14 problems

Ben-Or–Linial conjecture. Any Boolean function $f:\{0,1\}^n\to\{0,1\}$ with $\mathbb{E}(f)=1/2$ satisfies

Subset-sum conjecture. For any subset $\mathbb{T}\subset\mathbf{X}^D_K$ with $|\mathbb{T}|=2^D\times n$ elements,

Cumulative-XOR conjecture. After substitution, the cumulative XOR yields a string of ones across all paths or dimensional components, regardless of the subset size or dimension.

Orthotope sum conjecture. For every orthotope $\mathcal{O}\subset\mathbb{T}$,

Transformation sum conjecture. For any valid subset $\mathbb{T}\subset\mathbf{X}^D_S$,

Monotonicity conjecture. For given values of $D$ and $K=n+1$, the minimum number of subsets of different sizes in $\mathbf{X}^D_K$ is at least equal to the number of subsets of dif…

Structural conjecture. Every minimum universal set $M$ of 12 cubes satisfies the following rules:

Uniqueness conjecture. The 10 minimum universal sets are the only collections of 12 cubes that can build all 30 target cubes.

A threshold function is a map $f:\Sigma^N\to\Sigma$, where $\Sigma$ is the binary state space. A function is strongly neutralizable when it has the property that its associated Isi…

Let $d_n$ denote the discrepancy associated with the grid of size $n$. Power-index conjecture. Let $a$ be an odd natural number. If $a$ is not divisible by $21$, then for all…

Let $d_n$ denote the discrepancy associated with the grid of size $n$. The discrepancy-2 conjecture. There are infinitely many $n$ such that $d_n=2$. In particular, for all…

Let $\mathrm{SH}[k]$ denote the firing squad synchronization problem for squares with $k$ holes, and let a minimal-time solution mean a finite automaton that synchronizes every nod…

The scout-number conjecture. For $d\geq 3$, any effective scout protocol on $\mathbb{Z}^d$ requires at least $d+1$ scouts.

A pizza is a circular arrangement of pieces with positive sizes, and Alice and Bob alternately eat pieces according to the rules of the pizza-eating game. Winkler's four-ninths con…