Thursday, August 9, 2012

The Beauty of Maths

Sequential Inputs of numbers with 8
1 x
8 + 1 = 9
12 x
8 + 2 = 98
123 x
8 + 3 = 987
1234 x
8 + 4 = 9876
12345 x
8 + 5 = 98765
123456 x
8 + 6 = 987654
1234567 x
8 + 7 = 9876543
1234567
8 x 8 + 8 = 98765432
1234567
89 x 8 + 9 = 987654321
......................................

Sequential 1's with 9
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 + 10 = 1111111111
......................................

Sequential 8's with 9
9 x 9 + 7 =
88
9
8 x 9 + 6 = 888
9
87 x 9 + 5 = 8888
9
876 x 9 + 4 = 88888
9
8765 x 9 + 3 = 888888
9
87654 x 9 + 2 = 8888888
9
876543 x 9 + 1 = 88888888
9
8765432 x 9 + 0 = 888888888
......................................

Numeric Palindrome with 1's
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
......................................

Without 8
12345679 x 9 = 111111111
12345679 x 18 = 222222222
12345679 x 27 = 333333333
12345679 x 36 = 444444444
12345679 x 45 = 555555555
12345679 x 54 = 666666666
12345679 x 63 = 777777777
12345679 x 72 = 888888888
12345679 x 81 = 999999999
......................................

Sequential Inputs of 9
9 x 9 = 81
99 x 99 = 9801
999 x 999 = 998001
9999 x 9999 = 99980001
99999 x 99999 = 9999800001
999999 x 999999 = 999998000001
9999999 x 9999999 = 99999980000001
99999999 x 99999999 = 9999999800000001
999999999 x 999999999 = 999999998000000001
......................................

Sequential Inputs of
6
6 x 7 = 42
66 x 67 = 4422
666 x 667 = 444222
6666 x 6667 = 44442222
66666 x 66667 = 4444422222
666666 x 666667 = 444444222222
6666666 x 6666667 = 44444442222222
66666666 x 66666667 = 4444444422222222
666666666 x 666666667 = 444444444222222222
......................................

A Magic Square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. Building on the work of Ahmad ibn ‘Ali al-Buni, in about 1510 Heinrich Cornelius Agrippa wrote De Occulta Philosophia expounding on the magical virtues of seven magical squares of orders 3 to 9, each associated with one of the astrological planets.

Saturn = 15
4 9 2
3 5 7
8 1 6
Jupiter = 34
4 14 15 1
9 7 6 12
5 11 10 8
16 2 3 13
Mars = 65
11 24 7 20 3
4 12 25 8 16
17 5 13 21 9
10 18 1 14 22
23 6 19 2 15
Sol = 111
6 32 3 34 35 1
7 11 27 28 8 30
19 14 16 15 23 24
18 20 22 21 17 13
25 29 10 9 26 12
36 5 33 4 2 31
Venus = 175
22 47 16 41 10 35 4
5 23 48 17 42 11 29
30 6 24 49 18 36 12
13 31 7 25 43 19 37
38 14 32 1 26 44 20
21 39 8 33 2 27 45
46 15 40 9 34 3 28
Mercury = 260
8 58 59 5 4 62 63 1
49 15 14 52 53 11 10 56
41 23 22 44 45 19 18 48
32 34 35 29 28 38 39 25
40 26 27 37 36 30 31 33
17 47 46 20 21 43 42 24
9 55 54 12 13 51 50 16
64 2 3 61 60 6 7 57
Luna = 369
37 78 29 70 21 62 13 54 5
6 38 79 30 71 22 63 14 46
47 7 39 80 31 72 23 55 15
16 48 8 40 81 32 64 24 56
57 17 49 9 41 73 33 65 25
26 58 18 50 1 42 74 34 66
67 27 59 10 51 2 43 75 35
36 68 19 60 11 52 3 44 76
77 28 69 20 61 12 53 4 45

The order-4 magic square in Albrecht DÃ¼rer's engraving Melencolia I is believed to be the first seen in European art. The sum 34 can be found in the rows, columns, diagonals, each of the quadrants, the center four squares, and the corner squares (of the 4 x 4 as well as the four contained 3 x 3 grids). This sum can also be found in the four outer numbers clockwise from the corners (3 + 8 + 14 + 9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle, the two sets of four symmetrical numbers (2 + 8 + 9 + 15 and 3 + 5 + 12 + 14), the sum of the middle two entries of the two outer columns and rows (5 + 9 + 8 + 12 and 3 + 2 + 15 + 14), and in four kite or cross shaped quartets (3 + 5 + 11 + 15, 2 + 10 + 8 + 14, 3 + 9 + 7 + 15, and 2 + 6 + 12 + 14). The two numbers in the middle of the bottom row give the date of the engraving: 1514. The numbers 1 and 4 at either side of the date correspond to, in English, the letters 'A' and 'D' which are the initials of the artist. DÃ¼rer's image was distributed widely, and travelled as far away as India, where the Mughal miniaturist Farrukh Beg referenced the work in his 1615 miniature The Old Sufi.