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Site last modified: 2021-Oct

Welcome to Relativistic Quantum Reality: Virtual worlds of imaginary particles: The dreams stuff is made of: Life,the eternal ghost in the machine...
This site is dedicated to the quest for knowledge and wisdom, through science, mathematics, philosophy, invention, and technology. 
May the treasures buried herein spark the fires of imagination like the twinkling jewels of celestial light crowning the midnight sky...

Quantum Mechanics is derivable from Special Relativity
See SRQM - QM from SR - Simple RoadMap (.html)
See SRQM - QM from SR - Simple RoadMap (.pdf)
See SRQM - Online SR 4-Vector & Tensor Calculator
See Online Complex-capable RPN Scientific Calculator

***Learn, Discover, Explore***

See also these Math/Physics Javascript programs:

Math: Magic Squares

square #: check

row pos: col pos:

pattern:


Magicsqr

sum


Select some of the text:
Method of algebra

Magic squares can be generally solved by assigning variables to each cell and doing the algebra.

1x1 Magic Square, with Magic Constant=1M
This square is trivally magic, since it is a single number. It can be a "normal" magic square by setting M=1, so that the magic constant M = (1+1)/2*(1/1) = 1.
M


2x2 Magic Square, with Magic Constant=2M
This square is not really considered magic, since all entries are the same number M.
It can be a sort of "semi-normal" magic square by setting M=2.5, so that the magic constant 2M = (1+4)/2*(4/2) = 5
MM
MM


3x3 Magic Square, with Magic Constant=3M
This square is the first non-trivial magic square. It can be a "normal" magic square by setting M=5, so that the magic constant 3M = (1+9)/2*(9/3) = 15, and setting the other variables appropriately, ex. a=2, c=4.
It can be a trivial panmagic square by setting all values to M.
a3M-a-cc
M-a+cMM+a-c
2M-ca+c-M2M-a


4x4 Magic Square, with Magic Constant=4M
This particular algebraic solution is also panmagic and most perfect. It can be a "normal" magic square by setting M=8.5, so that the magic constant 4M = (1+16)/2*(16/4) = 34, and setting the other variables appropriately, ex. a=4, b=14, e=5, g=8.
Since all cell values with M are multiples of 2, the magic constant can be integer.
aba+e-g4M-2a-b-e+g
e4M-a-b-ega+b-g
2M-a-e+g-2M+2a+b+e-g2M-a2M-b
2M-g2M-a-b+g2M-e-2M+a+b+e


5x5 Magic Square, with Magic Constant=5M
This particular algebraic solution is also panmagic. It can be a "normal" magic square by setting M=13, so that the magic constant 5M = (1+25)/2*(25/5) = 65, and setting the other variables appropriately, ex. a=20,b=8,c=21,d=14,g=4.
The 5x5 Magic Square also has the property that every "+", every "x", every extended "+", and every extended "x" also sum to 5M.
abcd5M-a-b-c-d
3M-a-bg3M-b-dM+b-c+d-g-2M+a+b+c
4M-a-c-g-M+a+b+d-gM3M-a-b-d+g-2M+a+c+g
M-d+gM-b+c-d+g-M+b+d2M-g2M-c+d-g
-3M+a+b+c+d5M-a-b-c-g2M-c-M+a+c-d+g2M-a


6x6 Magic Square, with Magic Constant=6M
This particular algebraic solution is also panmagic, which for 6x6 means that it cannot be "normal" (having consecutive number from 1..36) A "normal" 6x6 magic square has a magic constant 6M = (1+36)/2*(36/6) = 111, giving M=18.5
However, there are cells with 3M and 5M, meaning non-integer.
This particular 6x6 also has many symmetric 2x3 and 3x2 subsets that sum to 6M
ab3M-a-b2M-s2M-ts+t-M
gh3M-g-h2M-y2M-zy+z-M
3M-a-g3M-b-ha+b+g+h-3Ms+y-Mt+z-M5M-s-t-y-z
st3M-s-t2M-a2M-ba+b-M
yz3M-y-z2M-g2M-hg+h-M
3M-s-y3M-t-zs+t+y+z-3Ma+g-Mb+h-M5M-a-b-g-h


