The Worlds Largest Magical Hexagon

For your consideration, I present to you an astounding structure!

Some might have thought it improbable, A few would even have said "Impossible!"

However, What you will see here is something truly amazing.

It doesn't slice, nor does it dice. What does it do?

Absotively, posilutely Nothing!

But it doesn't start with 1, and it is the biggest one that doesn't in the world today!

gasp!

Previously seen only by up to 6 Non paying observers, this Structure had existed, available for 15 US Dollars, as the inside back cover of "Magic Hexes; Why One? copyright February 2004, Louis Hoelbling, [email protected]", but due to the overwhelming nonresposiveness of the world at large I offer you this image for...
Not $10, Not $5, you don't even have to send me a single cent!

ITS FREE!!!

Yes, you now can see a 6 Ring Magical Hexagon, that uses all the integral numbers from 21 thru 111 inclusively! Its Magic Number is 546, and every one of its rows, in any of the 3 major hexagonal directions adds to this magical number.

Certainly, I have generated thousands of 5 Ring Magical hexagons on My PC, and millions of 4 Ring magical hexagons, and lets not forget I have every {56} single 3 Ring Magical Hexagon that are individually unique, {ie... any other 3 ring magical hexagon is a transformation via rotation, reflection, multiplication, or any combination thereof} But This 6 Ring is My Largest, 1 of several Hundred that I've generated so far.

To Quote the Famous Gentleman, a man of High standing in the Mathematical world at large, Mr. Martin Gardner, This is part of "Some new path-breaking results on Magic Polygons"

So, Come One, Come All!

refreshements can be found in the back.



http://www.vbforums.com/attachment.p...postid=1691099

:wave:
-Lou

-PS: I would have posted a 7 ring version by now, but I had to quit developing my Progie. It generates the 3 rings in under a second, it starts iterating all the 4 rings within seconds, it hits the first 5 Ring within 2 minutes, and with the proper tweaking I can get it generating 6 ring Magical Hexagons within 30 minutes. But, with so many numbers in a 7 ring, I need a new strategy for analyizing my triangulating equations, to get it to return a solution within a mortals lifespan.

Actually it's not that I have too much to do, more that I would hav no idea where to start.
and it seems quite boring. At least until you get the program working fully.

"Fully" is a relative term.
How far should I develop it? It Theoretically will return every single Magical Hexagon of any size that doesn't exceed the limits of VBNet, so My progie IS fully developed in that respect. However, anything > 5 Rings takes ever increasingly greter periods of time, and 7 rings takes that timespan to a limit that exhausts even my patience.

However, I would put my progie up against anyone elses in a speed and flexability competiton right now. And yet, what I'd really like to do is get some corporate sponsership, have some Uni of Programming or Engineering get involved too, and get this competition give out real prizes, such as scholorships for programming, and/or money.

If I were to try to develop such a competiton, I'd have to keep myself out of the actual competing part of it. So, All I could offer to you and any others would be what I know at this time about streamlining the mathematical processing of the generic equations of N ring Magical Hexagons, including my analysis of Families, Sub Families, and their child cells, and their mathematical interrelationships, plus optimized set theory analysis of packing families with available integers, and methods to determine wether packed families will lead to a successful Magical Structure.

So, thats what I can offer, if people are interested, and perhaps with enough of a showing we could get some sort of sponsership, and offer a true yearly international competition.

:wave:
-Lou

{BTW, I also voted "Not enough time" or however I phrased it.}
:bigyello:

I realised as I wrote it that i shouldn't have used the work fully, I just posted anways, knowing people would understand anway. If you want a Uni (or anyone else) to back this up. Contact them, if the place is local, go see them in person as that will seem infinately more impressive than an email which will just get redirected to their junkmail anway. I'm not entirely sure how you'd go about asking them for the thing though. You might want to find a maths only forum and post around there first.

While posting the previous post, I (in a different tab as I sse FX) was searching for a maths only forum. I just typed in www.mathrus.com (Maths R Us) hoping it would exist. Go there, I got a half decent laugh from it.

Does the magical hexagon have any point beyond itself? Is the only point of making a magical hexagon the thrill of finally figuring it out?

Or is there some way they're applied?

Why not use a genetic algorithm? Should be dead fast then.

Quote:

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Originally posted by alkatran
Does the magical hexagon have any point beyond itself? Is the only point of making a magical hexagon the thrill of finally figuring it out?

Or is there some way they're applied?

---

Just like Magic Squares, Magical Hexagons have no apparent application to the real world.

However, since they are a series of ordered integers with pseudo random properties, I've thought that they could be used for keys to encrypted files, or perhaps they can lend themselves somehow to compression.

Feel free to contribute possible uses!
:wave:

Quote:

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Originally posted by wossname
Why not use a genetic algorithm? Should be dead fast then

---

Now THAT is why I love VBForums!

Its an absolutely phenominal Brain Trust!

So, wossname, I could go and google "genetic algorithm", but since you brought it up, perhaps you could describe them a little bit?

Thanks

:wave:
-Lou

Ahhh, I see. Good Starting Point

I do incorporate a bit of that, but I couldn't get my head around a pure GA approach, since it seems that it would only produce a single result.

Perhaps you could develop a sample app to process simple 3 or 4 ring hexagonal structures, or pseudocode a hypothetical approach using GA, which would iterate every single solution?

Later today, I'll try to explain the approach I use.



:wave:

Did you ever manage to make a seven one then?

