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Old 2006-08-27, 13:33   #12
ATH
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I checked all sums up to 120 for 4x4 squares and didn't find any solution until 120, it might be a typo with 102?

Going through all possibilities (I think), I found 128 squares with sum 120 including the one Kees found, but many of them are mirror images:

120.txt
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Old 2006-08-29, 15:03   #13
mfgoode
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Lightbulb Prime magic square 4x4

Quote:
Originally Posted by ATH View Post
I checked all sums up to 120 for 4x4 squares and didn't find any solution until 120, it might be a typo with 102?

Going through all possibilities (I think), I found 128 squares with sum 120 including the one Kees found, but many of them are mirror images:

120.txt

You are possibly right ATH: Excellent work!
If you count the rotations and reflections they should for a 4X4 come up to the region of 880 different squares. This is for 4x4 magic squares made up of the first 16 digits (1-16).

Have you tried making this square? Its a worthy exercise! If you only use pencil and paper and not a computer, and if you can do it in a day, then you are pretty good at such squares and above average.

This square is one of the finest created by man and has numerous properties.
If you present it here then I will post it s various and amazing properties.

Also from the 128 squares you have given if you want to really make sure they are different then join up, in straight lines, the primes in ascending order i.e from the lowest to the next lowest and last to the highest.
You will find distinct patterns formed which a computer can be programmed to find out, as either similar or different.

These patterns can be utilised in textile designs and put to a lot of other uses.

Try it out and best of luck!
Mally
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Old 2006-08-30, 08:57   #14
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Myabe 4x4 with c=102 is possible only if 1 is used.
3x3, c=111 is not possible without 1, right?
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Old 2006-08-31, 01:23   #15
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Quote:
Originally Posted by Kosmaj View Post
Myabe 4x4 with c=102 is possible only if 1 is used.
3x3, c=111 is not possible without 1, right?
You are right! I tried 4x4 squares including 1 to the list of primes and I tried all sums and the lowest sum working was 102, mystery solved:

102.txt

Though it's not a "prime" magic square when 1 is included I guess.


1 is also needed for the 3x3 with sum 111. Without 1 the smallest sum is 177:

3x3.txt

Last fiddled with by ATH on 2006-08-31 at 01:59
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Old 2006-08-31, 06:44   #16
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ATH, nice magics, congratulations!

As mfgoode said, when Dudeney worked on his magics, 1 was considered prime. Now it's not prime by convention, I think, just like Pluto is not a planet since a few days ago It will be nice if somebody in the know can tell us more about this (I mean about 1, not about Pluto )

BTW, books by Dudeney are out of copyright, and some are available free on the web. I found his "Amusements in mathematics" here ready for download. But this one doesn't have a lot on magic squares.

Also by surfing the web I found that the 12x12 magic with minimum constant 4514 is actually composed of first 144 odd consecutive primes (from 1 to 827) which I found really amazing! Maybe next time when you have some free time you can try to construct one of them...

Last fiddled with by Kosmaj on 2006-08-31 at 06:46
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Old 2006-08-31, 16:15   #17
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Thumbs up Using 1 in the prime magic square.

Quote:
Originally Posted by ATH View Post
You are right! I tried 4x4 squares including 1 to the list of primes and I tried all sums and the lowest sum working was 102, mystery solved:

102.txt

Though it's not a "prime" magic square when 1 is included I guess.


1 is also needed for the 3x3 with sum 111. Without 1 the smallest sum is 177:

3x3.txt

Great computing ATH.! You deserve all the credit for your very original work.

Even Martin Gardner skips it with a vague mention and thats why I thought its worth giving it a try and belief in history did the trick!

I suggest you make an entry in Wikipedia as these squares are not mentioned
therein.

Its a pity the definition of the number 1 has been changed to 'not prime' in the 20th century! The former definition was sufficient.

Now I might be jumped upon by prime enthusiasts for this remark so Im ready for a grip around my neck but would welcome comments on the pros and cons

Have you had the inclination to have a go at the 4x4 of the first 16 digits?
With whatever method you are using it should be a piece of cake.

Hint: First work out the magic constant and then arrange the numbers accordingly. There is a slight difference in the method though.
Of course, if you surf the net, you will easily get it but its better to be able to construct it independently at any time with just pencil and paper

Best of luck

mally
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Old 2006-08-31, 16:59   #18
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Thumbs up Thanks for the tip off.

Quote:
Originally Posted by Kosmaj View Post
ATH, nice magics, congratulations!

