Exploring the
Decathlon
Mathematics Topics
MARY GARNER
PROFESSOR EMERITUS OF MATHEMATICS
K ENNESAW STATE UNIVERSITY
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?
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?
An Indian Folktale
1, 2, 4, 8, 16, 32, 64, 128, …
This is a geometric sequence.
The nth term can be written as
fn = 2fn-1 in recursive form.
And f(n) = 2n (beginning with the
zeroth term)
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?
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.
fn = fn-1 + 100
f(n) = 100n+1
Speaking of Zero…
Brahmagupta 598-668 AD
Or Bhaskara I 600-680 AD (?)
http://www.storyofmathematics.com/indian_brahmagupta.html
Speaking of Zero
https://www.youtube.com/watch?v=fojaO_3OL2I
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.
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-ragalist/
Position 1 2
3
4
5
6
7
Sa
Ga1
Ga2
Ga3
Ma1
Ma2
Pa
Da1
Da2
Da3
Ni1
Ni2
Ni3
Ri1
Ri2
Ri3
Ragas
Position 1 2
3
4
5
6
7
Sa
Ga1
Ga2
Ga3
Ma1
Ma2
Pa
Da1
Da2
Da3
Ni1
Ni2
Ni3
Ri1
Ri2
Ri3
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 6th and 7th positions.
So how many possible ragas are there?
Ragas
Position 1 2
3
4
5
6
7
Sa
Ga1
Ga2
Ga3
Ma1
Ma2
Pa
Da1
Da2
Da3
Ni1
Ni2
Ni3
Ri1
Ri2
Ri3
So there are 1 x 6 x 2 x 1 x 6 = 72 possible ragas
Ragas
No.
01.
02.
03.
Melakarta Raga Name
kanakAngi S R1 G1 M1 P D1 N1 S
rathnAngi S R1 G1 M1 P D1 N2 S
gAnamUrthi S R1 G1 M1 P D1 N3 S
S N1 D1 P M1 G1 R1 S
S N2 D1 P M1 G1 R1 S
S N3 D1 P M1 G1 R1 S
http://www.ragasurabhi.com/carnatic-music/surabhi-post/post-post0001--raga-classification-and-structure.html
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?
Pascal’s Triangle
http://www.cecm.sfu.ca/organics/papers/granville/support/pascalform.html
Pascal’s Triangle
(a+b)2 = 1a2 + 2ab + 1b2
(a+b)3 = 1a3 + 3a2b + 3ab2 + b3
(a+b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + b4
Why?
Pascal’s Triangle
(a + b) (a + b) (a + b) (a + b)
(a + b) (a + b) (a + b) (a + b)
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
(a + b) (a + b) (a + b) (a + b)
(a + b) (a + b) (a + b) (a + b)
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
Pascal’s Triangle
(a + b) (a + b) (a + b) (a + b)
(a + b) (a + b) (a + b) (a + b)
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
(a + b) (a + b) (a + b) (a + b)
(a + b) (a + b) (a + b) (a + b)
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
Pascal’s Triangle
(a + b) (a + b) (a + b) (a + b)
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
(a + b) (a + b) (a + b) (a + b)
1a4 + 4a3b + 6a2b2 + 4ab3 + b4
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?
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