User:Hakerh400/Conjectures

Here are some conjectures made by User:Hakerh400. At the time of writing this article, none of the conjectures are resolved.

Conjectures

Conjecture 1: Special sokoban map

Is there a finite sokoban level such that there does not exist a finite sequence of moves after which no box can be moved anymore?

This is conjectured to be false, but nobody has proved it yet. It has been asked in 2017. on MSE^[1]^[2]. MSE user Ingix^[3] has proved^[4] that there is no such map with exactly one box. However, the case with multiple boxes remains unsolved.

Conjecture 2: Traversing planar island

Is there a FSM that can traverse all tiles of any finite island on a planar square grid?

This is also conjectured to be false. The idea appears in an esolang that is published in 2020. It has been proved that there is a FSM that can traverse any island without holes. If the island contains holes, it is probably impossible, but not proved yet.

Conjecture 3: Bit spiral

Let S be the concatenation of binary representations of all natural numbers S="0 1 10 11 100 101 110 111 1000 ...". Let L be the Ulam spiral, but instead of having marked prime numbers, it has marked only numbers which index a 1 in S (1-indexed). Conjecture:

Does every finite binary string appear somewhere in a row in L reading from left to right? Does it appear in a column reading from top to bottom?

This is conjectured to be true. There were no attempts so far to formally prove it.

Conjecture 4

Informal: The task involves cleaning a whiteboard measuring 2x2 units as efficiently as possible. A very thin eraser, measuring 1 unit in length, may be positioned anywhere on the board to initiate cleaning. However, once placed, the eraser cannot be removed until the board is completely clean, meaning it must reach every point on the surface. The eraser can be freely translated or rotated, but each point of the eraser can move a maximum of one unit per time unit.
Conjecture: The board cannot be cleaned in less than 5 time units.

Formal: For all sets $X$ and $Y$ , let $X\mapsto Y$ denote the set of all functions from $X$ to a subset of $Y$ (subset is denoted by $\subseteq$ and proper subset is denoted by $\subset$ ). Let $\mathbb {R} ^{+}=\left\{x\mid x\in \mathbb {R} \land x\geq 0\right\}$ . Let $\mathbb {R} _{1}=\left\{x\mid x\in \mathbb {R} ^{+}\land x\leq 1\right\}$ . Let $A=\left\{\left\langle x,y\right\rangle \mid x\in \mathbb {R} \land y\in \mathbb {R} \land 0\leq x\leq 2\land 0\leq y\leq 2\right\}$ . Let $F_{0}\left(\left\langle \left\langle x,y\right\rangle ,\alpha \right\rangle ,k\right)=\left\langle x+k\cos \alpha ,y+k\sin \alpha \right\rangle$ . Let $S=\left\{\left\langle \left\langle x,y\right\rangle ,\alpha \right\rangle \mid \alpha \in \mathbb {R} \land 0\leq \alpha <2\pi \land \varphi _{1}\left(x,y,\alpha \right)\right\}$ where $\varphi _{1}\left(x,y,\alpha \right)\iff \forall k\in \mathbb {R} _{1}\left(F_{0}(\left\langle \left\langle x,y\right\rangle ,\alpha \right\rangle ,k)\in A\right)$ . Let $F=\left\{f\mid f\in \left(\mathbb {R} ^{+}\mapsto S\right)\land \varphi _{2}\left(f\right)\right\}$ where $\varphi _{2}\left(f\right)\iff \forall i\in \left\{1,2\right\},t_{i}\in \mathbb {R} ^{+},k\in \mathbb {R} _{1},x_{i},y_{i}\left(\left\langle x_{i},y_{i}\right\rangle =F_{0}\left(f\left(t_{i}\right),k\right)\to {\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}\leq \left|t_{1}-t_{2}\right|\right)$ .
Conjecture: $\forall f\in F,T\in \mathbb {R} ^{+}\left(\left(\forall p\in A\exists t\in \mathbb {R} ^{+},k\in \mathbb {R} _{1}\left(t\leq T\land F_{0}\left(f\left(t\right),k\right)=p\right)\right)\to T\geq 5\right)$ .