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p.357, icon at Example 2 #1. Consider an infinite checkerboard of squares, where all squares are white other than an initial set Bo of n black squares; we call Bo the initial generation of black squares. We define new generations of black squares recursively. Subsequent generations of black squares B1, B2, ... are defined by the rule that a square is in B₁ if and only if at least two of this square itself, the square directly above it, and the square directly to its right are in B-1. That is, a square on the checkerboard is in a new generation of black squares, if in the previous generation of black squares, there are more black squares than white squares among the square itself, the square above it, and the square to its right. Use strong induction to prove that B₁ = 0, that is, after n steps (where n is the number of initial black squares), no squares are black.

p.357, icon at Example 2 #1. Consider an infinite checkerboard of squares, where all squares are white other than an initial set Bo of n black squares; we call Bo the initial generation of black squares. We define new generations of black squares recursively. Subsequent generations of black squares B1, B2, ... are defined by the rule that a square is in B₁ if and only if at least two of this square itself, the square directly above it, and the square directly to its right are in B-1. That is, a square on the checkerboard is in a new generation of black squares, if in the previous generation of black squares, there are more black squares than white squares among the square itself, the square above it, and the square to its right. Use strong induction to prove that B₁ = 0, that is, after n steps (where n is the number of initial black squares), no squares are black.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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p.357, icon at Example 2 #1. Consider an infinite checkerboard of squares, where all squares are white other than an initial set Bo of n black squares; we call Bo the initial generation of black squares. We define new generations of black squares recursively. Subsequent generations of black squares B1, B2, ... are defined by the rule that a square is in B₁ if and only if at least two of this square itself, the square directly above it, and the square directly to its right are in B-1. That is, a square on the checkerboard is in a new generation of black squares, if in the previous generation of black squares, there are more black squares than white squares among the square itself, the square above it, and the square to its right. Use strong induction to prove that B₁ = 0, that is, after n steps (where n is the number of initial black squares), no squares are black.
Transcribed Image Text:p.357, icon at Example 2 #1. Consider an infinite checkerboard of squares, where all squares are white other than an initial set Bo of n black squares; we call Bo the initial generation of black squares. We define new generations of black squares recursively. Subsequent generations of black squares B1, B2, ... are defined by the rule that a square is in B₁ if and only if at least two of this square itself, the square directly above it, and the square directly to its right are in B-1. That is, a square on the checkerboard is in a new generation of black squares, if in the previous generation of black squares, there are more black squares than white squares among the square itself, the square above it, and the square to its right. Use strong induction to prove that B₁ = 0, that is, after n steps (where n is the number of initial black squares), no squares are black.
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