Do numbers have shapes, and if these shapes exist – “How can we apply it in our world?”

This exercise started a few years back over an idea that numbers in a grid, a surface area – had shapes.

The rules are simple:

A 1x1 grid has only 1 number = 1

1


A 2x2 grid has a maximum of 4, a minimum of 1 and is equally distributed into a grid on a surface

1

2

3

4


A 3x3 grid has a maximum of 9, a minimum of 1 and is equally distributed into a grid on a surface

1

2

3

4

5

6

7

8

9


I think you can see where I am goind with this. The grid is equally distributed on the horizontal axis, and on the veritcal axis.

The grids continue to grow in this fashion 4x4, 5x5, 6x6, and so on.

The numbers begin to grow at a power of 2 since we are only working on grids: 62, 72, 82, 92 and so on.

All this is pretty basic, and are the rules to an ever growing grid, infinitum.

Since we are expanding our grid, we will only use the addition operator in our analysis.

Since we have a defined x and y axis : this number will be our specific constant.

The goal is to use the “specific constant” to add numbers so that they equal a “specific number” …

You cannot repeat the same number in the equation, but might use a similar number in other equations ( a middle set or number ).

It is probably easier if I just get on with it. Agreed!

Lets ignore 1, since there is nothing we can do with one 1 + ( nothing ) = 1

However, with a 2x2 grid, we do get to do something…

1

2

3

4


Specific Constant = 2 also equals the grid size
We need to use all the grid values

1+4 = 5
3+2 = 5

We needed to add two numbers to define the “Grid Constant” = 5

So, a 2x2 grid has a specific constant of 2 and a grid constant of 5

Two basic shapes appeared, and these two shapes will be a basis for all future grids – infinitum




























2 Diagonals, that when combined represent all the numbers of the grid. Sounds easy enough.

Lets start with a more robust grid – the 3x3 grid

1

2

3

4

5

6

7

8

9


Specific Constant = 3
We need to add three numbers to attain our “Grid Constant” = 15
So, a 3x3 grid has a spefic constant of 3 and a grid constant of 15

In order to use all the numbers in a grid, the most efficient method would be:

7+5+3=15
1+5+9=15
2+5+8=15
4+5+6=15

Their corresponding shapes for this grid are:

























































































We also can define a new number that we will call a “grid efficient variable” … which is basically how many shapes have been found given the grid size.

It should be noted that as we are working with a grid, a power of 2, our grid shapes should be symmetrical and have the least resistance possible …
in other words, any “jackass” can find three numbers to add to fifteen – but were they symmetrical?
Someone might say, well I can add 7+4+4=15 – but this is neither symmetrical and disobeys a rule, that no number will be used twice in an equation.

As stated before, the two diagonal shapes were also observed in the 2x2 grid – and it will be to noones suprise that they will appear on every subsequent grid.

All still pretty basic… right! But grids start to grow, sort of like the universe, molecules, and cells.

Lets look at the 4x4 grid

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16


Specific Constant = 4
We need to add three numbers to attain our “Grid Constant” = 34
So, a 4x4 grid has a spefic constant of 4 and a grid constant of 34

13+10+7+4=34
1+6+11+16=34
6+7+10+11=34
1+4+13+16=34
2+3+14+15=34
5+9+8+12=34

Their corresponding shapes for this grid are:























































































































































































1







2







3







4







5







6



The most efficient system, or use the least amount of shapes to attain the usage of all grid numbers, would be eliminating shape (3) and shape (4) … leaving us with:

13+10+7+4=34
1+6+11+16=34
2+3+14+15=34
5+9+8+12=34


























































































































When you overlap all these shapes, all the numbers of the grid are consumed, and we get the next constant… the “zero sum grid constant”.
So lets recap:


SPECIFIC CONSTANT
( grids x and y axis )

GRID CONSTANT
( Diagonal Addition )

ZERO SUM GRID
( Patterns Found )

2x2 GRID

2

5

2

3x3 GRID

3

15

4

4x4 GRID

4

34

4





Continue on, to the next grid, which is the 5x5 grid … look ma, I can count!

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25


This grid is probably one of my favorites, It has a lot of interesting stuff in it, probably worth noting:

How does 8 come into play? It is the efficient number of shapes to find in a 5x5 grid.

Specific Constant = 5
We need to add three numbers to attain our “Grid Constant” = 65
A 5x5 grid has a spefic constant of 5 and a grid constant of 65
It also has a “Zero Sum Variable” of 8… Prove it!




































































































































1





2





3





4































































































































































5





6





7





8



Shape 1 = 21+17+13+9+5=65
Shape 2 = 1+7+13+19+25=65
Shape 3 = 11+12+13+14+15=65
Shape 4 = 3+8+13+18+23=65
Shape 5 = 2+7+13+19+24=65
Shape 6 = 22+17+13+9+4=65
Shape 7 = 6+7+13+19+20=65
Shape 8 = 16+17+13+9+10=65

On a side note there are two more grid shapes, just like before that were discarded in the 4x4 grid shapes…. They are (9) and (10) below.



































































9






10



When I see these two shapes, two words come to my mind: “micro” and “macro”

And these two shapes resemble a lot, in my opinion, of the 4x4 grid shapes discarded … hmm a pattern, sweet!






















































There are shapes, but are they expanding to more complex forms? Lets take a look at the 6x6 grid:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36


Specific Constant = 6
We need to add six numbers to attain our “Grid Constant” = 111
So, a 6x6 grid has a spefic constant of 6 and a grid constant of 111
I wonder what the “zero sum variable” will be? I guess 8!






















































































































































































































































































































































































Yes… 8 apparently seems to be the shortest route to use all the numbers of the grid and follow a symmetry logic.

31+26+21+16+11+6=111
1+8+15+22+29+36=111
13+14+15+22+23+24=111
19+20+21+16+17+18=111
3+9+15+22+28+34=111
4+10+16+21+27+33=111
25+32+15+22+5+12=111
7+2+21+16+35+30=111

All numbers used.

But as our numbers increase, so do our possible permutations. For example, the 6x6 grid has the above 8 patterns, but I could also accomplished the task with the following patterns:






















































































































































































































































































































































































Again this solution requires 8, to accomplish the same task. Is one more elegant than the other?

I guess, I should pause here, and as they would say “cover my six” … before we head on to the next grid.

7x7 Grid Continue >