Farmer Klaus and the Mouse:

Overview: Farmer Klaus and the Mouse is a team game of chance played by young children in Germany, with the children teaming up to get the grain into the barn before the mice get it. The object of this project is to determine the frequency with which the children can expect to win the game, approximately (via simulation) and if possible exactly (via dynamic programming).
Mathematical and Computing Concepts: Dynamic programming, simulation
Programming Language: C++, Mathematica suggested. The language of choice for dynamic programming is Mathematica, which can be used as a functional programming language that can return any kind of object, and which can automatically store previously-found solutions to subproblems.
Prerequisites: CS 195.
Status: Jason Green is working on this, Spring 2001.

Maverick Solitaire:

Overview: In an episode of the old Maverick television show, Brett Maverick claims that 49 times out of 50 you can arrange 25 drawn cards into 5 pat poker hands. The object of this project is to confirm or disconfirm this claim by designing a program that tries to find 5 pat poker hands in 25 cards, then simulating draws of 25 cards.
Mathematical and Computing Concepts: Simulation
Programming Language: open
Prerequisites: CS 195.
Status: Not started

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Vess and Other Matching Puzzles

Overview: Various popular puzzles require the solver to place 9 pieces in a 3-by-3 grid so that the side of one piece matches (in head-and-tail fashion) the adjoining sides of the adjacent pieces (see first figure below). How many essentially different such puzzles are there? Generalizations involve other shapes (triangular or lozenge-shaped pieces, etc.-see second figure below) or symmetry of heads and tails (e.g., just colors). Such puzzles can be solved by brute force with backtracking, but is there a more efficient strategy, in terms of max/expected number of comparisons of potential fits of adjacent sides? How can information special to the puzzle best be used? For example, the cereal-box puzzle below features on each small square the front or the back part of each of four German houses; 6 of the small squares have the tops of all roofs pointing CCW, 3 squares have one exceptional house half pointing CW. Such information should narrow the search space considerably; here, each square can fit against a given side of another square in at most 1 way. In particular, is it efficient to build up the solution to a larger puzzle be built up by dynamic programming from smaller collections of fitted-together pieces?
Mathematical and Computing Concepts: Dynamic programming
Programming Language: Mathematica. The language of choice for dynamic programming is Mathematica, which can be used as a functional programming language that can return any kind of object, and which can automatically store previously-found solutions to subproblems.
Prerequisites: CS 195, Math 200.
Status: Not started.

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Headlamp Direction:

Overview: I once uncritically accepted the diagram and caption below (from a book I am intimately familiar with) about how a headlamp works:

Caption: "In an automobile headlight, a parabola directs rays of light outward in parallel lines: straight out, for the high beam source located at the focus, and down and to the right, for the low beam source located above and to the left of the focus."

This claim is not true, as Eileen Fernandez and students at Montclair State University brought to my attention. It is not hard to find rays that leave the headlamp in an upward direction. Nevertheless, the figure shows the design of headlamps. What is the actual geometry of distribution of the light rays when the light source is placed away from the focus, and what effect does it have that the light source is not a point source?
Mathematical and Computing Concepts: Analytic geometry of a parabola
Programming Language: Mathematica.
Prerequisites: Math 115, CS 111.
Status: Not started.

Basketball:

Overview: In basketball, the backboard provides an invaluable aid, helping some shots to go in that would otherwise not and allowing easy almost-sure lay-up shots. When, if ever, should a player in fact aim for the backboard instead of the hoop? Previous research suggests how the player should shoot if aiming at the hoop: Shoot high, and (for a free throw) shoot underhanded.
Mathematical and Computing Concepts: Analytic geometry
Programming Language: Programming may or may not be desirable; if used, it would be for simulation.
Prerequisites: College physics, preferably including Dynamics; perhaps mathematical probability.
Status: Not started.