Here are some suggestions for science project ideas. To give students lots of options, a short outline (sometimes with an example) is usually enough. The problem you describe is not always the project. These ideas are meant to spark thought and are not necessarily in a particular order.

1. Sometimes charities contact households to offer to buy used items. They are often limited to a single area. Once they have identified the pick-up locations, their problem is to determine the best route to collect the items. You can learn more about their methods and how you could improve them. The same question can also be asked about snowplows clearing streets and garbage collection. Refer to: Euclidean Tours, Chinese Postman Problem – Information can be found in most books about graph theory. However, the introduction to graph theory by G. Chartrand is a particularly useful book.

2. How can one locate an ambulance station to serve the community best? This article may be of assistance.

3. An International Food Group is made up of twenty couples that meet four times per year to share a meal. Four couples meet in each of five homes on each occasion. The group is very friendly and gets along well. However, some couples do not get along with each other all the time. Can you plan four evenings that allow couples to meet only once? This is just one example of many. These are known collectively as combinatorial plans. You should also investigate other designs.

4. How does NBA calculate the basketball schedules? How do you create such a schedule taking into account distances between games and home advantage, etc. What schedule could you create for your local competition?

5. How are major hospitals able to schedule operating theatres? Is it possible to do as many operations per day as possible in the most efficient manner?

6. Investigate “big” numbers. What is the definition of a large number? These examples may help you in your investigations. One million loonies are taken from a bank. How long would that take? How heavy would it weigh? What would be the space it would take up? What size pool is needed to hold all the blood of the world? Is 10100 a very large number? What is your record for the largest number? How did this number get there?

7. Make a phsical model using dissections in order to prove Pythagorean theorem. A Pythagorean exhibit can be built with “The semicircle around the hypotenuse

8. What is the minimum number required to colour a map? This is because no two countries sharing a border can use the same color. This was discovered by who? Is this proof of the truth interesting? Imagine Mars being divided into different areas, so that each country owns a portion of it. The same rule applies to them, but the colours must be chosen according to their country. What are the colors that are needed now? Refer to Joan Hutchinson.

9. The golden mean is a symbol of the unity of art, architecture, and geometry. It also has a connection to continued fractions and fibonacci numbers. Find out what else you can find. What is The Golden Mean?

10. Learn about the properties and geometry of regular solids (Archimidean and platonic), and their natural appearances in nature (e.g. virus shapes, fullerene molecules, crystals). Build models.

11. The history and tautochrone of the cycloid arc curve is worth studying. Build models.

12. Infinity can come in many sizes. What does that mean? What does this mean? References: Refer either to the Dover paperbacks Theory of Sets”, Theory of Sets”, Theory of Serial Order”, Huntington, or any other book about Set Theory.

13. Visual representations of finite numbers can be explored. If p has 100 digits and is therefore a prime, then 1 and 6 are on the same length of line. With p being six inches away from 1, then p1/2, which is the square root p, is 10-50 inches away. It is also less than an atom from 1, so p1/2 can be used to prove that p is not the sum of two squares. By inspecting the lattice point in the p1/2xp1/2 array, one can show that p = the sum of 2 squares. Continue investigating.

14. Discover the 17 different’ wallpapers. (Think of how wallpaper patterns repeat. How does this relate to Escher’s work? This problem’s history is available. References: G.C. Shephard Additive Frieze Patterns and Multiplication Tables”, Math. Gaz. 60(1976) p179-184; H.S.M. 60(1976) p179-184; H.S.M. Conway, H.S.M. Coxeter. Triangulated Ploygons und frieze pattern”, Math. Gaz.
57(1973) p87-94 (questions), 175-183 (answers).

15. Learn about winning strategies and game play. Analyze subtraction (nim-like games where the players alternately grab a number from a pile, with the numbers being limited to a particular subtraction set). References: E.R. Berlekamp, J.H. Conway, R.K. Guy, “Winning Ways”, Academic Press, London (this book includes hundreds of other games for which no analysis has been done). Toads & Frogs), ; R. Guy (editor), Combinatorial Game” Proceeedings of Symposia Applied Math publication (pay attention to section last where there are many questions).

