Zodiac Signs at the Table

Question

"And we find ourselves at another one of Ophiuchus' little themed puzzle dinners," you say while glancing at the other seven people.

"I really like its puzzle dinners, honestly," Sagittarius admits, "though I was never good at figuring out its secret themes in the past."

"One just needs to be observant," Gemini smiles. "Ophiuchus always designs the seating plan to fit the theme. Speaking of which, we did not receive a fixed seating plan this time, did we?"

"Nope. This time, Ophiuchus decided to make the seating plan a puzzle as well itself. Ophiuchus does like adding puzzles to its already puzzling dinners." It is well-known among the group that Pisces is not a fan of puzzles. "As if guessing the theme of the dinner is not difficult enough. Speaking of the seating plan, what are the rules again?"

"Wait, I'll remind everyone of the rules," Virgo read the email Ophiuchus sent to everyone an hour earlier. "Here they are..."

1. There is one person sitting right between Gemini and Sagittarius.
2. Cancer sits right next to Libra.
3. Libra does not sit right next to Virgo.
4. Sagittarius does not sit on the immediate right of Aries.
5. Pisces sits directly across from Libra.
6. Gemini sit directly across from Virgo.

"And you," Virgo turns to you with eyes sparkling...

7. You are sitting on my immediate right.

Aries is the fastest solver among the group. "I can work out two solutions. Any preferences which one we should go for?" it asks, while proudly showing its solutions on a piece of napkin.

"I prefer the side further from the fireplace," Cancer mutters. "After all, our group tends to get too warm very quickly."

---

"Weird theme, isn't it?" Virgo asks after delicately pulling in its chair. "I wonder what Ophiuchus thinks we eight have in common..."

You politely nod. Your eyes glance from left to right. How does this seating plan reflect the theme...?

"Why are there only eight people here in this party?" you overhear Cancer asking Libra. "I know Ophiuchus never attends these parties it organizes, but what about the other four?"

"Well, two of them were never even invited in the first place," Libra replies. "It is a bit mean, but Ophiuchus told them they do not fit the theme."

"And what about the last two who did get invited?" Cancer inquires.

"Oh, those two turned down the invitation very early. One of them already had plans to go diving in the sea it lives close to. The other is the ruling zodiac sign tonight, so it is too busy sorting out the paperwork and meeting up with the other gods. Surprisingly, Ophiuchus is quite pleased about all this." Libra's grin grows larger. "Ophiuchus told me this fits the theme even better, and brings an element of balance to the table, which as you know, is something I cannot complain about."

You stand up abruptly. Seven pairs of eyes stare back at you.

"I know the theme of this dinner party," you smugly announce.

---

Question: Who is the ruling zodiac sign tonight?

Answer 1

Like @WeatherVane, I contemplated various solutions for the 8 guests on round and square tables but kept reaching contradictions. However, I was able to find a solution when considering that...

...there could be two empty places for the two guests who were also invited but have not come!

I have also then worked out most of the context for the puzzle, namely that:

1. Taurus is the ruling zodiac sign tonight,
2. The narrator (me) is Scorpio,
3. The theme for the seating arrangement is the Periodic Table.

However, with my solution it very much seems like the seating arrangement is a completely separate puzzle to the resolution of the theme - so there may exist a way to link them both a bit better.

To explain the seating arrangement then:

Step 1:

Consider a round table with ten places set, for all ten invited guests, including the two no-shows. Let's place the three knowns: Virgo and Gemini opposite each other, and 'me' to Virgo's right.

Step 2:

Consider all of the places that other guests could sit, to fulfil the other rules considered one at a time:

Importantly there are two spaces where Aries (underlined) can sit but only if Sagittarius is not sat in the adjacent seat to the right.

