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September 22, 2019

Warm-up

       Let’s get your brain warmed up for our Problem with a challenge.  Lisa and her friends are standing in a circle in the schoolyard, getting ready to play a game.  Each child in the circle has two neighbors, one on either side, and it just so happens that the neighbors of each child always have the same gender.  There are 6 girls in the circle, and at least one boy. How many boys are there?

Problem

A farmer with a gigantic apple orchard has hired you to pick apples for him.  He has given you 8 boxes, which happen to be arranged in a circle:

Your job is to place apples in the boxes, making sure that there are no empty boxes.  Easy enough, but unfortunately for you, the farmer is also a retired mathematician!  To test your math prowess, he instructs you to place the apples so that one simple rule is satisfied: the number of apples in any pair of adjacent boxes must differ by exactly 1.  In other words:  


The Rule

Whenever two boxes are next to each other, one box must have exactly 1 more apple than the other.


  1. Is it possible to place apples in the 8 boxes following this rule?  (If you decide that this is possible, see if you can find a few different ways of doing it.  What is the simplest way you can think of to distribute the apples?  However, if you conclude that it isn’t possible, try to explain why it won’t work!)
  2. What if there were 100 boxes?
  3. What if there were 3 boxes?
  4. What if there were 11 boxes?
  5. What patterns are you noticing?
    1. How could you answer the apple placement question if you knew only that there was an even number of boxes, but not the exact number?
    2. How could you answer the apple placement question if you knew only that there was an odd number of boxes, but not the exact number?

Ready for more?