16. Many computers can now handle sound in one or more ways. The sound is stored in a series of numbers. Many numbers. You could get 40,000 every second. What happens if you mess with these numbers? eg. Add 10 numbers. Multiply each amount by 10. Divide each number by 10. Divide by 10. Add one sound to another (i.e. Add the sequences to get the corresponding numbers. Multiply these numbers. Separate them. Add one sound to make shifted copies. Repeat the process. Reverse the order. Every third number should be thrown out. Divide the sine into the numbers. These numbers can be summed. You can listen to the sound produced by each mathematical operation and see the change in the computer speakers. You can create some amazing effects with a little programming. Next, try to understand this using some kind of theory about signal processing.

17. Research self-avoiding random walking and their natural locations. Refer to: G. Slade. Random Walks, American Scientist, March-April 1996.

18. Investigate what happened to the space shuttle when it tried to place a satellite tethered into orbit.

19. Draw curves. List any interesting properties. ).

20. For the ambitious, you can create a family from polyhedra. Refer: You can see any Coxeter rev of Rouseball’s Mathematical Recreations and Essays” (which contains many great ideas). Also, DuVal and Flather, Petrie and The 59 Icosahedra” U of Toronto Press; Magnus J Wenninger Polyhedron Models” Cambridge, 1971; Doris Schattschneider & Wallace Walker M.C. Escher Kaleidocycles”, Pomegranate Art Books, 1987.

21. You can find as many triangles that have integer sides, and as many as possible with a linear relationship between them. What about right-angled triangles?

22. Learn all you can about Fibonacci Numbers 0, 1, 2, 3, 4, 5, 8,…

23. Find out everything you can about Catalan Numbers 1, 2, 5, 14 and 42.

24. What is Morley’s triangular? Draw the 18 Morley triangles that are associated with ABC. Find 18 more triangles for ABC: CHA, BH, AHB. H is ABC’s orthocentre. Explore the relationship between the 9 points circle and deltoid. (Envelope of Simson/Wallace line).

25. What is the hexaflexagon, exactly? As many as possible. What’s the deal? Reference: Martin Gardner, “Hexaflexagons, and Other Mathematical Diversions”, Univ. The University of Chicago Press published the work in 1988.

26. Investigate trianglar numbers. You can also explore pentagonal numbers and hexagonal numbers. Conway & Guy, “The Book of Numbers”, Springer, Copernicus series, 1996, Chapter 2.

27. Ten frogs perch on a log. Five green and five brown are on each side. The empty seat between them separates. They decide to swap places. They are not permitted to jump over another frog or into empty spaces. What is the limit on how many moves you can do? What if there was 100 frogs per side? The results reveal interesting patterns depending on how you look at the colour of the frogs, their type of moves, and empty space. It is fun to prove that it works. Although it can lead recursion it is also fascinating. Other questions are similar. The Tower of Hanoi is a famous example.

28. Investigate how secret codes were created (ciphers). You can find out their uses today! They are also used in a variety of ways. Have a look at the history. Use prime numbers to build yours.
M. Fellows, N. Koblitz. “Kid krypto.” Proc. CRYPTO ’92 Springer-Verlag Lecture Notes Computer Science vol. 740 (1993), 371-389.

29. This is an example of illustrating the binomial pattern. Marbles are dropped through the top, and then drop into cells. The pin positions can be changed to produce other types of distributions such as approximately rectangular, bimodal, and skewed. Explore.

30. Use both non-rigid and rigid geometric shapes. These are just a few of the possibilities. What are the uses of rigid structures? You might find unusual uses. An example of this could be the parallelogram formed by the midpoints in a quadrilateral’s sides (even though the quadrilateral may not be planar). Is there a similar thing in three dimensions?

31. Create a scale model to show the solar system. Illustrate where it would be located in your area. Maybe even do so! !

