Some More Examples

1. What is the next term in the sequence? In each case, think of at least three different ``plausible'' answers.
   1. .
   2. .
   3. .
2. What is the next term in the sequence? In each case, try to find a formula f(n) such that the sequence is given by .
   1. .
   2. 
   3. .
   4. .
   5. .
3. What is the formula?
   1. What is the maximum number of pieces into which a pancake can be cut by n straight cuts, each of which crosses each of the others?
   2. What is the maximum number of pieces that can be produced by n simultaneous straight cuts of a flat figure shaped like a crescent moon?
   3. How many pieces of cheesecake can be produced by n simultaneous plane cuts of a cylindrical cake?
   4. Into how many parts can the plane be divided by intersecting circles of the same size? Of different sizes? By intersecting ellipses of different sizes?
   5. Into how many regions can space be divided by intersecting spheres?
   6. With an unlimited supply of toothpicks of n different colors, how many different triangles can be formed on a flat surface, using three toothpicks for the three sides of each triangle? (Reflections are considered different, but not rotations.) How many different squares?
   7. How many different tetrahedra can be produced by coloring each face a solid color and using n different colors? (Two tetrahedra are the same if they can be turned and placed side by side so that corresponding sides match in color.) How many cubes with n colors?
   8. What is the maximum number of pieces that can be produced by n simultaneous plane cuts through a doughnut?