More questions for the triangle of circles:
1. What if the triangle had a more circles on each side? What would be the smallest and largest possible sums?
2. Is there a pattern to the smallest possible sum for the different-sized triangles?
3. What would happen if I began with a square instead of a triangle?
4. What if I changed the number of circles on each side of the square?
5. What if I began with other polygons like pentagons or hexagons?
6. What it my triangles, squares, etc. did not have the same number of circles on each side?
7. Could I find solutions that involve other whole numbers? Fractions? Negative numbers?
8. What if I changed the problem to making the product of each side the same?
9. Is there a formula for the minimum (maximum) sum based on the number of circles on each side of the triangle?
10. Is there a formula like this that also takes account of the number of sides on the shape?
See Triangle Sums if you would like a more structured version of the problem or if you would like to see some solutions and suggestions for conversation.