I tried to implement that which was discussed under "Magic to the square"
Well, I must not have implemented it properly.
BUT... I tweaked what I thought it was supposed to do, and when it couldn't find a better, umm, what I've called health of the system, It used the best health available.
In a half hour of programming, I created something that returned the attached in 2 minutes.
Granted, I've been concentrating on iteration methods, but thru iteration, the next is almost identical to the lasst.
So, a Random Search method seems fruitful.
WhatDaYaThink?
[edit]So Now I'm invesigating optimzation of my Magic/LowNum Limit methods, and hoepfully will be able to implement them such that I can accomplish large ring Magic determination in a timely fashion, and in tandom with proper family limitaions, I might be able to tie all that into a search method that will allow me to create a Magic Mesh of order 2 or 3 that is larger than my 8 ring Magic Hex.
Last edited by NotLKH; Jun 10th, 2008 at 09:26 PM.
I liked you better when you were too busy to visit us.
[serious]My lack of knowledge and total feeling of inadequacy towards your higher intellect forced me to diss you as a natural self-defence mechanism.[/serious]
I have to admit i'm not actually planning to open any of your attachments. Maybe you should tell me what it's supposed to do and i'll just whip up something better
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Maybe you should tell me what it's supposed to do and i'll just whip up something better
Those PDF files represent what I have termed "Magic Meshes"
The simplest definition of "Magic Mesh" would be the following:
A Magic Mesh is a graphical representation of a solution of a series of linear equations that all add to the same magic number. There are a total of N variables used in the set of linear equations, so collectively there is a continuous range of N integers which must be used once and only once per solution that together solve the system.
A Magic Mesh visually consists of a series of Regular Polygons whose sides each are built of a set of cells. The Cells represent a particular variable used in the set of linear equations.
The Magic Mesh is identified by #Cells per side of the most external Regular Polygon {Or "Ring"}, and by the #Sides that build the polygons shape.
Furthurmore, a Magic Mesh is also identified by its order number.
I have standardized on the following:
An Order 2 Magic Mesh represents those Magic meshes where the #Cells per side of each Polygon varies from that of the one above it by 2, decreasing from outside to in,
While An order 3 Magic mesh varies the #Cells per Side of each Nested Polygon by only 1, decreasing, from the outermost to innermost.
All the equations of an Order 2 Magic mesh have the same # of variables, while those of an Order 3 do not.
A representative Common Order 2 Magic Mesh is the simple Magic Square.
Similarly, A representative Order 3 Magic Mesh is a Magic Hexagon.
Every "Standard" Equation is represented visually in either an Order 2 or An order 3 in the following manor:
For every non-diagonal equation, the central sets of variables are the set of cells of a side of one of the nested polygons. The equation graphically extends from each vertic cell of that polygon side outwards, first continuing into the cell 1 spot to the left of the left vertice of the side of the polygon immediately above it, and also into the cell 1 spot to the right of the right vertice of the side of the polygon immediately above it.
It then continues outward, intersecting the next polygons cells that are 2 spots to the left of the left vertice and 2 cells to the right of the right vertice, and so on, out to the most outermost external polygon.
And, due to how the #Cells per side of a nested polygon vary as you go from inside to outside, The cells of an order 2 Magic mesh generally are used in only 2 equations, and those of an order 3 are generally used in only 3 equations.
Attached are 2 pdf files representing an order 2 and an Order 3 magic mesh. They both are pentagonal in shape, but you can easily see their differing aspects.
In the order 2, all the rows have only 5 cells each, including what I call 5 Diagonoid rows, which eminate from an external Vertice to the middle cell of the opposite external side, all of which intersect in the center cell of the entire figure. Every Cell of these Diagonoids exceptionally are used by a total of 3 Rows, except, again the Center Cell, which is used by all 5 Diaganoids.
In the order 3, the Row equations use either 4, 5, or 6 cells, while what I call the Half-Diagonals, which start at one of the external vertice, and absolutely terminate at the center cell, only use 4 cells each. In this, the only cell that is used in more than 3 rows is the center cell. All the others are used only 3 times each.
The Order 2 attachement {C0_0_1.pdf} uses the numbers 0 to 30 once and only once, and the Magic Sum is 77.
The order 3 Attachement uses the numbers 11 to 41 once and only once, ands adds to 125.
Last edited by NotLKH; Jun 17th, 2008 at 04:43 PM.
Also, let me illustrate the actual equations for those 5 sided order 2 and order 3 Magic Meshes.
Attached are 2 maps displaying the location of the variables for both systems.
The Order 2 figure represents 31 variables of which I have labeled C_0 thru C_30.
The Order 3 figure also uses 31 variables similarly labeled C_0 thru C_30.
The Equations have been normalized in the following manor:
LOWNUM represents the starting value, the lowest number that must be used across the range of the 31 distinct integers.
LOWNUM has been subtracted from each variable so that the variables actually use the numbers 0 thru 30 once and only once.
MAGIC is that value that each equation adds to if you add LOWNUM back into each variable.
So These are the standard Equations for a 5 sided, 5 cells per external side, 3 ring order 2 Magic Mesh:
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Originally Posted by BillGeek
