7 problems

A $1 \times 1$ Rush Hour instance has labeled pieces that move only horizontally or only vertically; Subway Shuffle is the corresponding colored-token puzzle on a graph. The comple…

Hypercup minimum-moves conjecture. The minimum number of moves is $r$, where $j_\sigma$ is the smallest nonnegative integer such that $r\in\mathbb{Z}$. This conjecture proposes an…

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.

For an $m \times n$ Numbrix puzzle, call a set of clues defining exactly one puzzle a defining clue set, and let the minimum number of clues be the smallest size of such a set. Min…

Let an $m \times n$ Numbrix puzzle be a puzzle on an $m$-by-$n$ board, and let the minimum number of clues required to define such a puzzle be the quantity bounded above in the pap…

Nonexistence conjecture. There are no such puzzles with only $11$ two-letter words.