Playing With The Multiplication Table

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This is going around on Facebook now, and it got me to thinking about when I had time on my hands how I used to play with it.

You can do the whole table like that in a 100×100 grid; I used to do it for fun. You will find many other patterns if you do. In this example, if you will note that there is a difference of 1 between the 2nd column and the 3rd one from left to right. If you diagonal up (from the 10 to the 8) the difference is 2. This stays 2 as you go up the column (9 to the 7, etc.). Then increase the jump from 10 to 7 and the difference is 3, 9 to 6, etc.

They are more fun patterns you can find in numbers:

1 2 3 4 5 6 7 8 9 0
2 3 4 5 6 7 8 9 0
3 4 5 6 7 8 9 0
4 5 6 7 8 9 0
5 6 7 8 9 0
6 7 8 9 0
7 6 5 4 3 2 1
8 7 6 5 4 3 2 1
9 8 7 6 5 4 3 2 1
0 9 8 7 6 5 4 3 2 1

Note how the number line up differently in the top half than they do in the bottom half. All the 2s, 3s, 4s, etc., line up diagonally from left to rig going up. Then from the 7 down point they start angling from left to right again, but going downwards instead of upwards (did you notice that I reversed the order of the count at 7 to a down count?). To can fill a 100×100 grid, I will settle with a 10×10 matrix for this post, do the multiplication table within it and you will find many more patterns.

What it looks like without the inversion of the count until reaching 0:

1 2 3 4 5 6 7 8 9 0
2 3 4 5 6 7 8 9 0
3 4 5 6 7 8 9 0
4 5 6 7 8 9 0
5 6 7 8 9 0
6 7 8 9 0
7 8 9 0
8 9 0
9 0
0 0
1 0
2 1 0
3 2 1 0
4 3 2 1 0
5 4 3 2 1 0
6 5 4 3 2 1 0
7 6 5 4 3 2 1 0
8 7 6 5 4 3 2 1 0
9 8 7 6 5 4 3 2 1 0
0 9 8 7 6 5 4 3 2 1 0

Below is a 10×10 grid of counting from 1 to 10 up and down the matrix:

1 2 3 4 5 6 7 8 9 0
2 3 4 5 6 7 8 9 0 1
3 4 5 6 7 8 9 0 1 2
4 5 6 7 8 9 0 1 2 3
5 6 7 8 9 0 1 2 3 4
6 7 8 9 0 1 2 3 4 5
7 8 9 0 1 2 3 4 5 6
8 9 0 1 2 3 4 5 6 7
9 0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 0

Note how the 0s cut the table in half, and how all the other numbers line up diagonally. If you start with the 1 in the upper left corner and go diagonally down to the 0 on the lower right you will see the increment is by 2s. If you add number at equal distances from the 0 at the upper right in a column to its corresponding row the sum will be 10.

Below is the count by 1s, 2s, 3s, etc., which is the multiplication table. Pick any number and you will see that its column matches its row.

1  2    3      4   5    6    7     8   9  10
2 4    6      8  10 12  14   16 18  20
3 6    9      12 15 18 21   24 27  30
4 8   12    16 20 24 28  32 36  40
5 10 15    20 25 30 35 40 45   50
6 12 18    24 30 36 42 48 54   60
7 14  21   28 35 42 49  56  63  70
8 16  24  32 40 48 52  64  72  80
9  18 27  36 45 54 63  72  81   90
10 20 30 40 50 60 70 80 90 100

If you can ignore the misalignment you will note that as you go down a column, regardless of its number, that it will correspond to the same increment as you go across a row with the same number. You will see that the down count on the left column is mirrored on the right column, but starting one row down and with 2 instead of 1. If you look at the 9’s column you will see that the right side of the numbers are a down count from 9 to 1. The right side of the 8’s column from the 5’s row is an up count by 2s to the top, and a down count by 2s to the bottom. The 5’s row and column make a cross of 5’s from left to right and top to bottom.

If you start with the 10 at the lower left and work up diagonally to the 0 on the upper right, you find this pattern: (1)0, 18,24, 28, 30, 30, 28, 24, 18, 0. If you fold between the 30s the numbers all lay on top of each other. There are many more patterns in the table, look at it and tell me what you can find.

The point of this is to point out the symmetry found in math, and while math can be used in modeling nature, there is no such symmetry in nature above the atomic and molecular level.