15 problems

Let $ET(n)$ denote the number of equilateral triangles in $\{0,1,\ldots,n\}^3$, and let $\zeta=n-1$. For each $n$, write the associated counting polynomial as $p_n(\zeta)$. Cubic-i…

For fixed indices $p,k\geq1$, let $n_p(k)$ denote the proposed disk numbers whose packings are arranged in a matrix indexed by $p$ and $k$. A row class is obtained by fixing $p$ an…

Let $\Delta(k)=\frac{k(k+1)}{2}$, and let $\delta(n)=L(n)-t(n)$, where $L(n)=1/d(n)$ and $t(n)$ is the corresponding triangular-packing baseline used in the paper. Consider the inf…

Let $d(n)$ be the largest possible minimum distance between the centers of $n$ equal disks packed in an equilateral triangle. For an infinite class $n=n(1),n(2),\ldots$ of disk num…

Let $\Delta(k)=\frac{k(k+1)}{2}$, and consider packings of equal nonoverlapping disks in an equilateral triangle. Let $d(n)$ denote the largest possible minimum distance between th…

Let $R=(5,8,7)$ be the triangular tile with the side of length $7$ facing the angle $\pi/3$, and let $T$ be an equilateral triangle. Minimality conjecture. The construction in the…

Possible tile-count conjecture. The possible values of $N$ are exactly

Let $(a,b,c)$ be the side lengths of the triangular tile, and say that a positive real number $X$ is equiconstructible by $(a,b,c)$ when an equilateral triangle with side length…

Let $T_6(n)$ denote the maximum number of equilateral triangles determined by a set of $n$ points in $\mathbb{R}^6$. Erdős's conjecture. Any set of $n$ points in $\mathbb{R}^6$ can…

Conway–Soifer conjecture. The triangle $T$ cannot be covered by $n^2+1$ unit equilateral triangles.

Nonexistence conjecture.

Triangular-row conjecture. For most $n$, there is an optimal configuration with $N$ rows. The $j$th row has $j$ points for $j\leq J$. If $n>N(N+1)/2$, the rows with $j>J$ each have…

Abrego–Fernández conjecture. This construction is asymptotically optimal, and

Four-dimensional tetrahedron-extension conjecture. Every equilateral triangle in $\mathbb Z^4$ can be extended by a fourth point of $\mathbb Z^4$ to a regular tetrahedron in…

Exhaustive parametrization conjecture. Every such solution is given, for some $x,y,z,t\in\mathbb Z$, by