## Senior Theses

2019

Aaron Williams

Jackson Liscombe

#### Abstract

This thesis is a study on games and puzzles, specifically that of the paper-and-pencil variety. Our examination has a focus on the puzzle Sto-Stone, created by Nikoli. A Sto-Stone puzzle consists of an m-by-n grid with its squares partitioned into smaller subgrids or rooms. These subgrids often have a number associated with them which is indicated in a cell within the room. The solver shades in the associated number of squares in the room making stones. Stones are subgrids based on orthogonal connectivity. The goal is to shade squares and create stones so that (a) each room contains exactly one stone, (b) stones do not cross between rooms, (c) numbered rooms contain one stone with exactly the indicated number of squares, and (d) when the stones “drop” they perfectly fill the bottom half of the grid. This final rule (d) is shown to have variations called Sto-Silt where (d) is ignored and Sto-Sand where (d) is weakened. We will analyze some of the puzzles published by Nikoli and puzzle fans, as well as my own. After discussing these puzzles, the backtracking algorithm is introduced along with our Python implementation of this algorithm, which is used to solve Sto-Stone puzzles. Then, we see the results that came from this implementation. We conclude our study of Sto-Stone by proving its hardness with a reduction of planar monotone rectilinear 3SAT, an NP-complete decision problem, to SILT. SILT is then reduced to SAND and SAND to STONE. Finally, we state our theorem: Determining if a Sto-Stone puzzle is solvable is NP-Complete.

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