Nope. Like I think I said, I've been too busy since February to bash my brains around a certain area that needs optimization when #Rings > 6.

How about You?

If you're willing to give it a go, attached you'll find a series of equations that, when satisfied, builds values of variable cells from a set of control cells and control families. When all cells are filled with a complete, continuous range of integers, satisfying the equations, then the 7 Ring Magical Hexagon adds to the indicated Magical Number.

The Equations are the most general possible, so if you targeted a 7 Ring Magical hexagon, Adds to 0, with -1* Symmetry across the center cell, then the equations could collapse some more, cutting the # Control cells by about half. Thus speeding up whatever progie you use geometrically.

Also, in my next post, I've attached a screenshot, so you can see where the cells are in the hex structure.

:wave:
-Lou

just a screenshot.

hmm, and not a very good one at that.
Let me try again in a minute.

Ok. Thats better.
The Blue cells are the control cells, and the red cells are the variable, built from mathematical operations on the control cells and control families.

Have fun!

:wave:
-Lou

What is the order of your algorithm?

Quote:

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Originally posted by Darkwraith
What is the order of your algorithm?

---

Sorry, I just woke up.

What do you mean?

:wave:
-Lou

The order of the algorithm... The Big-O of the algorithm... the efficiency...

For example, a linear search has an order of O(n) in the worst case. This means in order to find a given element, the algorithm would have to search through n elements in the worst case.

Hmmm, I see.

Alright. The goal is to rearrange N distinct integers to create the desired Magical property, in an Hexagonal arrangement. Certainly iterating thru every N! Possible arrangement would be how some would program it, however there are ways to speed up the process.

I've mentioned "Families", ie.. a family is a group of 6 cells which are rotationally equivalent. There are relationships between families, or more explicitely, the sum of the elements of a family entertain formulaic relationships with the sums of the elements of the other families.

Since the order of the elements in the families do not impact their sum, Ie.. 3 + 4 + 5 = 3 + 5 + 4 = 5 + 4 + 3, then What I've done is sped up the process to determine valid sum values of the families, and using whats been determined as valid target sums, then build sets of possible elements for each family that satisfies the families sum value.

Then, with the sets of possible elements, I start iterating thru their permutations.

This certainly reduces the number of iterations remarkably from N!, but I am not mathematically capable of expressing the exact figure.

One other thing, as seen in my text file of formulas, one doesn't have to iterate on Every element of the structure, because about half of the elemnts are derived from the other half. All one has to do is make sure that, as one iterates the control elements, the "variable" elements remain possible, and are not gravitating beyond the Lownum or Highnum of the range of integers, nor are they approaching a range of integers that have already been pulled.

So, there's alot of "Looking Ahead" in the formulas, watching when formulas are becoming impossible, and when they do, determining which control element previously determined needs to be adjusted, and by how much.

So, to answer your question,

"What is the order of your algorithm?"

Ummmh, Sorry!

I have no Idea.
:blush:


:wave:
-Lou

Quote:

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But, with so many numbers in a 7 ring, I need a new strategy for analyizing my triangulating equations, to get it to return a solution within a mortals lifespan.

---

Then how do you know that this is true? It might take a few days, but you might actually get a 7 ring.

Quote:

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Originally posted by Darkwraith
Then how do you know that this is true? It might take a few days, but you might actually get a 7 ring.

---

I was running it on my work computer.
I ran it for 4 days.
It got to a valid family sequence sometime after 24 hrs.
Using that family sequence, after the next 3 days, it still hadn't gotten to the first complete solution.

did I mention, I tweaked my progie to return a 6 ring solution before I turned old and grey?

The tweak was making it incriment thru possible family values in units of 100. It skipped {I Assume} Many viable solutions. But I got it to return a solution, once it discovered a viable set of family sums, within 2 hours.

That was after I gave up onm letting it sequence thru the family sums normally.

Again, I have a family sum that I know will solve a 7 ring, {well, I can get back to it after 24 hrs, I never recorded it}, but still, with the # of elements, my techniques don't create a solution from that point after 3 or 4 days. And so, I'm not satisfied. Esp since I "cheat". And, the reson for skipping, is there are certyain families that are linked, ie.. hmmm, well I won't get into it, except to say I need to revamp my calculations to find the sum of a certain set of groups of families before I find their individual values.

But I don't have the time to do that at this moment.

But enough about me.
Whers YOUR hex???

Here I am, I've shown you a 5 ring AND a 6 ring. What can YOU do???


Hmmm?

:wave:
-Lou

Man, no need to get snitty on me. I was just trying to understand the problem.

However, because I am such a nice guy, I'll tell you my observations. :) I think it could be possible to do this while iterating through all possible solutions in an efficient manner without a lot of fancy coding.

Quote:

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The tweak was making it incriment thru possible family values in units of 100. It skipped {I Assume} Many viable solutions.

---

This is a very important assumption. You are saying that you are skipping through family values at a haphazard increment, (well, its not random but you are not using heuristics either) but this increment is constant.

Therefore, if you could get ahold of 100 computers (its easier than you think) and have them iterate at an increment of 100 each starting at a different offset, then you should be able to cover all possible solutions within your iterate.

But that is just my thought.

Quote:

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Originally posted by Darkwraith
Man, no need to get snitty on me. I was just trying to understand the problem.

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Sorry. Didn't mean to sound snitty. I was trying to sound more like I was giving a playful jibe.

:wave:
-Lou