The Leftovers
3 friends, Anna, Brad, and Claire were asked to solve a problem. Here is the problem:
" You are given a very long rope. If you cut the rope at the middle, you will have 2 parts of the rope. If you fold the rope once, bringing one of the ends to the other, you will have 3 parts of the rope (1 of each end of the rope and the curved part). So, suppose that you have folded the rope 10 times in total, always bringing one part of the rope to the other, and cut it at the middle. How many pieces will you have?
All the friends were clueless as they were not good in Math or Science (very sad). But they tried to guess.
Anna said: "There seems to be an arithmetic progression, so the answer to this problem should be 2 + 10 x 1 = 12 , as for every fold, there is an extra 1 part you will get."
Brad said: "This should be based on logic, so the answer to this problem should be 3 , as no matter how many times you fold the rope, there will always be the 2 ends of the rope and the curved part between the 2 ends , which makes 3."
Claire said: "Hidden in this problem there seems to be a Fibonacci Sequence, so the answer to this problem should be the 10th Fibonacci number starting from 3, which is 233.
Who have guessed correctly?
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1 solution
Suppose you have folded the rope 2 times, there will be the 2 ends on a side, 1 curved part on the same side, and 2 curved parts on the other side (the 2 ends of the rope are connected to the curved part, so there are 2 curved parts on the other side). So when you cut at the middle, there will be altogether 5 parts of the rope.
Note: Try picturing the image in your brain. If not, you can always draw on a piece of paper to help you visualise!
Now suppose you have folded the rope 3 times, but remember, when we folded the rope 2 times just now and cut at the middle, there were 4 lines of the rope being cut. So if we continued folding them and they were not cut yet, there will be 4 curved parts on a side . The 2 curved parts from the previous case are now brought to the other side by folding. So now when you cut at the middle, there will be:
Altogether, there are 2 + 3 + 4 = 9 parts of the rope.
Let's look at the pattern for the number of folds to the number of parts:
See any pattern? Yes, the number of parts that we will get is related to the powers of 2! So, the formula is:
2^(The number of folds) + 1 = The number of parts (Note that 2^0 is 1, if you didn't know it before)
Now, we can easily solve the problem by substituting 10 into "The number of folds":
2^10 + 1 = The number of parts = 1024 + 1 = 1025
Goodness! The number of parts that we will get will be over a thousand! Seems like the 3 friends's guesses were not accurate, so the answer is None of them.
Hope you liked my explanation!