Download Geometrical Recursion Activities

Geometrical Recursion Activities
Kurt and Carol
Directions:
1. Set up a sequence of diagrams depicting dots, squares, regions, lines,
intersections, and/or regions.
2. Find a recurrence formula for this sequence.
3. Describe this recurrence formula in closed form.
Problems.
1. Suppose that for some positive integer, n, there are n lines in the Euclidean plane
such that no two are parallel and no three meet at the same point. Determine the
number of regions into which the plane is divided by these n lines.
2. The Towers of Hanoi game is played with a set of disks of graduated size with
holes in their centers and a playing board having three spokes for holding the
disks. The object of the game is to transfer all the disks from spoke A to spoke C
by moving one disk at a time without placing a larger disk on top of a smaller one.
The recursive formula for doing this is mn  2mn1  1 for n  2 with m1  1 .
Find the closed form for the relationship.
3. The triangular numbers, allegedly studied by Pythagoras of Samos (c. 572 – 497
BC) and his school of Pythagoreans are given by the number of dots in the
following sequence of triangles.
1st triangle
2nd triangle
3rd triangle
4th triangle
How many dots will be in the 5th triangle and 6th triangle?
4. There are n (n  1) ovals drawn in a plane. An oval must intersect each of the
other ovals at exactly 2 points and no three ovals meet at the same point. Using a
recurrence algorithm, develop a conjecture for the number of regions that will be
created. (Count the exterior region. Therefore one oval creates 2 regions)
5. Suppose that for some positive integer, n, there are n lines in the Euclidean plane
such that no two are parallel and no three meet at the same point. Predict the
number of intersections created by these lines.
6. Consider the following geometric sequence. Predict the number of black squares
in the nth figure.
…….
1st
2nd
3rd