Intereting Posts

Prove this limit without using these techniques, and for beginner students: $\lim_{x\to0} \frac{e^x-1-x}{x^2}=\frac12$
How to solve the equation $3x-4\lfloor x\rfloor=0$ for $x\in\mathbb{R}$?
Solving a set of recurrence relations
Logistic function passing through two points?
irreducibility of polynomials with integer coefficients
Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is…
product of densities
Computing $ \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}$
Show that $\inf(\frac{1}{n})=0$.
Given $XX^\top=A$, solving for $X$
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
Why does this distribution of polynomial roots resemble a collection of affine IFS fractals?
Sum of these quotient can not be integer
How to find a total order with constrained comparisons
Least squares and pseudo-inverse

I enjoy the card game Set, and have come up with a few variants based on the concept of assigning card “values” to stacks of cards (that is, each stack of cards is considered equivalent to a particular card) as follows:

- A stack containing a single card is equivalent to that card
- A stack containing multiple cards is equivalent to the third member of the set containing the top card and the card-equivalent of the rest of the stack (unless the rest of the stack is equivalent to the top card, in which case the whole stack is equivalent to the top card). This works because there is exactly one set containing any given pair of cards.

While there are a couple of variants, the basic idea is as follows: the top two cards of the deck are played into a stack, and players attempt to add cards to the stack so as to make the entire stack equivalent to the bottom card of the stack.

For example (I represent cards as $4$-tuples of $\{a,b,c\}$ signifying three options for four properties):

- Order of conjugate of an element given the order of its conjugate
- What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?
- Give an example of a nonabelian group in which a product of elements of finite order can have infinite order.
- Criterion for being a simple group
- $p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
- If I know the order of every element in a group, do I know the group?

If the starting stack (from the bottom up) is $(a,b,c,c,),\;(a,a,a,a)$ (equivalent to card $(a,c,b,b)$) one possible winning play would be

- $(b,a,a,b)$ (making the stack equivalent to $(c,b,c,a)$)
- $(b,b,c,c)$ (making it equivalent to $(a,b,c,b)$)
- $(a,b,c,a)$ (making it equivalent to $(a,b,c,c)$)

My question is this: given a version of this game where cards have $n$ properties (for a standard Set deck $n=4$) and the deck contains exactly one copy of each possible card, how many cards do you have to draw to guarantee that your hand contains at least one solution for any starting stack of $2$ cards. It’s at least $3^{n-1}+1$ since if your starting stack was $(a,\dots),\;(b,\dots)$, you could draw all $3^{n-1}$ cards with $(c,\dots)$ and still have no solution, but drawing any additional card is then guaranteed to produce a solution. This feels like the worst case to my intuition, but I’m not sure how to show that, or even if it’s correct.

**Note**: I’ve asked related questions before here and here, though there I tried to be more abstract, treating the cards as elements of a magma.

- On solvable quintics and septics
- Characterizing all ring homomorphisms $C\to\mathbb{R}$.
- Why are two permutations conjugate iff they have the same cycle structure?
- If $$ is finite, then $ = $ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
- Is the product of closed subgroups in topological group closed?
- Decomposition of polynomial into irreducible polynomials
- Transcendence degree of $K$
- If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$
- Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible
- Show that any prime ideal from such a ring is maximal.

- Are there any decompositions of a symmetric matrix that allow for the inversion of any submatrix?
- Find the values of $p$ such that $\left( \frac{7}{p} \right )= 1$ (Legendre Symbol)
- Is it possible to write the curl in terms of the infinitesimal rotation tensor?
- Differentiable Manifold Hausdorff, second countable
- Category of Field has no initial object
- Find: The expected number of urns that are empty
- Hahn-Banach to extend to the Lebesgue Measure
- How to compute the $n_{th}$ derivative of a composition: ${\left( {f \circ g} \right)^{(n)}}=?$
- Finding a (small) prime great enough that there are at least m elements of order m
- Number of permutations which are products of exactly two disjoint cycles.
- Simplifying an expression $\frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)}$ if we know $x+y+z=0$
- Interpolating the primorial $p_{n}\#$
- Proving $(A\le B)\vee (B\le A)$ for sets $A$ and $B$
- A Curious binomial identity
- Three pythagorean triples