MATHEMATICS RESEARCH PROJECT IDEAS
SUITABLE FOR HIGH SCHOOL AND COLLEGE STUDENTS

This list is a copy of the list Possible Science Fair Mathematics Projects which is maintained by Afton H. Cayford , at The University of British Columbia.

What follows is a selection of ideas for science projects. In most cases only a very brief outline is presented (sometimes with a reference) in order to leave students plenty of scope for what they do. It is not expected that the problem stated will necessarily be the project. These are ideas intended to get people thinking (they are in no particular order).

1. At certain times charities call households offering to pick-up used items for sale in their stores. They often do a particular geographical area at a time. Their problem, once they know where the pick-ups are, is to decide on the most efficient routes to make the collection. Find out how they do this and investigate improving their procedure. A similar question can be asked about snow plows clearing city streets, or garbage collection. References: Euclidean tours, chinese postman problem - information can be found in most books on graph theory but one of particular interest is ``Introduction to Graph Theory'' by G. Chartrand.

2. How should one to locate ambulance stations, so as to best serve the needs of the community? The reference given above may help.

3. An International Food Group consists of twenty couples who meet four times a year for a meal. On each occasion, four couples meet at each of five houses. The members of the group get along very well together; nonetheless, there is always a bit of discontent during the year when some couples meet more than once! Is it possible to plan four evenings such that no two couples meet more than once? There are many problems like this. They are called combinatorial designs. Investigate others.

4. How does the NBA work out the basketball schedule? How would you do such a schedule bearing in mind distances between locations of games, home team advantage etc.? Could you devise a good schedule for one of your local competitions?

5. How do major hospitals schedule the use of operating theatres? Are they doing it the best way possible so that the maximum number of operations are done each day?

6. Investigate ``big'' numbers. What is a big number? The following examples might guide your investigation. A bank is robbed of 1 million loonies. How long it would take to move that many? How much it would weigh? How much space would it take up? How big a swimming pool do you need to contain all the blood in the world? Is 10100 very big? What is the biggest number anyone has ever written down (check the Guiness book of world records over the last few years)? How did this number come about?

7. Build a phsical model based on dissections to prove the Pythagorean Theorem. Build an exhibit on the Pythagorean theorem but with "The semicircle on the hypotenuse ..."

8. What is the fewest number of colours needed to colour any map if the rule is that no two countries with a common border can have the same colour. Who discovered this? Why is the proof interesting? What if Mars is also divided into areas so that these areas are owned by different countries on earth. They too are coloured by the same rule but the areas there must be coloured by the colour of the country they belong to. How many colours are now needed? Reference: Joan Hutchinson, ...

9. Study the golden mean, its appearance in art, architecture, biology, and geometry, and it's connection with continued fractions, fibonacci numbers. What else can you find out? What is the Golden Mean?

10. Study the regular solids (platonic and Archimidean), their properties, geometry, and occurance in nature (e.g. virus shapes, fullerene molecules, crystals). Build models.

11. Study the cycloid curve: its tautochrone and brachistochrone properties and its history. Build models.

12. Infinity comes in different ``sizes''. What does this mean? How can it be explained? References: Refer to either of the Dover paperbacks, ``Theory of Sets'' by Kamke, and ``The Continuum and other types of Serial Order'' by Huntington, or any book on Set Theory.

13. Investigate visual representations of different finite numbers. For example, if p is a prime with 100 digits, then if 1 and p are on the same line segment, with p say 6 inches to the right of 1, then p1/2, the square root of p, is about 10-50 inches to the right of 1, less than one atom away. (And it's by inspecting the lattice points in the p1/2 x p1/2 array that one proves that p is the sum of two squares!) Investigate further.

14. Discover all 17 ``different'' kinds of wallpaper. (Think about how patterns on wallpaper repeat.) How is this related to the work of Escher? Discover the history of this problem. References: G.C. Shephard, ``Additive Frieze patterns and multiplication tables'', Math. Gaz. 60(1976) p179-184; H.S.M. Coxeter, ``Frieze Patterns'', Acta Arithmetica, XVIII(1971) p297-310; and J.H. Conway and H.S.M. Coxeter, ``Triangulated Ploygons and frieze patterns'', Math. Gaz. 57(1973) p87-94 (questions), 175-183 (answers).

