1. Exploring the Decathlon Mathematics Topics MARY GARNER PROFESSOR EMERITUS OF MATHEMATICS KENNESAW STATE UNIVERSITY

2. An Indian Folktale One Grain of Rice by Demi How many grains of rice on day 1? Day 2? Day 3? Day 10? Day 20?

3. An Indian Folktale 1, 2, 4, 8, 16, 32, 64, 128, … What kind of sequence is this? How would you write the nth term of the sequence? What kind of function can be used?

4. An Indian Folktale 1, 2, 4, 8, 16, 32, 64, 128, … This is a geometric sequence. The nth term can be written as f n = 2f n-1 in recursive form. And f(n) = 2 n (beginning with the zeroth term)

5. An Indian Folktale What would the sequence look like if you added 100 grains of rice each day? What kind of sequence is this? How would you write the nth term of the sequence? What kind of function can be used?

6. An Indian Folktale What would the sequence look like if you added 100 grains of rice each day? 1, 101, 201, 301, 401, … An arithmetic sequence. f n = f n-1 + 100 f(n) = 100n+1

7. Speaking of Zero… http://www.storyofmathematics.com/indian_brahmagupta.html Brahmagupta 598-668 AD Or Bhaskara I 600-680 AD (?)

8. Speaking of Zero https://www.youtube.com/watch?v=fojaO_3OL2I

10. Back to India and Sequences … There are two types of Indian classical music: ◦Hindustani ◦Carnatic (South Indian Classical Music) A note in Carnatic music is called a swara and the note names are Sa Ri1,2,3 Ga1,2,3 Ma1,2 Pa Da1,2,3 Ni1,2,3 Each of these notes can take one of 7 positions to form a “Raga.” Ragas are special and specific note arrangements. There are certain rules, however, about how to place the notes.

11. Ragas “ A raga is a scientific, precise, subtle and aesthetic melodic form with its own peculiar ascending and descending movement consisting of either a full seven note octave, or a series of six or five notes (or a combination of any of these) in a rising or falling structure called the Arohana and Avarohana…” – Ravi Shankar http://www.hitxp.com/keyboard-music-notes/articles/carnatic-melakartha-raga- list/ http://www.hitxp.com/keyboard-music-notes/articles/carnatic-melakartha-raga- list/ Position 1234567 SaRi1 Ri2 Ri3 Ga1 Ga2 Ga3 Ma1 Ma2 PaDa1 Da2 Da3 Ni1 Ni2 Ni3

12. Ragas Position 1234567 SaRi1 Ri2 Ri3 Ga1 Ga2 Ga3 Ma1 Ma2 PaDa1 Da2 Da3 Ni1 Ni2 Ni3 Rules: Sa must be in position 1 and Pa must be in position 5. In position 2 and 3, the second note must be ½ note higher than the R note. So if Ra1 is picked the possible notes for the third place are any of the 3 G notes, but if Ra2 is picked, you can only have Ga2 or Ga3, and if Ra3 is picked, you can only follow with Ga3. Likewise for the 6 th and 7 th positions. So how many possible ragas are there?

13. Ragas Position 1234567 SaRi1 Ri2 Ri3 Ga1 Ga2 Ga3 Ma1 Ma2 PaDa1 Da2 Da3 Ni1 Ni2 Ni3 So there are 1 x 6 x 2 x 1 x 6 = 72 possible ragas

15. http://www.ragasurabhi.com/carnatic-music/surabhi-post/post-- post0001--raga-classification-and-structure.html No.Melakarta Raga Name 01.kanakAngiS R1 G1 M1 P D1 N1 SS N1 D1 P M1 G1 R1 S 02.rathnAngiS R1 G1 M1 P D1 N2 SS N2 D1 P M1 G1 R1 S 03.gAnamUrthiS R1 G1 M1 P D1 N3 SS N3 D1 P M1 G1 R1 S Ragas

16. Combinations What if I just want to pull 5 notes (skip the first and fifth position) from 14, and once I have the notes, I put them in order? How many ragas would I have?

18. Pascal’s Triangle http://www.cecm.sfu.ca/organics/papers/granville/support/pascalform.html

19. Pascal’s Triangle (a+b) 2 = 1a 2 + 2ab + 1b 2 (a+b) 3 = 1a 3 + 3a 2 b + 3ab 2 + b 3 (a+b) 4 = 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 Why?

20. Pascal’s Triangle (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4

21. Pascal’s Triangle (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4

22. Pascal’s Triangle (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 (a + b) (a + b) (a + b) (a + b) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4

23. Combinatorics and Probability Suppose 2 students need to be selected from a team of 5 students to participate in a special competition. Suppose also 3 of the 5 students are female and 2 are male. What’s the probability that the selected team would be male only?

24. Summary  The Indian folktale One Grain of Rice, arithmetic and geometric sequences, linear and exponential functions.  India saves the world (algebra) by inventing nothing (zero)!  Carnatic music and the multiplication principle.  Pascal’s triangle, number sequences, combinatorics, probability. Thank you! mgarner@kennnesaw.edu