One chocolate bar, forty-eight squares, and an irresistible temptation to find the perfect sequence of breaks.
The Chocolate Bar
Riddle statement
You have a rectangular chocolate bar divided into 48 small squares.
In each move, you choose one existing piece and break it in two along a line between squares.
You may not stack pieces or break more than one piece at once.
What is the minimum number of moves needed to separate all 48 squares?
Show solution
Solution
Answer: exactly 47 moves are required.
At the beginning, the whole chocolate bar is a single piece.
Each move replaces one piece with two. The total number of pieces therefore increases by exactly one every time:
To finish with all 48 squares separated, there must be 48 pieces. If there are \(1+n\) pieces after \(n\) moves, then
Therefore,
This proves not only that the task cannot be completed in fewer than 47 moves. It proves the stronger statement that every valid sequence of breaks takes exactly 47 moves. The order, the shapes of the pieces, and the chosen strategy cannot change the answer.
Key idea: every move creates exactly one new piece; going from one piece to 48 requires creating 47 additional pieces.