7x7 Magic Square, with Magic Constant=7M
This particular algebraic solution is also panmagic. It can be a "normal" magic square by setting M=25, so that the magic constant 7M = (1+49)/2*(49/7) = 175.
abcd-a-e-n-b-f-j+w-c-2g-k-p+t-d-h-l+x-i-r+14Me+n-s-v+b+2f+3j+o-w+c+4g+3k+3p+d+2h+3l+2q+u-x+i+m+2r-28Ms+v-b-f-2j-o-c-2g-2k-2p-t-d-h-2l-2q-u-m-r+21M
efghia+e+n+b+j-w+g+p-t+d+l+u-m+r-7M-a-2e-n-b-f-j+w-2g-p+t-d-h-l-u-i+m-r+14M
-f-j-o-c-g-2k-p-t-h-l-q-m+14Mjklm-j-g-k-p-d-h-2l-q-u-i-m-r+14Mf+j+o+c+2g+2k+2p+t+d+2h+2l+2q+u+i+m+r-21M
nopqr-e-n+s-b-f-2j-o+w-c-3g-2k-3p-d-2h-3l-2q-u-i-m-2r+28Me-s+b+f+2j-w+c+3g+2k+2p+d+2h+3l+q+u+i+m+r-21M
s-e-n-b-f-2j-o-2g-k-2p-d-h-2l-q-u-i-r+21Mtue+n-s+b+f+2j-w+c+3g+k+2p-t+d+h+2l-x+i+r-14Ms+v-b-f-j+w-c-2g-k-p-l+x-r+7M-s-v+b+f+j+o+g+k+p+l+q+r-7M
vwe+b+f+j+g-t+d+h+l+i-7Mx-e-v+o+k+t+q-i-a-e-n-b-f-j-g+t-d+m+7Ma+e+n-o-w-k-t-h-l-q-x-m+7M
-a-e-n-s-v+f+j+o+c+g+2k+p+t+h+l+q+m-7Me+n+j-w+2g+k+2p+d+h+2l+q+u+i+r-14M-e-b-f-j-c-2g-k-p-d-h-l-i+14M-d-h-l-q-u-x+7Ma+e+s+v-j-o-g-k-p-t-l-q-m-r+7M-s+b+f+j+c+2g+k+p+d+h+2l+q+i+m+r-14Ms-f-j+w-c-2g-2k-2p-h-2l-q+x-i-m-r+14M


8x8 Magic Square, with Magic Constant=8M
This particular algebraic solution is also panmagic. It can be a "normal" magic square by setting M=32.5, so that the magic constant 8M = (1+64)/2*(64/8) = 260.
Since all cell values with M are multiples of 2, the magic constant can be integer.
abc4M-a-b-c2M-j2M-k2M-l-2M+j+k+l
def4M-d-e-f2M-m2M-n2M-o-2M+m+n+o
ghi4M-g-h-i2M-p2M-q2M-r-2M+p+q+r
4M-a-d-g4M-b-e-h4M-c-f-ia+b+c+d+e+f+g+h+i-8M-2M+j+m+p-2M+k+n+q-2M+l+o+r10M-j-k-l-m-n-o-p-q-r
jkl4M-j-k-l2M-a2M-b2M-c-2M+a+b+c
mno4M-m-n-o2M-d2M-e2M-f-2M+d+e+f
pqr4M-p-q-r2M-g2M-h2M-i-2M+g+h+i
4M-j-m-p4M-k-n-q4M-l-o-rj+k+l+m+n+o+p+q+r-8M-2M+a+d+g-2M+b+e+h-2M+c+f+i10M-a-b-c-d-e-f-g-h-i


8x8 Super-Magic Square, with Magic Constant=8M
This particular algebraic solution is also panmagic and most perfect. It can be a "normal" magic square by setting M=32.5, so that the magic constant 8M = (1+64)/2*(64/8) = 260.
Since all cell values with M are multiples of 2, the magic constant can be integer.
This particular 8x8 also has many symmetric 1x4 and 4x1 subsets that sum to 4M
Being most perfect means that every 2x2 subset sums to 4M.
a-a+j+k+a-j+l4M-a-k-l2M-j2M-k2M-l-2M+j+k+l
d4M-d-j-k+d+j-l-d+k+l-2M+a+d+j2M-a+k-d-2M+a+l+d6M-k-a-l-d-j
g-g+j+k+g-j+l4M-g-k-l2M-a+g-j2M-g+a-k2M+g-a-l-2M+k+a+l-g+j
4M-a-d-ga+d+g-j-k4M-a-d-g+j-la+d+k+g+l-4M2M-d-g+j-2M+d+k+g2M-d+l-g2M-j-k+d-l+g
jkl4M-j-k-l2M-a2M+a-j-k2M-a+j-l-2M+k+a+l
4M-a-d-ja+d-k4M-a-d-lk+a+d+j+l-4M2M-d-2M+d+j+k2M-d-j+l2M-k+d-l
a+j-g-a+g+k+a-g+l4M-a+g-j-k-l2M-g2M+g-j-k2M-g+j-l-2M+k+g+l
d+g-j4M-d-g-k+d+g-l-d-g+j+k+l-2M+a+d+g2M-a-d-g+j+k-2M+a-j+l+d+g6M-a-l-d-k-g