15. Study games and winning strategies - maybe explore a game where the winning strategy is not known. Analyze subtraction games (nim-like games in which the two players alternately take a number of beans from a heap, the numbers being restricted to a given subtraction set). References: E.R. Berlekamp, J.H. Conway, R.K. Guy, ``Winning Ways'', Academic Press, London (this book contains hundreds of othr games for which the complete analysis is unknown eg. Toads and Frogs) ; R. Guy (editor), ``Combinatorial Games'', Proceeedings of Symposia in Applied Math, AMS publication (pay special attention to the last section where lots of questions are asked).

16. Most computers these days can handle sound one way or another. They store the sound as a sequence of numbers. Lots of numbers. 40,000 per second, say. What happens when you play around with those numbers? eg. Add 10 to each number. Multiply each number by 10. Divide by 10. Take absolute values. Take one sound, and add it to another sound (i.e. add up corresponding pairs of numbers in the sequences). Multiply them. Divide them. Take one sound, and add it to shifted copies of itself. Shuffle the numbers in the sequence. Turn them around backwards. Throw out every third number. Take the sine of the numbers. Square them. For each mathematical operation, you can play the resulting sound on the computers speakers, and hear what change has occurred. A little bit of programming, and you can get some very bizarre effects. Then try to make sense of this from some sort of theory of signal processing.

17. Investigate self-avoiding random walks and where they naturally occur. Reference: : G. Slade, ``Random walks'', American Scientist, March-April, 1996.

18. Investigate the space shuttle's failed attempt to put a tethered satellite into orbit.

19. Draw, and list any interesting properties of, various curves: evolutes, involutes, roulettes, pedal curves, conchoids, cissoids, strophoids, caustics, spirals, ovals, ... Reference: Cundy and Rollett, ``Mathematical Models'', Oxford (which has lots of other ideas, too); E.H. Lockwood, ``A Book of Curves'', Cambridge; and there's also a book by Yates, ``Curve Tracing''(?).

20. Make a family of polyhedra, e.g., the Archimedean solids, or Deltahedra (whose faces are all equilateral triangles), or equilateral zonohedra, or, for the very ambitious, the 59 Isocahedra. Reference: See any Coxeter revision of Rouse Ball's ``Mathematical Recreations and Essays'' (which is full of many ideas). There's also Coxeter, DuVal, Flather and Petrie, ``The 59 Icosahedra'', U of Toronto Press; Magnus J Wenninger, ``Polyhedron Models'', Cambridge, 1971; and Doris Schattschneider and Wallace Walker, ``M.C. Escher Kaleidocycles'', Pomegranate Art Books, 1987.

21. Find as many triangles as you can with integer sides and a simple linear relation between the angles. What about the special case when the triangle is right-angled?

22. Find out all you can about the Fibonacci Numbers, 0, 1, 1, 2, 3, 5, 8, ...

23. Find out all you can about the Catalan Numbers, 1, 1, 2, 5, 14, 42, ...

24. What is Morley's triangle? Draw a picture of the 18 Morley triangles associated with a given triangle ABC. Find the 18 more for each of the triangles BHC, CHA, AHB, where H is the orthocentre of ABC. Discover the relation with the 9-point circle and deltoid (envelope of the Simson or Wallace line).

25. What is a hexaflexagon? Make as may different ones as you can. What is going on? Reference: Martin Gardner, ``Hexaflexagons and other Mathematical Diversions'', Univ. of Chicago Press, 1988.

26. Investigate trianglar numbers. If that's not enough, do squares, pentagonal numbers, hexagonal numbers, etc Venture into the third and even the fourth dimension. Reference: Conway and Guy, ``The Book of Numbers'', Springer, Copernicus Series, 1996, Chapter 2.

