Property With Specific Properties

Question

I am thinking of making a game (for the mathematicians that study numbers) in which players try to construct a set with certain properties. Which properties satisfy the following:

Formally, every finite set with the property has a finite super-set that satisfies it, and one that does not satisfy it.

Also it should not be frequent. Formally:

If for any set, one element can be added to make it satisfy the property, it is a bad property.

I think I would need at least two such properties. It does not matter which number system you chose (as long as it is common enough.) I can shape the game around it.

Note: I might make cards with different properties on it, and each player is tring to satisfy a different property.

Note 2: Also, with possibly modified requirements, I could make something like Ghost.

Answer 1

That game sound really cool. Have you considered a number theory style of Mao? But with only number theoretic rules being valid? Of course under the assumption that each card takes on it value and ace,jack,queen,king being $1,11,12,13$ respectively? You could also let the sign of the card change the value of it in some conceiled way.

Or make your own cards with different numbers. The Idea is you need to look at the number theoretic properties of the working numbers and make assumptions from there.

Answer 2

Let me start the list with my favorite property-with-super-set property:

The product of the set's numbers $+1$ is a prime.

This works because for any $n \in \Bbb N$ there are infinitely many primes $\equiv 1 \bmod n$, a fact that I think isn't loved enough.