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Prove by strong induction that it will take exactly mn rate the chocolate bar into individual pieces. - 1 breaks to completely sepa-

Prove by strong induction that it will take exactly mn rate the chocolate bar into individual pieces. - 1 breaks to completely sepa-

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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You have a bar of fancy artisanal European chocolate divided into a grid of m × n pieces (minimum 1 piece), and you want to savor each piece individually. You must do so by breaking the chocolate bar along the grid lines, until all the pieces of chocolate are separated. You can only break a single connected block of chocolate at a time. For instance, the following each count as a single break for a 4 x 3 chocolate bar: Prove by strong induction that it will take exactly mn - 1 breaks to completely sepa- rate the chocolate bar into individual pieces. Hint: There are two variables and we can only induct on one quantity at a time. You'll need to induct on m+n. This means: • Your base case should be when m+ n = 2, since that's one piece of chocolate. • Your inductive hypothesis should consider when m+n≤k for some k. • Your inductive step should look at a chocolate bar where m+n=k+1, and carefully figure out how to split into smaller cases.
Transcribed Image Text:You have a bar of fancy artisanal European chocolate divided into a grid of m × n pieces (minimum 1 piece), and you want to savor each piece individually. You must do so by breaking the chocolate bar along the grid lines, until all the pieces of chocolate are separated. You can only break a single connected block of chocolate at a time. For instance, the following each count as a single break for a 4 x 3 chocolate bar: Prove by strong induction that it will take exactly mn - 1 breaks to completely sepa- rate the chocolate bar into individual pieces. Hint: There are two variables and we can only induct on one quantity at a time. You'll need to induct on m+n. This means: • Your base case should be when m+ n = 2, since that's one piece of chocolate. • Your inductive hypothesis should consider when m+n≤k for some k. • Your inductive step should look at a chocolate bar where m+n=k+1, and carefully figure out how to split into smaller cases.
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