This sample offers just a little taste of what we have to offer.

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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?

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.

- 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!) - What if there were 100 boxes?
- What if there were 3 boxes?
- What if there were 11 boxes?
- What patterns are you noticing?
- 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?
- 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?