# Math Circles at Home: Adding Boxes

June 8, 2020Uncategorized

Suppose you place two numbers in a row of boxes like so: Then we generate the next number by adding those first two boxes together. We continue in this fashion to complete the row: On the other hand, suppose I give you this: What numbers should be in the boxes between the 5 and the 16?

Some follow up questions:

1. A simpler case: If there are only 3 boxes, and I give you the first and the last boxes, what goes in the middle?
2. What happens if I switch the first and last number?
3. Can you give a general rule for finding the numbers between the first and last box when there are 5 boxes?
4. What if you had 4 boxes? Or 6? Or n? Can you come up with a general rule for finding the numbers in between when you have any number of boxes?
5. If the first and last numbers are whole numbers, will the numbers in between be whole numbers? If not, under what conditions will those numbers be whole numbers?
6. How can you change the rules? What questions could you ask then?

Box on!