1. Ways to Construct Triangles in an m × n Array of Dots Kulawat Udomwongsup Natcha Techachainirun Poompipat Nuenkhaekul Mentors: Suwat Sriyotee 1 , Rungsima Sairattanathongkum 1 and Prof. Dinesh G. Sarvate 2 1 Mahidol Wittayanusorn School and 2 Mathematics Department, College of Charleston, Charleston, South Carolina, U.S.A.

2. Acknowledgement Prof. Dinesh G. Sarvate Mathematics Dept., College of Charleston, Charleston, South Carolina, U.S.A. Suwat Sriyotee and Rungsima Sairattanathongkum Mahidol Wittayanusorn School Salaya, Putthamonthon, Nakhon Pathom, Thailand

3. Introduction Background: The transition from place to place can also be viewed as a journey from one dot to another, while tracks traveled can be represented by lines. When these lines are connected, forming a closed shape, they make polygons. Shapes created by the connection of three dots with three lines are called triangles. Significance: This project is to serve as a foundation for future studies.

4. Objective To find the relation leading to the number of triangles obtainable in an m × n array of dots.

5. Methodology 1.) Counting stage 2.) Formularizing stage 2.1) 3× n array of dots 2.2) 4× n array of dots 2.3) 5× n array of dots 1. Note the method to obtain the number of triangles. 2. Break the rows into groups of three. 3. Consider each grouped row for the relation of the number of straight lines. 4. Record the result. 5. Check the relation by substituting values of m and n .

6. Methodology 1.) Counting stage 1. Draw an array of any size. 5. Record the results. 2. Calculate ways to connect three dots together. 3. Consider any three-dotted connection that does not result in a triangle. 4. Remove the number of non-triangles from the number of ways to connect three dots.

7. For a 3×5 array of dots Number of triangles: Number of straight lines with three dots Methodology 1.) Counting stage (continued) 5 3

8. For a 3×5 array of dots Number of triangles: Methodology 1.) Counting stage (continued) 5 3

9. For a 3×5 array of dots Number of triangles: Methodology 1.) Counting stage (continued) 5 3

10. For a 3×5 array of dots Number of triangles: Methodology 1.) Counting stage (continued) 5 3

11. For a 3×5 array of dots Number of triangles: Methodology 1.) Counting stage (continued) 5 3

12. For a 3×5 array of dots Number of triangles: Methodology 1.) Counting stage (continued) 5 3

13. For a 3×5 array of dots Number of triangles: Methodology 1.) Counting stage (continued) 5 3

14. For a 3×5 array of dots Number of triangles: 455 – 43 = 412 Methodology 1.) Counting stage (continued) 5 3

15. The table shows the number of triangles obtained in an m × n array of dots, where m and n are integers. Methodology 1.) Counting stage (continued) n m 2 3 4 5 6 7 2 4 18 48 100 180 294 3 18 76 200 412 738 1,200 4 48 200 516 1,056 1,884 3,052 5 100 412 1,056 2,148 3,820 6,176 6 180 738 1,884 3,820 6,772 10,930 7 294 1,200 3,052 6,176 10,930 17,616

16. 2.1) 3× n array of dots 2.2) 4× n array of dots 2.3) 5× n array of dots Methodology 2.) Formularizing stage 1. Note the method to obtain the number of triangles. 2. Break the rows into groups of three. 3. Consider each grouped row for the relation of the number of straight lines. 4. Record the result. 5. Check the relation by substituting values of m and n .

17. Methodology 2.1) 3×n array of dots (1, c 1 ) (2, c 2 ) (3, c 3 ) n

18. The equation is formed by using slope equality. Methodology

19. Substitution into the equation results as follows: However, the number obtained is only for one-direction inclination. Indeed, there are two-direction inclinations. If n = 1, the number of diagonals is 0. If n = 2, the number of diagonals is 0. If n = 3, the number of diagonals is 1. If n = 4, the number of diagonals is 1 + 1. If n = 5, the number of diagonals is 1 + 1 + 2. If n = 6, the number of diagonals is 1 + 1 + 2 + 2. If n = 7, the number of diagonals is 1 + 1 + 2 + 2 + 3. · · · · · · Methodology Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1

20. Relations are obtained from the format: Multiply the above relation by 2 to get the total number of diagonals, both leaning to the left and right to get: Methodology Case 1: Case 2: Case 1: Case 2:

21. Methodology 2.2) 4×n array of dots (2, c 2 ) (3, c 3 ) (4, c 4 ) (1, c 1 ) n

22. Relations leading to number of diagonals are as follows: Methodology 2.2) 4×n array of dots (continued)

23. Methodology 2.3) 5×n array of dots (3, c 2 ) (4, c 4 ) (5, c 5 ) (2, c 2 ) (1, c 1 ) n

24. Relations are obtained from the format: Methodology 2.3) 5×n array of dots (continued)

25. Each term to be taken off consists of two sub-terms. The front term varies depending on the change of m , but the back term always remains the same. Methodology

26. Result: Relation between size of array of dots (m×n) and the number of triangles available in it.

27. Conclusion Progress has been made in finding the relation leading to the number of triangles available in an m × n array of dots.

28. Thank you.