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Definition
A magic tesseract is a four-dimensional array, equivalent to the magic cube and magic square of lower dimensions, containing the numbers 1, 2, 3, …, m^4 arranged in such a way that the sum of the numbers in each of the m^3 rows, m^3 columns, m^3 pillars, m^3 files and in the eight major quadragonals passing through the center and joining opposite corners is a constant sum S, called the magic sum, which is given by: S = m(m^4+1)/2 and where n is called the order of the tesseract.
We will consider the magic tesseract of order m = 3.
On this page
http://www.magic-squares.net/magic_tesseract_index.htmyou see the classic magic tesseract of order 3.
I found unconventional magic tesseract of order 3 from different natural numbers:
You not want to obtain the magic tesseract of order 3 of prime numbers?
I recommend the article:
Keh Ying Lin. “Magic cubes and hypercubes of order 3”
http://yadi.sk/d/1EGOJXcyKExTiAnd also the theme at a forum in Russia:
http://e-science.ru/forum/index.php?showtopic=20641&st=0Here you can see an scheme of order-3 magic tesseract projected onto the plane:
This general formula magic tesseract of order 3 with a magic constant s = 3*k/2 (k – constant associativity tesseract):
x10=s-x1-x6 x11=s-x2-x7 x12=s-x4-x8 x13=s-x5-x9 x14=(10*s)/3-2*x1-x2-x4-x5-x6-x7-x8-x9 x15=-((2*s)/3)+x5+x7+x9 x16=-((2*s)/3)+x5+x8+x9 x17=(4*s)/3-2*x5-x9 x18=-((2*s)/3)+x7+x8+x9 x19=(4*s)/3-2*x7-x9 x20=-((5*s)/3)+2*x1+x2+x4+x5+x6 x21=s/3-x5+x7 x22=s/3-x5+x8 y1=s-x1-x2 y2=s-x4-x5 y3=s-x1-x4 y4=s-x2-x5 y5=-s+x1+x2+x4+x5 y6=s-x6-x7 y7=s-x8-x9 y8=s-x6-x8 y9=s-x7-x9 y10=-s+x6+x7+x8+x9 y11=s-x10-x11 y12=s-x12-x13 y13=s-x10-x12 y14=s-x11-x13 y15=-s+x10+x11+x12+x13 y16=s-x14-x15 y17=s-x18-x19 y18=4s/3-2*x18-x19 y19=s-20-x21(You must look at the scheme.)
According to this formula is easy to find the solution of distinct positive integers.
For example:98 280 435 308 466 39 407 67 339 341 454 18 353 1 459 119 358 336 374 79 360 152 346 315 287 388 138 311 379 123 278 151 384 224 283 306 266 175 372 377 271 165 170 367 276 236 259 318 158 391 264 419 163 231 404 154 255 227 196 390 182 463 168 206 184 423 83 541 189 524 88 201 203 475 135 503 76 234 107 262 444 S=813However, to find a solution of distinct primes difficult.
I have not managed to do it.I invite you all to solve this problem.
This is the minimal unconventional order-3 tesseract of different natural numbers:
23 37 66 29 70 27 74 19 33 40 78 8 82 3 41 4 45 77 63 11 52 15 53 58 48 62 16 35 67 24 71 25 30 20 34 72 79 9 38 1 42 83 46 75 5 12 50 64 54 59 13 60 17 49 68 22 36 26 31 69 32 73 21 7 39 80 43 81 2 76 6 44 51 65 10 57 14 55 18 47 61 S=126I found such an approximation of prime numbers:
8297 33563 8609 18149 11213 21107 24023 5693 20753 10529 7703 32237 18077 30983 1409 21863 11783 16823 31643 9203 9623 14243 8273 27953 4583 32993 12893 21419 16253 12797* 26627 13883 9959* 2423 20333 27713* 23117 20903 6449 155* 16823 33491* 27197 12743 10529 5933* 13313 31223 23687 19763 7019 20849 17393 12227 20753 653 29063 5693 25373 19403 24023 24443 2003 16823 21863 11783 32237 2663 15569 1409 25943 23117 12893 27953 9623 12539 22433 15497 25037 83 25349 S=50469In this solution the six elements are not prime numbers, and many items are repeated.
