Induction: Breaking Chocolate

You have a delightful bar of chocolate that's divided into 50 squares, arranged in a 5x10 grid. You can only break the bar along straight lines that go from one edge to the other, and stacking pieces to break multiple at once is not allowed. What is the minimum number of breaks you need to divide the bar into 50 individual squares?

Answer

To determine the minimum number of breaks needed to divide the chocolate bar into 50 individual squares, we start by looking at smaller cases. For 2 squares, 1 break is needed; for 3 squares, 2 breaks are required. This pattern shows that for $$ n $$ squares, the minimum number of breaks needed is $$ n - 1 $$ .

Therefore, for 50 squares:

\[ b(50) = 50 - 1 = 49 \]

Notes and comments

Comment: By observing the pattern from smaller numbers of squares, we can generalize that dividing a chocolate bar into $$ n $$ individual squares requires $$ n - 1 $$ breaks.

Comment 2: Confirm this choice of $$49$$ by brute force calculations.