27. Ten frogs sit on a log - 5 green frogs on one side and 5 brown frogs on the other with an empty seat separating them. They decide to switch places. The only moves permitted are to jump over one frog of a different colour into an empty space or to jump into an adjacent space. What is the minimum number of moves? What if there were 100 frogs on each side? Coming up with the answers reveals interesting patterns depending on whether you focus on colour of frog, type of move, or empty space. Proving it works is interesting also - it can lead to recursion, there is also a simple proof that is not immediately obvious when you start. Look for and explore other questions like this - one of the most famous is the Tower of Hanoi.

28. Investigate the creation of secret codes (ciphers). Find out where they are used (today!) and how they are used. Look at their history. Build your own using prime numbers. Reference: M. Fellows and N. Koblitz. ``Kid krypto.'' Proc. CRYPTO '92, Springer-Verlag, Lecture Notes in Computer Science vol. 740 (1993), 371-389.

29. There is a well-known device for illustrating the binomial distribution. Marbles are dropped through the top and encounter a number of pins before dropping into cells where they are distributed according to the binomial distribution. By changing the position of the pins one should be able to get other kinds of distributions (bimodal, skewed, approximately rectangular, etc.) Explore.

30. Build rigid and nonrigid geometric structures. Explore them. Where are rigid structures used? Find unusual applications. This could include an illustration of the fact that the midpoints of the sides of a quadrilateral form a parallelogram (even when the quadrilateral is not planar). Are there similar things in three dimensions?

31. Build a true scale model of the solar system - but be careful because it cannot be contained within the confines of an exhibit. Illustrate how you would locate it in your town. Maybe even do so!!

32. Build models to illustrate asymptotic results such as Stirling's formula or the prime number theorem.

33. What is/are Napier's bones and what can you do with it/them?

34. Covering a chessboard with dominoes so that no two dominoes overlap and no square on the chessboard is uncovered. Consider (a) a full chessboard, (b) a chessboard with one square removed (impossible - why?), (c) a chessboard with two adjacent corners removed, (d) one with two opposite corners removed (possible or impossible?), (e) A chessboard with any two squares removed. What about using shapes other than dominoes (eg 3 one-by-one squares joined together)? What about chessboards of different dimensions? Reference: ``Polyominoes'' by Solomon W. Golumb, pub. Charles Scribner's Sons

35. Build models showing that parallelograms with the same base and height have the same areas (is there a 3-dimensional analogue?). This can lead to a purely visual proof of the Pythagorean theorem. The formula for the area of a circle can also be presented in this way. R eference: H. R. Jacobs,`` Mathematics a Human Endeavor'', 3rd ed, p 38)

36. Use Monte Carlo methods to find areas (rather than using random numbers, throw a bunch of small objects onto the required area and count the numbers of objects inside the area as a fraction of the total in the rectangular frame) or to estimate pi.

37. Find pictures which show that 1 + 2 + ... + n = (1/2)n(n+1),
that 12 + 22 + ... + n2 = (1/6)n(n+1)(2n+1)
and that 13 + 23 + ... + n3 = (1 + 2 + ... + n)2.
How many other ways can you find to prove these identities? Is any one of them ``best''?

38. What is fractal dimension? Investigate it by exaomining examples showing what happens when you double the scale to (a) lines (b) areas (c) solids (d) the Koch curve.

39. Knots. What happens when you put a knot in a strip of paper and flatten it carefully? When is what appears to be a knot really a knot? Look at methods for drawing knots.

40. Is there an algorithm for getting out of 2-dimensional mazes? What about 3-dimensional? Look at the history of mazes (some are extraordinary). How would you go about finding someone who is lost in a maze (2 or 3 dimensional) and wandering randomly? How many people would you need to find them?

41. Investigate the history of pi and the many ways in which it can be approximated. Calculate new digits of Pi - see Peter Borwein's homepage to discover what this means.

