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Part 1. Take a sheet of paper and tear it into eight pieces. Choose one of the resulting pieces and tear it into eight pieces. Continue this process: choose any one of the pieces you have and tear it into eight smaller pieces.
Can we produce exactly 2014 pieces of paper using this method? Why or why not? |
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Part 2. Take a sheet of paper and tear it into either four pieces or six pieces (you choose). Continue this process: choose any one of the pieces you have and tear it into either four or six smaller pieces.
Can we produce exactly 2014 pieces of paper using this method? Why or why not? |
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Follow-up ChallengesNote that neither of the paper-ripping methods allows us to create exactly 5 pieces. Are there other impossible numbers of pieces that neither of our two methods can create? Are there some numbers of pieces that can be created with one method but not the other?
Suppose we change the procedure from Part 1 to be: "Start with a piece of paper. At each step, choose a piece and tear it into 4 pieces." Could we create exactly 2014 pieces with this method? What numbers are possible or impossible to create? What if the procedure were to choose a piece and tear it into 10 pieces? N pieces? Could we get exactly 2014 pieces? Are there other possible or impossible numbers? |
Response VideosReach out to share your solution and we'll include it here! :)
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