2. Discrete Mathematics

3. Study of discontinuous numbers

4. Logic, Set Theory, Combinatorics, Algorithms, Automata Theory, Graph Theory, Number Theory, Game Theory, Information Theory

5. Recreational Number Theory

6. Power of 9s

7. 9 * 9 = 81

8. 8 + 1 = 9

9. Multiply any number by 9 Add the resultant digits together until you get one digit

10. Always 9 e.g., 4 * 9 = 36 3 + 6 = 9

11. Square Root of Palendromic Numbers

12. Square Root of 123454321 = 11111

13. Square Root of 1234567654321 = 1111111

14. Leonardo of Pisa, known as Fibonacci. Series first stated in 1202 book Liber Abaci

15. 0,1,1,2,3,5,8,13,21,34,55,89.. Each pair of previous numbers equaling the next number of the Sequence.

16. Dividing a number in the sequence into the following number produces the Golden Ratio 1.62

17. Debussy, Stravinsky, Bartók composed using Golden mean (ratio, section).

18. Bartók’s Music for Strings, Percussion and Celeste

19. Importance of number sequences to music. music. After all, music is a sequence of numbers.

20. Pascal’s Triangle

22. The sum of each row results in increasing powers of 2 (i.e., 1, 2, 4, 8, 16, 32, and so on). The 45 ° diagonals represent various number systems. For example, the first diagonal represents units (1, 1...), the second diagonal, the natural numbers (1, 2, 3, 4...), the third diagonal, the triangular numbers (1, 3, 6, 10...), the fourth diagonal, the tetrahedral numbers (1, 4, 10, 20...), and so on. All row numbers—row numbers begin at 0—whose contents are divisible by that row number are successive prime numbers. The count of odd numbers in any row always equates to a power of 2. The numbers in the shallow diagonals (from 22.5° upper right to lower left) add to produce the Fibonacci series (1, 1, 2, 3, 5, 8, 13...), discussed in chapter 4. The powers of 11 beginning with zero produce a compacted Pascal's triangle (e.g., 11 0 = 1, 11 1 = 11, 11 2 = 121, 11 3 = 1331, 11 4 = 14641, and so on). Compressing Pascal's triangle using modulo 2 (remainders after successive divisions of 2, leading to binary 0s and 1s) reveals the famous Sierpinski gasket, a fractal-like various-sized triangles, as shown in figure 7.2, with the zeros (a) and without the zeros (b), the latter presented to make the graph clearer.

23. $1 million prize to create formula for creating next primes without trial and error

24. The sum of each row results in increasing powers of 2 (i.e., 1, 2, 4, 8, 16, 32, and so on). The 45 ° diagonals represent various number systems. For example, the first diagonal represents units (1, 1...), the second diagonal, the natural numbers (1, 2, 3, 4...), the third diagonal, the triangular numbers (1, 3, 6, 10...), the fourth diagonal, the tetrahedral numbers (1, 4, 10, 20...), and so on. All row numbers—row numbers begin at 0—whose contents are divisible by that row number are successive prime numbers. The count of odd numbers in any row always equates to a power of 2. The numbers in the shallow diagonals (from 22.5° upper right to lower left) add to produce the Fibonacci series (1, 1, 2, 3, 5, 8, 13...), discussed in chapter 4. The powers of 11 beginning with zero produce a compacted Pascal's triangle (e.g., 11 0 = 1, 11 1 = 11, 11 2 = 121, 11 3 = 1331, 11 4 = 14641, and so on). Compressing Pascal's triangle using modulo 2 (remainders after successive divisions of 2, leading to binary 0s and 1s) reveals the famous Sierpinski gasket, a fractal-like various-sized triangles, as shown in figure 7.2, with the zeros (a) and without the zeros (b), the latter presented to make the graph clearer.

26. Magic Squares

27. Square Matrix in which all horizontal ranks all vertical columns both diagonals equal same number when added together

31. Musikalisches Würfelspiele

36. Number of Possibilities of 2 matrixes is 11 16 or 45,949,729,863,572,161 45 quadrillion

37. Let’s hear a couple

38. X n+1 = 1/cosX n 2

39. (defun cope (n seed) (if (zerop n)() (let ((test (/ 1 (cos (* seed seed))))) (cons (round test) (cope (1- n) test)))))

40. ? (cope 40 2) (-2 -1 -2 -4 -1 -11 -3 2 -1 10 1 -2 -1 2 -9 -2 1 2 29 1 -7 3 -9 -4 1 2 -2 -1 2 -1 3 1 -2 -1 2 4 1 2 -2 -1)

41. Tom Johnson’s Formulas for String Quartet

42. No. 7

43. Iannis Xenakis Metastasis Metastasis