Looking at this puzzle it's easy to be overwhelmed; there's lots of constraints to keep in mind and lots of details to keep track of. However there is actually a very finite amount of possile solutions to choose from. Billy can either have meat or fish, which makes two cases. Conny can either have meat or fish, and for each of those cases there is a case where Billy has fish and where Billy has meat, making it four possibilities. Daniel, Billy and Conny can have 8 possibilites, because Daniel can have meat or fish, and for each of these cases we already know Billy and Conny make up four possibilities.
Since there are six guests, there is a total of 2^6, or 64 possibilities, one of which is:
Billy: Meat
Conny: Fish
Daniel: Fish
Ellen: Fish
Fiona: Meat
Anna: Meat
If someone who ordered meat sits between an unmatched couple, or if someone who ordered fish sits between a matched couple, that person is lying. Since Fiona ordered meat and the two persons next to her ordered different things, that means she is indeed always lying. However, we can clearly see that she is telling the truth; Fiona sits next to Anna who sits next to Billy, all ordered meat, just like Fiona said. This is a contradiction and we can safely assume that the guests did not order their meals in this way.
Now, it might seem troublesome to have to do this 64 times, but if you know who is lying and who is telling the truth you simply don't have to print out those setups, For example, a lot of people quickly gather that Billy can't possibly be telling the truth. Someone who ordered fish is telling the truth only if exactly one neighbour ordered meat. If everyone ordered fish, then that possibility is excluded, and everyone must be lying. If everyone is lying, then Billy must also be lying, and not everyone did order fish. That's obviously a contradiction, so we must assume that Billy is lying.
So we just won't check out the solutions where Billy is lying; and since for every instance where Billy is lying there is one where he is telling the truth, it means we just halved the number of outcomes we have to check!
Another more obvious observation is that Daniel is accusing Ellen of lying. Either Daniel is telling the truth about Ellen's lying, or Ellen is telling the truth, which means Daniel is lying about Ellen's lying. So one is telling the truth and the other is lying. It might seem trivial, but it actually means we now have reduced the number of setups we have to check from 64 to 16!
You have less fingers and toes than that! Surely that can't be considered too much to handle!
I however will not show 16 setups in this guide. Instead I will show how you can deduce the truth-telling status, that is to say whether someone is telling the truth or lying, of all the guests.
Let's start with two observations:
First, assume four guests are sitting in line. The first one ordered the same thing as the fourth one, and the second ordered the same thing as the third one. It doesn't actually matter if they all ordered the same thing; they may have or they may not. We don't know that.
What we do know is that the second and the third both ordered the same thing and they both are sitting next to someone from the "inner pair" and someone from the "outer pair". So they must have the same truth-value!
Next, imagine two tables that are identical save for one guest. In one of the tables a guest ordered meat, and the equivalent guest in the other table ordered fish. All other corresponding guests ordered the same things. What we can see here is that not only does the deviating guest have different truth-value than the equivalent in the other table, but also his neighbours. In one table the neighbour of the deviating guest is having matching neighbours, and in the other doesn't. Thus if you change the dish of one guest you change the truth-value of three guests.
Now, in the first observation we saw that if a pair ordered the same as each other and the neighbours of a pair ordered the same as each other, then that pair has the same truth-value. But now we also know that you can't change the dish of one without changing the truth-value of both that guest and his neighbour, so the pair will have the same truth-value even if they have a different dish!
Now, we have learned that if a pair's neighbour ordered the same thing, that pair will definitely have the same truth value. And it turns out this observation is particularly valuable when dealing with this tpye of problem. For instance, we can observe that Ellen and Daniel are sitting next to each other, and we also know that they have different truth-values. If their neighbours ordered the same thing, then Ellen and Daniel must have had the same truth-value, but since that is not the case it must be that Conny and Fiona ordered different dishes! But, we still can't know what those dishes are, so how exactly is this information useful? Well, we do know that the only way a pair could have unmatching truth-values is if the pair's neighbour ordered different dishes, and since we know that Conny and Fiona ordered different dishes we can conclude that the guests between them, that is Billy and Anna, have different truth-values. But since we already know that Billy is a liar, Anna must be a truth-teller! And since Anna is a truth-teller, that must mean half the guests ordered meat and half of them ordered fish; this can be organized as three pairs of either a) Meat-Fish, Meat-Fish, Meat-Fish; or b) Meat-Meat, Fish-Fish, Meat-Fish. The way things are ordered in a and b we don't know who has what dish, or if the pairs are sitting next to each other. It's just neater to have things organized that way.
Let's assume Ellen is lying. That means she did order the same thing as Billy, and so Ellen and Billy makes up a matching pair. ( meat-meat or fish-fish ) Since any matching pairs does not appear in a, it must be the case of b. It also means that there will next be one matching pair and one unmatching pair. Let us review; in our assumption both Ellen and Billy are lying, that is they have the same truth-value. As a consequence they ordered the same dish. That must mean they are either both sitting between an unmatched or a matched pair; but we already established what we need is one of each. Otherwise it will contadict Anna's statement that there is an equal part meat or fish, which we know is true. Therefore it must not be true that Ellen is lying, and she is in fact telling the truth.
Since Ellen and Billy ordered different dishes, and the guests between two guests who ordered different dishes have different truth-values, Fiona and Anna must have different truth values. We know that Anna is telling the truth, o Fiona must be lying. The same thing can be applied to Daniel and Conny. Daniel accused Ellen of lying, but Ellen is in fact telling the truth. So Conny must in fact be telling the truth as well ( even though he didn't say anything. )
So now we know exactly who is telling the truth and who is lying. The list can be ordered as following:
Anna is telling the truth.
Billy is lying.
Conny is telling the truth.
Daniel is lying.
Ellen is telling the truth.
Fiona is lying.
Anna is telling the truth, which means there are three guests who ordered meat and three who ordered fish. That means the longest string of consecutive meat-dishes is three, but that would mean everyone who ordered meat would sit in a row, just like Fiona said. Since we know Fiona is lying, that can't be the case.
The longest string of consecutive meat-dishes is two or one. However, if the longest string is two, then that constitutes a pair. If at least one of the neighours ordered meat we would be talking about a string longer than two, so both neighbours did order fish. We already established that if a pair's neighbour ordered the same thing, the pair has the same truth-value. But we can see in our list that none of the guests have the same truth value as their neighbour, so it must be the case that the longest string of guests who ordered meat is at most three, but not three or two. That leaves a length of one.
Since this logic can be applied to fish-dishes as well, it means every other dish must be meat, and every other must be fish. That means there are only neighbours who ordered the same thing, and the ones who told the truth ordered meat, and the liars ordered fish. This gives us the list:
Anna ordered meat.
Billy ordered fish.
Conny ordered meat.
Daniel ordered fish.
Ellen ordered meat.
Fiona ordered fish.
So yes, Christine can figure out who ordered meat and who ordered fish, but one can't help but wonder if her talents can't be better applied elsewhere.
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