Step 3:

Importantly, this means that Sagittarius...

cannot sit two spaces anticlockwise from Gemini. The only potential occupants for the space in between them and Gemini would be Aries (who cannot sit that side of Sagittarius) and Cancer (who would then not be sat next to Libra, as is required by the rules). So Sagittarius must instead sit two spaces clockwise from Gemini:

Step 4:

Using the facts that Cancer must sit next to Libra, and Libra and Pisces must be opposite each other, we can narrow down a lot more options:

This now leaves just one possible place for Pisces and Libra to occupy:



And finally, we use the information that the seat between Gemini and Sagittarius must be occupied, to place Aries there:

This leaves Cancer with a choice of two spaces where to sit, as per all of the rules. Which means the fireplace must be behind either:

Sagittarius or 'me'. Right now it's hard to tell who exactly...

Which brings us to start considering the theme of the puzzle...

Explaining the theme:

There are a few hints that this seating arrangement is based on:

The Periodic Table - notably Libra's telling remark about bringing "an element of balance to the table".

Moreover, two of the zodiac signs were not invited at all. These would be:

Aquarius and Leo.

Why? Because...

...their names are the only ones which do not begin with the symbol of an element in the Periodic Table!

Aries (Argon)
Cancer and Capricorn (where one will be Carbon (C) and the other Calcium (Ca))
Gemini (Germanium)
Libra (Lithium)
Pisces (Phosphorus)
Sagittarius (Sulfur)
Scorpio (Scandium, since Sulfur is already taken)
Taurus (Tantalum)
Virgo (Vanadium)

There is no 'A' or 'Aq' element for Aquarius, and no 'L' or 'Le' element for Leo.

Removing these two uninvited zodiac signs leaves us with three to assign as 'me' and the two invited no-shows:

Capricorn, Scorpio and Taurus.

This is where we can use the extra information to differentiate between them... Let's take a look at the following image that might help:

Those in yellow are definitely at the table, two of the four in shades of orange are not. But who?

I would like to suggest that Taurus is the ruling sign tonight, as its elemental equivalent Tantalum is named after the mythological figure Tantalus, a son of Zeus, which would explain why it is so appropriate that it is 'meeting up with the other gods'.

Meanwhile, Libra (the scales, of course) is pleased about the two omissions leaving everything so nicely 'balanced'. I would suggest this means that of the two possibilities at the very edge or shore of the Periodic Table (i.e. by 'the sea') - Scandium and Carbon - it should be Carbon (i.e. Capricorn) that didn't make the event, since then we are left with four elements on the left and four on the right, which would balance nicely indeed!

This would make Cancer's equivalent element Calcium, and the narrator is left as Scorpio.

Like I say, it appears that my solution to the second part (which fits very nicely indeed) isn't really all that connected to the precise ordering of the diners around the table. It may be that I have missed something in the logical deduction of the table plan that does link the two more closely - perhaps someone else can point out what I may have missed!

Answer 2

Partial solution.

Attempt 1: logical deduction.

Considering a circular table
6. Gemini sit directly across from Virgo.
7. [Virgo says] You are sitting on my immediate right.



1. There is one person sitting right between Gemini and Sagittarius.
So there are two possible places for Sagittarius



3. Libra does not sit right next to Virgo.
So there are four possible places for Libra



5. Pisces sits directly across from Libra.
So Libra cannot be opposite Saggitarius or You
Removing Libra from three places, and adding Pisces opposite



2. Cancer sits right next to Libra.
Which also places Saggitarius



There is one empty chair, for Aries



4. Sagittarius does not sit on the immediate right of Aries.
But Sagittarius is sitting on the immediate right of Aries.

So a circular or octagonal table is impossible.

But there are other possible arrangements:
a) A circular or octagonal table
b) A square table with two seated on each side
c) A rectangular table with one or three seated on each side
d) A long rectangular table with four seated on each side

I followed a similar logic for these, and found one solution for (c) and apparently several solutions for (b), and didn't look at (d) until later. But I got into a terrible muddle, and there was a lot of explaining to do. Starting the logic with Virgo, there are two places it can sit at a square table, and four places at a rectangular table (but only one at the circular table).

So the second attempt is a program solution.