42. What is game theory all about and where is it applied?

43. Construct a Kaleidoscope. Investigate its history and the mathematics of symmetry.

44. Consider tiling the plane using shapes of the same size. What's possible and what isn't. In particular it can be shown that any 4-sided shape can tile the plane. What about 5 sides? Look for books and articles by Grunbaum and Shepherd, and check the Martin Gardner books.

45. Explore Penrose tiles and discover why they are of interest.

46. Investigate the Steiner problem - one application of which is concerned with the location of telphone exchanges to minimize costs.

47. Look for new strategies for solving the travelling salesman problem.

48. Explore egyptian fractions.

49. How do computer bar codes (the ones you see on everything you buy) work? This is an example of coding theory at work. Find others. Investigate coding theory - there are many books with titles like ``an introduction to coding theory'' (this is not about secret codes). References: Joe Gallian, `` How computers can read and correct ID numbers'', Math Horizons, Winter, 1993, p14-15; Joe Gallian, ``Assigning Driver's License Numbers'', Mathematics Magazine, 64 (1991), 13-22; and Joe Gallian, ``Math on Money'', Math Horizons, November, 1995, p10-11.

50. The Art Gallery problem: What is the least number of guards required to watch over all paintings in an art gallery? The guards are positioned at specific locations and collectively must have a direct line of sight to every point on the walls. Reference: Alan Tucker, ``The Art Gallery Problem'', Math Horizons, Spring, 1994, p24-26

51. The Parabolic Reflector Microphone is used at sporting events when you want to be able to hear one person in a noisy area. Investigate this; explaining the mathematics behind what is happening.

52. There is a traditional Chinese way of illustrating the Pythagorean theorem using paper. Investigate and make models.

53. Use PID (proportional-integral-differential) controllers and oscilloscopes to demonstrate the integration and differentiation of different functions.

54. Try the "Monty Hall" effect. Behind one of three doors there is a prize. You pick door #1, he shows you that the prize wasn't behind door #2 and then gives you the choice of switching to door #3 or staying with #1, what should you do? Why should you switch? Make an exhibit and run trials to ``show'' this is so. Find the mathematical reason for the switch.

55. Look at the ways different bases are used in our culture and how they have been used in other cultures. Collect examples: time, date etc. Demonstrate how to add using the Mayan base 20, maybe compare to trying to add with Roman numerals (is it even possible?)

56. Explore the history and use of the Abacus.

57. Investigate card tricks. Some of the best in the world were designed by the mathematician/statistician Persi Diaconis. Reference: Don Albers, `` Professor of (Magic) Mathematics'', Math Horizons, February 1995, p11-15

58. Explore magic tricks based in Mathematics (again see the article about Persi Diaconis).

59. Investigate compass and straight-edge constructions - showing what's possible and discussing what's not. For example, given a line segment of length one can you use them straight edge and compass to ``construct'' all the radicals?

60. Chaos and the double pendulum.

61. There are several books that have a variety of articles that can be used to generate projects:
* John Mason with Leone Burton and Kaye Stacey, ``Thinking Mathematically'', revised edition, Addison-Wesley , 1985
* Cliff Sloyer, ``Fantastiks of Mathematics: Applications of Secondary Mathematics'', Janson Publications, Inc., Providence, R.I., 1986. ISBN 0- 939765-00-4.
* Paul Hoffman, ``Archimedes Revenge'', Ballantine Books
* T.F Banchoff (in Steen, L.A. , ed.) (1990). ``On the shoulders of giants: new approaches to numeracy'', National Academy Press, Washington, D.C
* Nancy Casey and Mike Fellows (1993). `` This is mega-mathematics: stories and activities for mathematical thinking, problem-solving and communication'', The Los Alamos National Laboratory, Los Alamos, New Mexico
* Arthur L Loeb, "Concepts and Images, Visual Mathematics"
* Paulos, J.A. (1991). ``Beyond numeracy: ruminations of a numbers man'', Alfred A. Knopf, New York.
* And then of course there are all the Martin Gardner books.

Tony Phillips
1998-09-21