Tesseract Knight Tour

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On September 27, 2008, Awani Kumar [1] announced the discovery of the first Magic Knight Tour of a tesseract figure.

By November 21, 2008 he has enumerated 208 magic tours of knights in 4x4x4x4 hyperspace.
Sum of all the rows, columns, pillars and files is 514. Links to more on Knight-tours are at the end of this page.

This figure shows the 4-dimensional tesseract projected as 2-dimensions on a flat surface (this screen). In the following discussion, I will refer to the 4 dimensions as 4 directions.
To save space, distances are not the same in all directions, so the length of the L shaped steps will not be in consistent proportions.

All 64 rows, columns, pillars, and files of this figure sum to the same constant, 514.

However, this tesseract is not magic because none of the eight quadragonals sum to the constant 514. Their sums range from 472 to 556 but the total of these sums is 4072. I am surprised that this total is not 8 times 514!
(No diagonals or triagonals are required to sum correctly for a simple magic tesseract.)

The corners (which are the ends of the quadragonals) are obvious in this diagram. In the text listing following, I have indicated them by underlining the numbers.

For those having difficulty seeing the quadragonals, I show them here.

001   063   208   242   204   059   005   224
233   215   040   026   036   211   237   056
195   253   014   052   010   249   199   030
043   021   230   220   226   017   047   246
Sums are:
472   552   492   540   476   536   448   556

Unlike magic squares, cubes, tesseracts, etc., the space diagonals are not required to be magic, for a rectilinear figure to be a magic Knight Tour.

In studies of Knight tours, the requirement is that all integers in the figure are accessed by a series of identical steps, usually 2 positions in 1 direction, and 1 position in another direction. This example is such a case, with each move utilizing steps in two of the four available directions.

To make this easier to see, I have indicated the numbers 1 to 5 in red, and the numbers 6 to 10 in green.
The numbers are colored similarly in the text version, which follows, of this figure.

This particular knight tour is re-entrant. The first number in the series is exactly one move away from the last number, so on completion of the tour, the next move could be back to the beginning! It is also a four-fold cyclic magic tour. That is, it remains magic when the tour starts from 65, 129 and 193. Awani Kumar has compiled over 50 such tours.

I have not personally checked the orthogonal line sums of this tesseract, but they have been independently confirmed correct by Guenter Stertenbrink. [2]

The text representation of the above tesseract.
This is the form in which Awani Kumar announced this discovery in an email on September 27, 2008.

001  080  191  242    112  033  210  159    177  256  015  066    224  145  098  047
192  241  002  079    209  160  111  034    016  065  178  255    097  048  223  146
113  064  207  130    032  081  162  239    193  144  127  050    176  225  018  095
208  129  114  063    161  240  031  082    128  049  194  143    017  096  175  226

120  057  202  135    025  088  167  234    200  137  122  055    169  232  023  090
201  136  119  058    168  233  026  087    121  056  199  138    024  089  170  231
008  073  186  247    105  040  215  154    184  249  010  071    217  152  103  042
185  248  007  074    216  153  106  039    009  072  183  250    104  041  218  151

189  244  003  078    212  157  110  035    013  068  179  254    100  045  222  147
004  077  190  243    109  036  211  158    180  253  014  067    221  148  099  046
205  132  115  062    164  237  030  083    125  052  195  142    020  093  174  227
116  061  206  131    029  084  163  238    196  141  126  051    173  228  019  094

204  133  118  059    165  236  027  086    124  053  198  139    021  092  171  230
117  060  203  134    028  085  166  235    197  140  123  054    172  229  022  091
188  245  006  075    213  156  107  038    012  069  182  251    101  044  219  150
005  076  187  246    108  037  214  155    181  252  011  070    220  149  102  043

A 4x4x4x4x4 Magic Knight Tour  (added Oct.18, 2008)

Less then 24 hours after I posted this page, Awani Kumar announced to the Internet community by email that he had successfully constructed a dimension 5 magic knight tour. He said

I am happy to inform you that magic knight tour has been extended further into five dimensional hyperspace. A magic tour of knight has been constructed in 4x4x4x4x4 hyperspace. Sum of all the rows, columns, pillars, files and poles is 2050. Guenter Stertenbrink has checked it. Thank you very much Guenter, for checking and putting it on the computing magic knight tours web site.

As with the above tesseract, this figure is not considered magic because the 16 quintagonals do not sum correctly. I have again underlined the 32 corners of the figure, but leave it to the reader to figure out the quintagonals (if he is curious).

