# “Pickled Walnuts” Logic Problem

Here's a problem by puzzle composer Hubert Phillips called “Pickled Walnuts”. What conclusions (if any) can be drawn about armadillos and picked walnuts, assuming all statements are true?

1. Pickled walnuts are always provided at Professor Piltdown’s parties.
2. No animal that does not prefer Beethoven to Mozart ever takes a taxi in Bond Street.
3. All armadillos can speak the Basque dialect.
4. No animal can be registered as a philatelist who does not carry a collapsible umbrella.
5. Any animal that can speak Basque is eligible for the Tintinnabulum Club.
6. Only animals that are registered philatelists are invited to Professor Piltdown’s parties.
7. All animals eligible for the Tintinnabulum Club prefer Mozart to Beethoven.
8. The only animals that enjoy pickled walnuts are those who get them at Professor Piltdown’s. (You can assume he's referring to Professor Piltdown's parties.)
9. Only animals that take taxis in Bond Street carry collapsible umbrellas.
• Can armadillos go to the parties without being invited? Commented Sep 8, 2023 at 7:50
• Good question. That would be a problem however it's been assumed that all must have an invitation. Commented Sep 8, 2023 at 12:40

From these statements, we can conclude that

Armadillos do not enjoy pickled walnuts

This can be determined by:

Using the following chain of logic from the following statements, sometimes using De Morgan's theorem

I'm using ">" to mean "implies"

enjoys pickled walnuts > invited to party (8 and 1?)
invited to party > philatelist (6)
philatelist > has collapsible umbrella (4)
has collapsible umbrella > takes taxis in Bond streets (9)
takes taxis in Bond streets > prefers Beethoven to Mozart (2)
prefers Beethoven to Mozart > NOT eligible for Tintinnabulum club (7)
NOT eligible for Tintinnabulum club > doesn't speak Basque (5)
doesn't speak Basque > not armadillo (3)

This gives us:

enjoys pickled walnuts > not armadillo (or, using contraposition, armadillo > DOESN'T enjoy pickled walnuts)

Here is my take on it. In a more human-readable form and, well, I don't get the same conclusion.

3. All armadillos can speak the Basque dialect.
=> ok, armadillos can speak Basque

5. Any animal that can speak Basque is eligible for the Tintinnabulum Club.
=> armadillos are eligible for the club

7. All animals eligible for the Tintinnabulum Club prefer Mozart to Beethoven.

2. No animal that does not prefer Beethoven to Mozart ever takes a taxi in Bond Street.
i.e. animals that prefer Mozart never take a taxi in Bond Street
=> armadillos never take a taxi there

9. Only animals that take taxis in Bond Street carry collapsible umbrellas.
i.e. animals that do not take taxis there do not carry such umbrellas
=> armadillos do not carry collapsible umbrellas

4. No animal can be registered as a philatelist who does not carry a collapsible umbrella.
i.e. an animal that doesn't carry such an umbrella cannot register

6. Only animals that are registered philatelists are invited to Professor Piltdown’s parties.
=> armadillos are not invited to the parties

and

8. The only animals that enjoy pickled walnuts are those who get them at Professor Piltdown’s [parties].
=> armadillos who enjoy pickled walnuts get them at these parties

The conclusion is:

Armadillos who enjoy pickled walnuts get them at Professor Piltdown’s parties without being invited.

And you can note that sentence 1 is not necessary to reach that conclusion.

• Your conclusions for 8 is what I question. => "armadillos who enjoy pickled walnuts get them at these parties" I interpret this as: IF there exists an animal that enjoys pickled walnuts THEN he gets them at PP's parties. But we don't know that such an animal exists. So you could say "IF there are Armadillos who enjoy pickled walnuts they get them at Professor Piltdown’s parties without being invited." However this would be a weak conclusion. Commented Sep 9, 2023 at 7:36
• Just as the "If... " is implicit in sentence 8 it is implicit in the conclusion. The conclusion indeed doesn't say such armadillos do exist. Commented Sep 9, 2023 at 13:39
• The conclusion given by samm82 is "Armadillos do not enjoy pickled walnuts" or "IF armadillos exist THEN they do not enjoy pickled walnuts." But we know that armadillos exist but do not know that an animal that enjoys pickled walnuts exists. So samm82 would have the stronger conclusion. Commented Sep 9, 2023 at 18:05
• Well, we know that armadillos exist in real life, but we also know that armadillos don't speak basque in real life. If we rely purely on the information given in the statement, we cannot tell armadillos exist. And regarding my conclusion being weaker, this is because sam82 assumes that you need to be invited to get pickled walnuts from the parties, an assumption I do not take as a given. I wrote my answer before you commented that it is the case. Commented Sep 9, 2023 at 18:24
• This problem was poorly worded but it's not mine. Maybe some others would care to comment here... Commented Sep 9, 2023 at 22:18