers and symbols, but actual words that explain how the reader would convince themselves there is not any solution with less than 100% alcohol drinkers.

(Some people somehow get the idea that the fewer actual English words they use in their writing, the more impeccably mathematical will it be. They then leave out all of the explanations and produce just an impenetrable mess of formulas. This is exactly wrong).

hmakholm left over Monica
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  • Thank you for your reply! It cleared it up for me. You mentioned "However, for a full answer you would also need some kind of argument that this is the only possible situation.", I was wondering if my wording below the bullet points would have been sufficient in this case? – kit Aug 10 '18 at 11:50
  • @kit: The wording I think you're pointing to seems to be one that explains how to read your table, not an explanation of why it would be impossible to have a different table. – hmakholm left over Monica Aug 10 '18 at 12:00
  • @kit: You could have said something like: "We can see this must be the case for all solutions. Since $(100-90)+(100-80)+(100-70)+(100-60)=100$ the number of minuses in the table must be the same as the number of villagers, and we're also told that each villager has at least one minus -- so there are not enough minuses to go around that we can make anyone skip more than one beverage." – hmakholm left over Monica Aug 10 '18 at 12:03
  • HenningMakholm: Your last comment is essentially reproducing Arthur's solution, in which case the table is not needed at all. @kit: That is why I said that your answer is incorrect and essentially cannot be fixed, because proving why one cannot have a different table (as Henning said) up to permutation of rows is essentially as difficult as solving the original problem completely. – user21820 Aug 10 '18 at 12:06
  • @user21820: Without the table (or something similar) as an existence proof I wouldn't consider the question completely answered. Imagine that someone asks "$x$ is an integer whose square is $42$; what is $x$?" I can prove that the assumption implies that $x=0$, but I wouldn't think "0" is a good answer to the question, even when accompanied by my proof. – hmakholm left over Monica Aug 10 '18 at 12:11
  • @HenningMakholm: The question explicitly says that the people satisfy those constraints. If those constraints are inconsistent, it is fine (and even good) to point that out, but it is not at all wrong to prove anything. You surely know that. Like in your example, the problem lies with the problem setter, not with the answerer. In the case at hand, the constraints are not inconsistent, and so the correct answers provided by Acccumulation and Arthur are complete solutions. The table is unneeded, and even distracting. – user21820 Aug 10 '18 at 12:15
  • @user21820: If you allow yourself to infer that the constraints in the problem are consistent, simply because the problem was posed, then I don't see on which principle you refuse to also infer that they have a unique solution. Both of these are, as far as I can see, equally implied by setting the problem at all. If you assume both of them, then the table is a good answer. If you assume neither, then an answer of the kind you're championing is as incomplete as Kit's. – hmakholm left over Monica Aug 10 '18 at 12:22
  • @HenningMakholm: I think you don't understand the given solutions. None of them assume a unique solution! Also, the problem never requires you to infer a unique solution. If you are unaware, this is the standard way such problems are phrased in English; it translates to "given that you know this and that about a small village, deduce the percentage of people in that village who drink alcohol", which is a universal statement that can be proven. I'm not saying that I assume those constraints to be consistent; I meant that to prove the quantified claim we can (and must) assume they are true. – user21820 Aug 10 '18 at 12:27
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I commend you on asking this question. Let me say that it is always hard to explain why some purported writing/solution is wrong mathematics, because essentially the core reason is simply that mathematics is about purely logical reasoning that starts from accepted assumptions. Unlike science, to claim that a mathematical theorem is true, one needs to provide a proof (or at least a proof sketch with enough detail so that experts can be fully convinced that the formal proof can be constructed). There is no room for intuition in a proof. But I will try to give a specific and complete explanation for this particular example that you have brought up.

Firstly, playing around with examples, diagrams, tables and so on is a good thing. It helps one to get a feel for what the problem is like, observe patterns, come up with conjectures and so on. However, always remember that every single illustration of any sort remains a single instance.

If the claim is that some mathematical object has certain properties, and you can give an illustration that provides complete information about one such object, and the viewer can deterministically check that the object you have illustrated indeed has the claimed properties, then your illustration can be considered to be sufficient to establish the result. For example, if I claim that there is a planar graph where every vertex has exactly $5$ neighbours, it suffices to describe just one such graph, and it is acceptable and convenient to do so by drawing it (such as this diagram).

If, however, the claim has a universal quantifier in it, namely that it says something about a whole collection of mathematical objects, then a single illustration can never function as a proof per se. Just for example, if I claim that every planar graph has at least one vertex with less than 6 neighbours, then no matter how many illustrations I give, it does not constitute anything close to a proof. It is possible that by judicious or ingenious diagrams one can convey to another mathematician some key ideas that underlie a proper proof, but that is different from providing a proof.

I want to emphasize what I just said. The four-colour theorem is notorious for attracting fake proofs as well as incorrect proofs. Two well-known incorrect proofs (one by Kempe and another by Tait) were both based on diagrams and intuition, and each was thought by the mathematical community to be correct for a decade. See this article for a detailed explanation of these incorrect proofs, which I just found via Google.

In your case, the problem is to show that every collection of people whose drinking habits satisfy the given constraints must all drink whisky or gin. You did not at all show that; the table can only show that one collection of people whose drinking habits satisfy the given constraints do indeed all drink whisky or gin. At the most, the table can only represent certain kinds of collections of people whose drinking habits satisfy the given constraints. The table does not (and cannot) show that the illustrated kind of collection covers all the possible kinds that satisfy the given constraints.

This logical gap is not fixable, because you would have to show (roughly speaking) that no other kind of row can occur in the table, which is literally the same as showing that every person drinks $3$ of the drinks. And to prove that, the easiest ways are still one of the other posted methods, which means that the table becomes useless for the purpose of the final proof.

Secondly, your attempted explanation:

As is evident by this graph which holds strictly to the correct ratios of Tea:Coffee:Whiskey:Gin that there are no gaps where any percent of people drink no alcohol in this village and no percent of people drink all four beverages. Therefore 100% of the population drinks some form of alcohol.

is wrong for the reason given above. It is only evident that the kind of collection of people that is illustrated by your table all drink whiskey or gin. It is not evident that there are no other kinds.

user21820
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