Depth in Bingo Closure

TL;DR

This paper analyzes the closure process in Bingo on an n×n grid, determining the maximum number of steps needed to reach closure from any initial set of marked squares.

Contribution

It establishes the maximum number of steps required in the Bingo closure process for an n×n board, extending previous understanding from the 5×5 case.

Findings

Maximum steps for n×n Bingo closure

Closure process complexity analysis

Generalization from 5×5 to n×n boards

Abstract

Bingo is played on a $5\times 5$ grid. Take the 25 squares to be the ground set of a closure system in which square $s$ is dependent on a set $S$ of squares iff $s$ completes a line - a row, column, or diagonal - with squares that are already in $S$. The closure of a set $S$ is obtained via an iterative process in which, at each stage, the squares dependent upon the current state are added. In this paper we establish for the $n \times n$ Bingo board the maximum number of steps required in this closure process.