5x5x5x5x5 decimal:
   1  320  767  962    448  129  834  639    705 1024   63  258    896  577  386  191
 768  961    2  319    833  640  447  130     64  257  706 1023    385  192  895  578
 449  256  831  514    128  321  642  959    769  576  511  194    704  897   66  383
 832  513  450  255    641  960  127  322    512  193  770  575     65  384  703  898

 480  225  802  543     97  352  671  930    800  545  482  223    673  928   95  354
 801  544  479  226    672  929   98  351    481  224  799  546     96  353  674  927
  32  289  738  991    417  160  863  610    736  993   34  287    865  608  415  162
 737  992   31  290    864  609  418  159     33  288  735  994    416  161  866  607

 753  976   15  306    848  625  434  143     49  272  719 1010    400  177  882  591
  16  305  754  975    433  144  847  626    720 1009   50  271    881  592  399  178
 817  528  463  242    656  945  114  335    497  208  783  562     80  369  690  911
 464  241  818  527    113  336  655  946    784  561  498  207    689  912   79  370

 816  529  466  239    657  944  111  338    496  209  786  559     81  368  687  914
 465  240  815  530    112  337  658  943    785  560  495  210    688  913   82  367
 752  977   18  303    849  624  431  146     48  273  722 1007    401  176  879  594
  17  304  751  978    432  145  850  623    721 1008   47  274    880  593  402  175
------
------
 456  249  826  519    121  328  647  954    776  569  506  199    697  904   71  378
 825  520  455  250    648  953  122  327    505  200  775  570     72  377  698  903
   8  313  762  967    441  136  839  634    712 1017   58  263    889  584  391  186
 761  968    7  314    840  633  442  135     57  264  711 1018    392  185  890  583

  25  296  743  986    424  153  858  615    729 1000   39  282    872  601  410  167
 744  985   26  295    857  616  423  154     40  281  730  999    409  168  871  602
 473  232  807  538    104  345  666  935    793  552  487  218    680  921   90  359
 808  537  474  231    665  936  103  346    488  217  794  551     89  360  679  922

 824  521  458  247    649  952  119  330    504  201  778  567     73  376  695  906
 457  248  823  522    120  329  650  951    777  568  503  202    696  905   74  375
 760  969   10  311    841  632  439  138     56  265  714 1015    393  184  887  586
   9  312  759  970    440  137  842  631    713 1016   55  266    888  585  394  183

 745  984   23  298    856  617  426  151     41  280  727 1002    408  169  874  599
  24  297  746  983    425  152  855  618    728 1001   42  279    873  600  407  170
 809  536  471  234    664  937  106  343    489  216  791  554     88  361  682  919
 472  233  810  535    105  344  663  938    792  553  490  215    681  920   87  362
------
------
 765  964    3  318    836  637  446  131     61  260  707 1022    388  189  894  579
   4  317  766  963    445  132  835  638    708 1021   62  259    893  580  387  190
 829  516  451  254    644  957  126  323    509  196  771  574     68  381  702  899
 452  253  830  515    125  324  643  958    772  573  510  195    701  900   67  382

 804  541  478  227    669  932   99  350    484  221  798  547     93  356  675  926
 477  228  803  542    100  349  670  931    797  548  483  222    676  925   94  355
 740  989   30  291    861  612  419  158     36  285  734  995    413  164  867  606
  29  292  739  990    420  157  862  611    733  996   35  286    868  605  414  163

  13  308  755  974    436  141  846  627    717 1012   51  270    884  589  398  179
 756  973   14  307    845  628  435  142     52  269  718 1011    397  180  883  590
 461  244  819  526    116  333  654  947    781  564  499  206    692  909   78  371
 820  525  462  243    653  948  115  334    500  205  782  563     77  372  691  910

 468  237  814  531    109  340  659  942    788  557  494  211    685  916   83  366
 813  532  467  238    660  941  110  339    493  212  787  558     84  365  686  915
  20  301  750  979    429  148  851  622    724 1005   46  275    877  596  403  174
 749  980   19  302    852  621  430  147     45  276  723 1006    404  173  878  595
------
------
 828  517  454  251    645  956  123  326    508  197  774  571     69  380  699  902
 453  252  827  518    124  325  646  955    773  572  507  198    700  901   70  379
 764  965    6  315    837  636  443  134     60  261  710 1019    389  188  891  582
   5  316  763  966    444  133  838  635    709 1020   59  262    892  581  390  187

 741  988   27  294    860  613  422  155     37  284  731  998    412  165  870  603
  28  293  742  987    421  156  859  614    732  997   38  283    869  604  411  166
 805  540  475  230    668  933  102  347    485  220  795  550     92  357  678  923
 476  229  806  539    101  348  667  934    796  549  486  219    677  924   91  358

 460  245  822  523    117  332  651  950    780  565  502  203    693  908   75  374
 821  524  459  246    652  949  118  331    501  204  779  566     76  373  694  907
  12  309  758  971    437  140  843  630    716 1013   54  267    885  588  395  182
 757  972   11  310    844  629  438  139     53  268  715 1014    396  181  886  587

  21  300  747  982    428  149  854  619    725 1004   43  278    876  597  406  171
 748  981   22  299    853  620  427  150     44  277  726 1003    405  172  875  598
 469  236  811  534    108  341  662  939    789  556  491  214    684  917   86  363
 812  533  470  235    661  940  107  342    492  213  790  555     85  364  683  918

[1] Awani Kumarís 3-D knight tours are at  http://www.gpj.connectfree.co.uk/gpj43.htm   
[2] Guenter Stertenbrink and Jean Charles Meyrignac created a website  in 2003 that chronicles magic knight tour events and updates.
[3] Much more information on Magic Knight Tours may be seen on my  Knight Tours page, including the listing for a Kumar 3x3x3 magic knight tour. 
[4] More information on magic tesseracts are available at other pages on this site. See especially my Overview page. 

This page was originally posted October 2008
It was last updated October 20, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz