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Old 2010-04-13, 22:25   #1
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Default Magic Squares

Christian Boyer posted these puzzles Magic Squares on his site.

Thank you for your time

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Old 2010-04-26, 02:43   #2
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Toshihiro Shirakawa has solved 2 of the enigmas:

Main Enigma #5 solved!
Small Enigma #3a solved!

His solutions can be viewed here
Solutions
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Old 2010-05-03, 14:34   #3
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I didn't solve main enigma #1 but I found a semi magic square where 1 of the 2 diagonals also has the correct sum So halfway between semimagic and magic, is there a name for that?

1032 3022 3942
4462 2332 622
2182 3342 3132

Sum=257,049 (red diagonal sum = 162,867)

Last fiddled with by ATH on 2010-05-03 at 14:36
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Old 2010-05-03, 23:55   #4
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Quote:
Originally Posted by ATH View Post
I didn't solve main enigma #1 but I found a semi magic square where 1 of the 2 diagonals also has the correct sum So halfway between semimagic and magic, is there a name for that?

1032 3022 3942
4462 2332 622
2182 3342 3132

Sum=257,049 (red diagonal sum = 162,867)
well could it fit demisemimagic square ?
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Old 2010-05-06, 01:23   #5
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Quote:
Originally Posted by ATH View Post
I didn't solve main enigma #1 but I found a semi magic square where 1 of the 2 diagonals also has the correct sum So halfway between semimagic and magic, is there a name for that?

1032 3022 3942
4462 2332 622
2182 3342 3132

Sum=257,049 (red diagonal sum = 162,867)
ATH,

Good work, unfortunately this has already been discovered. This is in the Lucas 3x3 Semi Magic Squares of Squares family:

with the following values

p=4
q=9
r=11
s=17

plugging those values in Lucas's equation and you will get your magic sum of 5072

Keep up the good work!

Thank you for your efforts

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Old 2010-05-11, 23:34   #6
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for the 3x3 magic square of squares solve a2 + b2=c2 such that a and b take 4 different values each
then fill in the center:

a,a,a,
b,x,a,
b,b,b,


so :

1) is there a a^2 + b^2 = c^2 such that c^2 can be represented 4 ways with a and b each being unique.
2) can you plug this value into a^2+b^2=c^2 to get a value you can plug in the center and still give a square ?

this might help

Last fiddled with by science_man_88 on 2010-05-12 at 00:29
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Old 2010-05-12, 13:29   #7
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See this research by Lee Morgenstern: http://home.earthlink.net/~morgenstern/magic/sq3.htm

We can represent the square like:

Code:
     c+a      c-(a+b)  c+b
     c-(a-b)     c     c+(a-b)
     c-b      c+(a+b)  c-a
So we need all these numbers to be square: c, c±a, c±b, c±(a+b), c±(a-b)

I made a program to search for a solution to this but no luck.

There are many square c which have many candidates a and b where c±a and c±b is also square, but the "hard" part is that c±(a+b) and c±(a-b) also need to be square, which they are not, so far.

Last fiddled with by ATH on 2010-05-12 at 13:32
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Old 2010-05-12, 14:36   #8
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a^2+b^2+d^2=e^2

a^2+b^2=c^2
c^2+d^2=e^2
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Old 2010-05-12, 14:43   #9
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this is basically what you have to solve in the 3*3 case and try for d such that one of the other three gets repeated in 4 but no others repeat except the d value. though I'm probably wrong.

Last fiddled with by science_man_88 on 2010-05-12 at 15:01
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Old 2010-06-09, 23:42   #10
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what I think is that most pythagorean triples (if not all) follow the formulae:

b=1/2n*a^2 + n^2/2n
c=1/2n*a^2 - n^2/2n

so basically to solve this (or have a chance with pythagorean triples) we need to find a c that works for 4 n values that hopefully will give 4 unique pairs of a and b that c can then be plugged in as b to find another a or as a and solve for a new b to go in the middle 25 is the lowest repeating c so far but I only found it twice in my lists. (and so the search continues).


is there software to check when multiple equations get the same y value ? if not should I try and make some ?


so far I can figure:

3,4,5 6,8,10 9,12,15 12,16,20 15,20,25
5,12,13 10,24,26 15,36,39 20,48,52 25,60,65
7,24,25 14,48,50 21,72,75 anyway I'm bored of typing it out

and more still

Last fiddled with by science_man_88 on 2010-06-09 at 23:48
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Old 2010-06-11, 00:15   #11
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sorry b equation should do - and c should do + messed up

Last fiddled with by science_man_88 on 2010-06-11 at 00:26
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