Murphy has a rope cutting puzzle. When he folds a rope once and cuts it in the middle, it becomes 3 pieces. Let's understand this pattern and solve for more complex cases.
To solve this systematically, we need to understand the pattern. Each fold doubles the number of layers. After n folds, we have 2 to the power of n layers. When we cut through all layers, each cut creates additional pieces equal to the number of layers.
Let's solve the first case. When Murphy folds the rope 3 times, he creates 8 layers. Making 2 cuts through all 8 layers means each cut adds 8 new pieces. Starting with 1 rope plus 16 additional pieces gives us 17 total pieces.
For the second case, Murphy folds the rope 4 times, creating 16 layers. Making 3 cuts through all 16 layers means each cut adds 16 new pieces. Starting with 1 rope plus 48 additional pieces gives us 49 total pieces.
In summary, the formula is: total pieces equals 1 plus cuts times 2 to the power of folds. For Murphy's problems: 3 folds with 2 cuts gives 17 pieces, and 4 folds with 3 cuts gives 49 pieces. This pattern works because folding doubles the layers each time, and cutting goes through all layers simultaneously.