Open-problem statements, with their sources attached.

The second Cuboid conjecture

Conjecture:* Are there infinitely many Leinster groups?

This asks whether there exist infinitely many (non-isomorphic) finite groups that are Leinster groups.

Formalized via the negation of "Does there exist an n such that all Leinster groups have order less than n".

The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite, $\aleph_0$ or $2^{\aleph_0}$.

The Gompertz constant $\delta$ is transcendental.

The third Cuboid conjecture

Gottschalk's surjunctivity conjecture* (1973): every group is surjunctive. That is, for every group G and every finite alphabet A, every injective cellular automaton on A^G is surjective.

$\Gamma(1/n)$ for n ≥ 2 is transcendental.

The Bing-Borsuk Conjecture: every $n$-dimensional homogeneous absolute neighborhood retract is a topological $n$-manifold. A topological space $X$ is an $n$-dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X). The hypothesis [MetrizableSpace X] implies T2Space X so this does not appear in the conclusion.

The Inverse Galois Problem: every finite group is isomorphic to the Galois group of a Galois extension of the rationals.

Any T2, Toronto space is discrete.

Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$.

Does the determinant of the sum $A + B$ of two $n \times n$ normal complex matrices $A$ and $B$ always lie in the convex hull of the $n!$ points $\prod_i (\lambda(A)_i + \lambda(B)_{\sigma(i)})$? Here the numbers $\lambda(A)_i$ and $\lambda(B)_i$ are the eigenvalues of $A$ and $B$, and $\sigma$ is an element of the symmetric group $S_n$.

Inscribed square problem* Does every Jordan curve admit an inscribed square?

Sendov's conjecture* states that for a polynomial $f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point.

There exists a Hadamard matrix for all $n = 4k$.

Inscribed rectangle problem* Does every Jordan curve admit inscribed rectangles of any given aspect ratio?

The smallest order for which no Hadamard matrix is presently known is $668 = 4 * 167$.

What is the smallest square that can contain 11 unit squares?

Reference: Wikipedia

What is the smallest square that can contain 17 unit squares?

Reference: Wikipedia

What is the smallest circle that can contain 3 unit squares?

Reference: Wikipedia