Question

Assume that a chocolate bar consists of n squares arranged in a rectangular pattern. The entire bar, a smaller rectangularpieceofthebar,canbebrokenalongavertical orahorizontallineseparatingthesquares.Assumingthat only one piece can be broken at a time, determine how manybreaksyoumustsuccessivelymaketobreakthebar into n separate squares. Use strong induction to prove your answer.

Answer 1

Step-by-step explanation:

Claim:

it takes n - 1 number of breaks to break the bar into n separate squares for all integers n.

Basic case -> n = 1

The bar is already completely broken into pieces.

Case -> n ≥ 2

Assuming that assertion is true for all rectangular bars with fewer than n squares. Break the bar into two pieces of size k and n - k where 1 ≤ k < n

The bar  with k squares requires k − 1 breaks and the bar with n − k squares

requires n − k − 1 breaks.

So the original bar requires  1 + (k−1) + (n−k−1) breaks.

simplifying yields,

1 + k − 1 + n − k − 1

1 - 1 + n - 1

n - 1

Therefore, we proved as we claimed that it takes n - 1 breaks to break the bar into n separate squares.