Attempt 2: computer code.

I wrote a C program to permute the dinner guests for each of the four types of table.
I found these results (no solution for the circular table, or 4-per side):

 Square table
       You Vir
    Can       Sag
    Lib       Pis
       Ari Gem

 Square table
       You Vir
    Ari       Sag
    Lib       Pis
       Can Gem

 Rectangular table
       Pis Sag You
    Gem           Vir
       Lib Can Ari

But Aries only found two solutions.

Looking at this wording more carefully -
2. Cancer sits right next to Libra.
In the middle solution of my three, that's not true. Cancer is sitting next to Libra, but on a different side of the table: not right next to Libra.

So the first and third are the two solutions.
At a square table:



At a rectangular table:

Answer 3

Regarding the shape of the table:

the table is rectangular with 4 seats on each side, because

Aries said there are two solutions, suggesting only 2-fold rotational symmetry, and

Cancer prefers the side further from the fireplace. The use of the comparative adjective suggests only 2 sides.

Admittedly, the above doesn't rule out a rectangular table with 3 seats on each long side, and 1 seat on each short side. However, the analysis is significantly more complicated in this case (since "diagonal-right" can also be "right" and "across the length of the table" can also be "across"), which I guess is not intended.

The following are the zodiacs, in order, prefixed with an asterisk if it is definitely included in the table (note: "You" doesn't have an asterisk yet), sometimes with a mapping. Justification of some mappings are shown later (under corresponding special symbols). The idea is mainly inspired by @Stiv.

The idea that the mapping is related to the periodic table comes from the fact that there are 10 invited zodiacs, and that exactly 10 zodiacs have names with initials uniquely assignable to symbols in the periodic table, and all the definitely invited zodiacs' names satisfy this rule.

*Aries = Ar = Argon
Taurus = Ta = Tantalum
*Gemini = Ge = Germanium
*Cancer = C = Carbon#
Leo
*Virgo = V = Vanadium
*Libra = Li = Lithium
Scorpio = Sc = Scandium
*Sagittarius = S = Sulfur
Capricorn = Ca = Calcium
Aquarius
*Pisces = P = Phosphorus

Justification of "#":

Because Cancer said that "our group tends to get too warm very quickly", and Carbon is usually flammable. Also, Cancer wanted to avoid the fireplace, which usually burns coal which is primarily carbon.

So the 2 uninvited zodiacs that didn't fit the theme are

Leo and Aquarius.

So we need to distinguish "You" vs the 2 who turned down the invite, among the 3 of

Taurus, Scorpio, and Capricorn.

The zodiac which had plans to go diving in the sea it lives close to is

Scorpio, because it is the only one of these 3 that has zodiac element (a.k.a. triplicity) water.

The identity of "You" needs to bring an element of balance to the table, so

You = Capricorn, because then the attendance of the zodiacs are periodic manner, i.e. observe the asterisks in the table earlier would be *_**_**_**_*, so every 3rd zodiac doesn't attend.

To be honest, I'm not satisfied with this idea, because if both Scorpio and Taurus attended, then the asterisks are still periodic (every 6th zodiac doesn't attend)

So finally the ruling zodiac is

Taurus

Notice how I haven't solved the seating plan yet (I will do so below). The seating plan seems completely independent of the ruling zodiac, which seems unintended. The solution path is

First notice that V is most restricted. Since there is rotational symmetry, we may first assume it is on the bottom row. After solving, we can then produce the rotationally symmetric solution. So by rule 6, Ge is on the top row. Then apply rules 7 and 1 to arrive at:

Ge/S | ? | Ge/S | not Ge
? | ? | ? | not V

Then notice by rule 2 that C and Li can only be in the bottom row. So C, Li, V, Ca are in the bottom row, and S, P, Ge, Ar are in the top row. Then we arrive at the following (note: rule 4 is not violated because the top row's zodiacs face downwards when seated):

Ge | Ar | S | P
V | Ca | C | Li