As mfgoode said, when Dudeney worked on his magics, 1 was considered prime. Now it's not prime by convention, I think, just like Pluto is not a planet since a few days ago It will be nice if somebody in the know can tell us more about this (I mean about 1, not about Pluto )

BTW, books by Dudeney are out of copyright, and some are available free on the web. I found his "Amusements in mathematics" here ready for download. But this one doesn't have a lot on magic squares.

Also by surfing the web I found that the 12x12 magic with minimum constant 4514 is actually composed of first 144 odd consecutive primes (from 1 to 827) which I found really amazing! Maybe next time when you have some free time you can try to construct one of them...

Your tip off Kosmaj gave me renewed hope on my hunch that the 4x4 did have a constant of 102. I spent a lot of time trying it out and came very close to the solution until I logged in today and found ATH has worked it out already.:surprised
You are right about the copyright and thats how I copied the summary as I also happen to have the book.
Yes Martin Gardner who has written a lot of books on Math recreation (I have most of them) does mention the 12 x12 consecutive prime magic square but eliminates 2 as he says it destroys the parity. He claims that the 12x12 is the smallest one and even publishes it. Its a collectors item in squares and I have his book too.
I liked the very original reason about 1 and Pluto that you gave. Well there are reasons why 1 is no longer considered a prime and like the case of Pluto
there are pros and cons on both sides of the coin.
If I have time I will post the controversy
Mally
BTW: Most books on math recreation have chapters or several on magic squares. There are even books devoted entirely on the construction. The credit goes to Dudeney for introducing this topic to the Math world and the U.S. in particular where it caught on like wild fire.
Needless to say these were well known from antiquity in India and China. There are certain squares even named after Nasik a town in India, and Lo Shu an emperor in China in days gone by, who found the square on a turtle's back!
Personally I have seen a very old square in Nandi in the Fiji islands which is the 3x3 order, and the basic square to start with.
I will be willing to entertain any query on magic squares.
The method for the 6x6 is a unique one and without a computer it is very difficult to construct one of the 36 initial digits it is made up off.
Your term 'magics' is a convenient one to use and you should register and coin it some place.
Mally
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Old 2006-08-31, 18:43   #19
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Well, I made a simple c++ program :) I'm not sure I'm very good with pencil and paper.

For the 4x4 square with numbers 1-16 the sum would have to be (1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16)/4 = 16*17/8 = 34.

Trying manually to make rows with sum 34, which is the "building blocks" of the square: sum34.txt I only made those starting with 1. Trying a few of these rows as the top row and leftmost collumn I found this:

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


Andreas.

Last fiddled with by ATH on 2006-08-31 at 18:43
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Old 2006-09-01, 04:12   #20
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Thumbs up Durer's square 4x4

Quote:
Originally Posted by ATH View Post
Well, I made a simple c++ program :) I'm not sure I'm very good with pencil and paper.

For the 4x4 square with numbers 1-16 the sum would have to be (1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16)/4 = 16*17/8 = 34. ........
Andreas.

You are on the right track Andreas.

Here I'm referring you to a m/square used by Albrecht Durer in his engraving titled 'Melancholia'

This contains one of the most fantastic 4x4 magic squares ever created by man. Moreover even more astounding, the year he made it is at the bottom row (the middle two numbers) 1514. I have made a full analysis of this square(so have others) Try this URL out and if it is too much to digest click straight on the title 'Melancholia' I have given as you scroll down. Look out for the m/square in the right hand corner and you will see the year 1514 A.D. the year he created it in.
Later I will give you my total analysis on this m/
square and I will show you how to construct it every time.

http://www.bemyastrologer.com/albrec...ic_square.html

Thats not coming thru so try this:

http://en.wikipedia.org/wiki/Melancholia_I


Regards,

Mally

Last fiddled with by mfgoode on 2006-09-01 at 04:19
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Old 2006-09-01, 14:23   #21
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1 (one) is not a prime, but a possible prime, which at same time is a square.
It is = (0*6 +1) * (0*6+1).
2 and 3 are not possible primes and should never be considered as primes.
If you want to know more about possible primes you can look up some of
my previous threads.
Y.s.
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Old 2006-09-01, 15:52   #22
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Quote:
Originally Posted by troels munkner View Post
1 (one) is not a prime, but a possible prime, which at same time is a square.
It is = (0*6 +1) * (0*6+1).
2 and 3 are not possible primes and should never be considered as primes.
If you want to know more about possible primes you can look up some of
my previous threads.
Y.s.

Im on the 1st chapter of your book troels where you mention this.
I'll comment when I finish it
Mally
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