9x9 Super Magic Square, with Magic Constant=9M
This particular algebraic solution is also panmagic. It can be a "normal" magic square by setting M=41, so that the magic constant 9M = (1+81)/2*(81/9) = 369.
This 9x9 Magic Square also has many symmetric 1x3 and 3x1 subsets that sum to 3M, which also means the main symmetric 3x3 subsets sum to 9M.
ab-a-b+3Mcd-c-d+3M-j+3M-o-h-i+j+3M+o-q-rh+i-3M+q+r
ef-e-f+3Mgf-l+n-f-g+l+3M-n2a+b-2c-d+e-g-h+j-k+3M-o-p+q+r-s3M-n-t-2a-b+2c+d-e+g+h-j+k-3M+n+o+p-q-r+s+t
-a-e+3M-b-f+3Ma+b+e+f-3M-c-g+3M-d-f+l+3M-nc+d+f+g-l-3M+n-2a-b+2c+d-e+g+h+k-3M+2o+p-q-r+sh+i-j-3M+n-o+q+r+t2a+b-2c-d+e-g-2h-i+j-k+9M-n-o-p-s-t
hi-h-i+3Mj-2a-b+2c+d-e+g+2h+i-2j+k-m2a+b-2c-d+e-g-2h-i+j-k+m+3M-a+3M-qa-c-d+3M-o-p+qc+d-3M+o+p
kl-k-l+3Mmn-m+3M-na-c-g+3M-o+q-s-f+l+3M-n-t-a+c+f+g-l-3M+n+o-q+s+t
-h-k+3M-i-l+3Mh+i+k+l-3M-j-m+3M2a+b-2c-d+e-g-2h-i+2j-k+m+3M-n-2a-b+2c+d-e+g+2h+i-j+k-3M+nc+g-3M+o+s-a+c+d+f-l-3M+n+o+p-q+ta-2c-d-f-g+l+9M-n-2o-p+q-s-t
op3M-o-pqr3M-q-r-c-h+3Ma-c-d+e-g-h-i+j-k+m+3M-a+2c+d-e+g+2h+i-j+k-m-3M
st3M-s-t-2a-b+2c+d-e+g+2o+p-2q-r+s-l+n+t2a+b-2c-d+e-g+l+3M-n-2o-p+2q+r-s-t-a+c-e+h-j-m+3M-f+3M-na-c+e+f-h+j+m-3M+n
3M-o-s3M-p-t-3M+o+p+s+t2a+b-2c-d+e-g+3M-2o-p+q+r-sl+3M-n-r-t-2a-b+2c+d-e+g-l-3M+n+2o+p-q+s+ta+e+j+m-3M-a+c+d-e+f+g+h+i-j+k-m-3M+n-c-d-f-g-h-i-k+9M-n

SiteMap of SciRealm | About John | Send email to John
John's Science,Math,Philosophy Stuff:
The Science Realm: John's Virtual Sci-Tech Universe | 4-Vectors | Ambigrams | Antipodes | Covert Ops Fragments | Cyrillic Projector | Forced Induction (Sums Of Powers Of Integers) | Frontiers | IFS Fractals | JavaScript Graphics | JavaScript Graphics Using Table | Kid Science | Kryptos | Philosophy | Photography | Prime Sieve | QM from SR | QM from SR-Simple RoadMap | QM from SR-Simple RoadMap (PDF) | SR 4-Vector &Tensor Calculator |Quantum Phase | Quotes | RuneQuest Cipher Challenge | Secret Codes &Ciphers | Scientific Calculator (complex capable) | Science+Math | Sci-Pan Pantheist Poems | Stereograms | SuperMagicSqr4x4 |Turkish Grammar |Melike Wilson Art

Welcome to Relativistic Quantum Reality: Virtual worlds of imaginary particles: The dreams stuff is made of: Life,the eternal ghost in the machine...
This site is dedicated to the quest for knowledge and wisdom,through science,mathematics,philosophy,invention,and technology. 
May the treasures buried herein spark the fires of imagination like the twinkling jewels of celestial light crowning the midnight sky...

Quantum Mechanics is derivable from Special Relativity
See SRQM-QM from SR-Simple RoadMap (.html)
See SRQM-QM from SR-Simple RoadMap (.pdf)
See SRQM-Online SR 4-Vector &Tensor Calculator
See Online Complex-capable RPN Scientific Calculator

***Learn,Discover,Explore***