Syllabus   —   Project   —   Week 1   —   Week 2 PROJECT

In addition to several “homework” problems assigned during the program, each student must complete a small project. There are potential project ideas indicated in this packet. I anticipate that most students will work with one partner; if you wish to work alone or in a group of three, please discuss this with me. Since only one group will be allowed to work on a given project, it is important to commit to a project as soon as possible. There are other possibilities for projects: expansions of problems worked in class, reflections on a theme in class, problems or models you have encountered. Expositions should include English text when appropriate and should be typed; mathematical expressions, which are difficult to type, may be written neatly. It is expected that proper English and mathematical notation be used, graphs and diagrams be neatly drawn, tables be clearly labeled. It is appropriate to include a discussion of your thought processes and a reflection of what you did and why it worked. Each person in the class must present at least one of the problems to the class during the final two weeks of the program.

Potential Project Ideas

Problem 1. Consider the function defined by f(x)=2x when 0≤x≤1/2, and f(x)=2(1-x) when 1/2<x≤1.

1. Begin with x=(1/5). Compute f(x) to get a new number (say y). Then compute f(y) to get another number. Repeat this process until a pattern emerges.

2. Repeat (a) using other fractions to initiate the process.

3. Determine the nature of patterns that result from the process described at (a) when one begins with various fractions between 0 and 1.

Problem 2. What is the radius of the smallest circle into which N unit circles can fit without overlap? Investigate for N=1,2,3,4,..,12. Then determine some estimates when N is large.

Problem 3. Begin with an integer in the range from 1 to 100 billion. Compute the sum of the squares of its digits to obtain a new number. Then compute the sum of the squares of its digits, and so on for a total of fifteen times or when you can predict with certainty subsequent numbers in the sequence. Repeat when the integers are represented in base 5. Then investigate other bases.

Problem 4.

1. Choose four integers in the range 0-1000.

2. Place the largest number at A, the second largest at B, third largest at C, and smallest at D.

3. At the midpoint of each side, place the difference of the entries from the adjacent corners (larger minus the smaller).

4. Connect the midpoints together to generate a new square.

5. Repeat steps c, d until a total of 5 nested squares result.

6. Comment on the result. Then try other orderings of A,B,C,D.

7. Repeat the idea of c-e with a triangle; do not worry about the order of the beginning numbers.

8. Repeat with a hexagon, beginning with 0 or 7 at each of the vertices.

Problem 5. Suppose n disks, black on one side and white on the other, are laid out in a straight line with a random arrangement of black sides up. You are playing a game of solitaire, in which a turn consists of removing a black disk and flipping over its immediate neighbors, if any. (Two disks are not considered immediate neighbors if there used to be a disk between them that is now gone.) You win if you succeed in removing all n disks. Describe all initial configurations of disks that are winnable, explain how to win them, and show that all other configurations are unwinnable.

Problem 6. Suppose you had a set of dice, each of a different color and each having spots 0-5.

1. Determine the number of ways of obtaining various sums using 2 dice; 3 dice.

2. Suppose you knew the number of ways of obtaining all possible sums using N dice; how could you determine the number of ways of obtaining all possible sums with N+1 dice?

3. Use a spreadsheet to generate the distribution of the sums of N dice (N=1,2,3,4) using the idea in (b). Then generate the distribution experimentally using the random number generator in the spreadsheet.

Problem 7. Let f(x)=bx(1-x) where b is a real number between 0 and 4. Generate a sequence by beginning with some x₀ between 0 and 1 and then computing x_n+1=f(x_n). Discuss the patterns that result.

Problem 8. A business women wishes to build an indoor driving range. Describe the shape of the roof assuming that (1) a golf ball can have an initial speed as high as V independent of the direction in which it is hit, (2) air resistance is negligible, and (3) the objective is for golf balls not to reach the roof.

Problem 9. Let f(x)=(1/x) and g(x)=1-x. Additional functions can be generated by using functional composition; for example, f(g(x)) is a function formed by composing f with g. Find all functions that can be generated using successive compositions of f,g.

Problem 10. A geoboard consists of 25 pegs arranged in a square array. How many squares can be formed using the pegs as vertices of the squares? Generalize to an n×n geoboard.

Problem 11. A social security number (9 digits) uses each of the digits 1-9 exactly once. The first two digits form a number that is divisible by 2; the first k digits form a number divisible by k. What is the number?

Problem 12. How many triangles are there in the figure shown? Generalize to the general case of n units on each side.

Problem 13. Some applications to astronomy.

1. Determine the maximum amount of time that a lunar eclipse can last in terms of the sizes of the earth and moon plus the distances between them.

2. What is the highest that Venus can appear at sunset? Use a model based upon distances from the sun to Venus and sun to the earth.

3. The tidal force due to an object is proportional to the mass of the object and inversely proportional to the third power of the distance between the earth and the object. Using only the fact that the apparent sizes of the sun and moon (as seen from earth) are equal, determine the ratio (tidal influence of the moon)/(tidal influence of the sun) in term of the average densities of the moon, sun.

Problem 14.

1. Consider the simple board game depicted below. Assume that one begins at square #1. Moves are determined via a flip of a coin — heads means advance 1 square, tails means advance 2 squares (clockwise). Determine the probability of being in each of the squares after 1, 2, 3 moves. What is the probability of being in square #1 after a large number of moves?

   3	go to 2
   2	1

2. The following is a simplified model of Jai-Alai scoring. Three players A, B, C compete. The first exchange is between A and B. The winner of each exchange then plays the next exchange with the player who had been sitting out. The game ends when one player accumulates 3 or more points. The winner of the first exchange is awarded one point; thereafter, winners are awarded two points. Determine the probabilities of each player being the overall winner. Assume that the players have equal ability. Then repeat when the first two exchanges are awarded one point (two points thereafter).

Problem 15. This project deals with polyhedra. We will use F to denote the number of faces, V the number of vertices and E the number of edges.

1. Complete the table for each of the following:

   Shape of Face	Number of Faces	Edges at Vertex	Total Edges	Total Vertices
   Triangle	4	3
   Triangle	8	4
   Triangle	20	5
   Square	6	3
   Pentagon	12	3

2. Determine F, V, E for a prism consisting of vertical sides and polygons with n sides as bases.

3. Determine F, V, E for a pyramid consisting of a polygon with n sides as a base.

4. Find a relationship among F, V, E that is consistent with all the data thus far.

5. Suppose that two polyhedra satisfy (d). Assume a face of one is congruent to a face of the other and the two polyhedra are joined by gluing together these congruent faces. Is (d) satisfied for the combined object?

6. Suppose that two polyhedra satisfy (d). Assume that there are two sets of congruent faces and that gluing the corresponding faces results in an object with a hole. Is (d) satisfied for the combined object?

7. Describe all polyhedra that satisfy (d), have three edges meet at each vertex and have faces that are either squares of triangles.

8. A polyhedron satisfies (d), consists of pentagons and hexagons, and has three edges meeting at each vertex. How many of the faces could be pentagons?

9. A polyhedron satisfied (d), consists of pentagons and triangles, and has four edges meeting at each vertex. What are the possible number of pentagons and triangles?

Problem 16.

1. Let ABC be a triangle, and let points D, E, F be on AB, BC, CA, respectively such that AD/DB=BE/EC=CF/FA=1/2. How does the area of triangle DEF compare to the area of ABC.

2. Consider a right triangle. Let S and T denote the areas of the two squares that can be inscribed in the triangle having their base on the longest leg and hypotenuse, respectively. Which is larger: S or T?

This document is maintained by Melissa Elaine Smith smith@math.fsu.edu Last modified: 21 May 2001