I’m trying to find the best solution.I have a new approach to solving
14563 33409 12241 21961 6271 31981 23689 20533 15991 32401 9973 17839 12451 34729 13033 15361 15511 29341 13249 16831 30133 25801 19213 15199 21163 24169 14881 20389 10831 28993 13309 33013 13891 26515 16369 17329 17011 25609 17593 20653 20071 19489 22549 14533 23131 22813 23773 13627 26251 7129 26833 11149 29311 19753 25261 15973 18979 24943 20929 14341 10009 23311 26893 10801 24631 24781 27109 5413 27691 22303 30169 7741 24151 19609 16453 8161 33871 18181 27901 6733 25579S = 60213
In this solution is not the same numbers, but ten elements are not prime numbers.
There is little progress.
In that solution only 9 elements are not prime numbers19379 19889 28949 35801 5639 26777 13037 42689 12491 26561 26297 15359 20507 35141 12569 21149 6779 40289 22277 22031 23909 11909 27437 28871 34031 18749 15437 18797 21599 27821 15809 44537 7871 33611 2081 32525 36467 3221 28529 14801 22739 30677 16949 42257 9011 12953 43397 11867 37607 941 29669 17657 23879 26681 30041 26729 11447 16607 18041 33569 21569 23447 23201 5189 38699 24329 32909 10337 24971 30119 19181 18917 32987 2789 32441 18701 39839 9677 16529 25589 26099S = 68217
In these solutions only 8 elements are not prime numbers.
25219 36037 42373 30757 18859 54013 47653 48733 7243 40357 35089 28183 17659 55837 30133 45613 12703 45313 38053 32503 33073 55213 28933 19483 10363 42193 51073 60397 40699 2533 23269 44617 35743 19963 18313 65353 39499 12157 51973 47017 34543 22069 17113 56929 29587 3733 50773 49123 33343 24469 45817 66553 28387 8689 18013 26893 58723 49603 40153 13873 36013 36583 31033 23773 56383 23473 38953 13249 51427 40903 33997 28729 61843 20353 21433 15073 50227 38329 26713 33049 43867S=103629
29423 27749 50321 70841 19763 16889 7229 59981 40283 67901 14741 24851 10631 52889 43973 28961 39863 38669 10169 65003 32321 26021 34841 46631 71303 7649 28541 34949 15731 56813 11621 50909 44963 60923 40853 5717 6599 60953 39941 69173 35831 2489 31721 10709 65063 65945 30809 10739 26699 20753 60041 14849 55931 36713 43121 64013 359 25031 36821 45641 39341 6659 61493 32993 31799 42701 27689 18773 61031 46811 56921 3761 31379 11681 64433 54773 51899 821 21341 43913 42239S=107493
26987 16889 67181 72161 31019 7877 11909 63149 35999 57809 32069 21179 9767 70901 30389 43481 8087 59489 26261 62099 22697 29129 9137 72791 55667 39821 15569 25601 59951 25505 37649 15137 58271 47807 35969 27281 38699 13037 59321 57641 37019 16397 14717 61001 35339 46757 38069 26231 15767 58901 36389 48533 14087 48437 58469 34217 18371 1247 64901 44909 51341 11939 47777 14549 65951 30557 43649 3137 64271 52859 41969 16229 38039 10889 62129 66161 43019 1877 6857 57149 47051S=111057
This latter approximate solution
Wrong complementary pairs:
(10909, 65017), (41473, 34453), (42583, 33343), (17041, 58885), (39289, 36637), (72829, 3097)Dear colleagues!
I invite you to take part in the discussion
http://mathoverflow.net/questions/225907/magic-tesseract-of-order-3-composed-of-prime-numbers
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