32. Models can be built to demonstrate asymptotic results like Stirling’s formula and the prime number theory.

33. What/are Napier’s bones and how can you use them/them?

34. It is important to cover a board with dominoes in such a way that they do not overlap. Also, make sure no squares remain unoccupied. Consider (a), the full chessboard; and (b), the chessboard where one square has been removed. (c) A chessboard that has two corners removed from it, (b) one with the opposite corners removed (possible but impossible). (e) A board with all two squares removed. How about shapes other that dominoes? For example, 3 one-square squares connected together. What about chessboards that are different in size? Refer to: “Polyominoes” by Solomon W. Golumb. Charles Scribner’s Sons

35. Create models that prove parallelograms with equal height and base areas. This could be used to visually prove the Pythagorean Theorem. This can also be used to present the formula that determines the area of any circle. R eferences to H. R. Jacobs in Mathematics a Personal Endeavor (pp 38).

36. Monte Carlo methods can help you find areas. Instead using random numbers, put a lot of small objects on the desired area. Then count how many objects are there as a fraction.

37. See photos that show 1 + 2 +… + n = (1/2),n+1
12 + 22 +… + n2 = (1/6)n+1(2n+1).
And that 13 + 23 +… + N3 = (1 + 2) +…+ n2 = ( + 2 +… + 1)2.
These identities can be proven in many different ways. Which one is the best?

38. What is fractal dimensions? Exercise the concept with examples of what happens if you double the scale to (a), areas (b), and (c), solids(d).

39. Knots. What happens if you tie a knot to a piece a paper and flatten the paper carefully? What is a knot, even if it looks like one? See the following methods to draw knots.

40. Is it possible to escape 2-dimensional mazes using an algorithm? What about the 3-dimensional mazes? You can see the incredible history of mazes. How do we find lost people in mazes (2-3 dimensional) who are wandering aimlessly? How many people are needed to find them

41. Find out the history and possible approximations of pi. Find out how to calculate new Pi digits – Peter Borwein has the answer.

42. What does game theory mean and how can it be applied?

43. Construct a Kaleidoscope. Analyse the history of this Kaleidoscope and the mathematics behind it.

44. Think about tiling a plane using the same shape shapes. What is possible? It is clear that any 4-sided form can tile the plane. How about five sides? You can find Grunbaum articles and Grunbaum books, and Martin Gardner books.

45. Discover why Penrose tiles are so interesting.

46. The Steiner problem is one example of how to find telphone exchanges in order to reduce costs.

47. Find new ways to solve the problem of the traveling salesman.

48. Explore the fractions of Egypt.

49. How does the computer barcode (the one you see on every purchase) work? This is one example of coding theory in action. Look for others. Look for books that describe coding theory. This is not about secret codes. References: Joe Gallian How computers can correct ID numbers” Math Horizons, Winter 1993, page 14-15; Joe Gallian Assigning Drivers License Numbers” Mathematics Magazine, 64 (1991), 13-22. Joe Gallian Math on Money” Math Horizons (November 1995, page 10-11.

50.
Problem in the Art Gallery: How few guards should be required to ensure that all paintings are protected? Guards must be placed at certain locations, and they must all have direct sight to all points on the walls. Reference: Alan Tucker. The Art Gallery Problem, Math Horizons. Spring 1994. pp24-26

51. Parabolic Reflector Microphones can be used to listen to one person at a time in noisy environments. You can investigate this and explain the math behind it.

52. Traditional Chinese methods of showing the Pythagorean Theorem with paper are available. Make models and investigate.

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

54. Try the “Monty Hall Effect.” The prize is behind each door. He shows you which door you should choose, and you are given the option to either switch to the third door or remain at the first. Why would you want to switch? To show’ that it is, create an exhibit. Find the mathematical explanation for the switch.

55. Seek out examples of how bases can be used in your culture and those used in other cultures. Collect examples: time, date etc. Demonstrate the ability to add using Mayan 20.

56. Learn about the history of the Abacus and its use.

57. Try out 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. Discover magic tricks that are based on Mathematics (refer to the Persi Diaconis article).

59. Discuss straight-edge and compass constructions. To take an example, what if one has a long length line segment? Can they be used straight edge and the compass to create all the radicals.

60. The double pendulum and chaos