# 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 map such that there does not exist a finite sequence of moves after which no box can be moved any longer?*

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

(Removed)