Knight Tours

In the last several years there has been a marked increase in activity concerning Knight Tours. Much of this is reported on Guenter Stertenbrink’s website. [1]

A knight tour is a series of steps performed by a chess knight on x by y array of cells. If x=8 and y=8 then we have a standard chess board.

If we assign a number to each step, is it possible to complete a knight tour such that these numbers form a magic square? This has been an open question for many years. In August 2003, this question was answered. No, it is not possible to have a magic knight tour of an 8x8 (chess) board. [1][2][3]

In much of the knight tour literature, a tour is called MKT (Magic Knight Tour) if only the rows and columns, but not diagonals, sum correctly. I (and some others) define a MKT as requiring that rows, columns and diagonals must sum correctly. This is the same definitions as used for a Magic Square (MS). This is the definition used when making the above claim that no MKT exists for an 8x8 board.

Some history (mostly from Games Ancient and Oriental ...[4] )

The attempt to cover all the squares of a chess board with knight moves, without going over the same square a second time, is perhaps, as old as the invention of the game itself. In a Persian paper written before the year 899 appeared these words;

“Finally, I will show you how to move a knight from any individual square on the board so that it may cover each of the remaining squares in as many moves…” [4]

Games Ancient and Oriental and How to Play Them, covers this subject on pages 309 to 356. It starts off with a 17 item bibliography of material dating between 1750 and 1854! It includes 66 8x8 tour diagrams with symmetrical features.

Leonard Euler worked on this problem (around 1750?)and developed a general routine where the tour could start on any cell. It was ingenious but sometimes complicated and laborious.

Here is an example of a pattern developed as a personal calling card by a Dr. Roget in 1840. This resulted from his method, much improved from Euler’s.Dr.

Roget’s method considered only one quadrant of the square, then copied the pattern to the other 3 quadrants. Because this pattern is re-entry, you may start this pattern on any of the 64 cells!

In this pattern, all columns sum to 260, but no rows or diagonals.

In much of the knight tour literature, a tour is called MKT (Magic Knight Tour) if only the rows and columns sum correctly. I (and some others) define a MKT as requiring rows, columns and diagonals must sum correctly. This is the same definitions as used for a Magic Square (MS). This is the definition used when making the above claim that no MKT exists for an 8x8 board.
If the starting and ending cells are a knight’s move apart, the tour is called reentrant.

The term SMKT (Semi-Magic Knight Tour) will be used for a tour where all rows and all columns, but not the 2 main diagonals, sum to the constant. J. C. Meyrignac’s program proved that there are no MKT’s on an 8x8 board, but there are 140 distinct SMKT’s.
Dan Thomasson has classified and attractively diagrammed these tours.[3]

The first semi-magic knight tour on a chess board?

All rows and columns sum to 260, but diagonals equal 210 and 282 so this is only semi-magic. All 2x2 squares sum to 130. The tour is not normally considered reentrant. However, if you consider wrap-around, as in a pandiagonal magic square, then yes, cell 1 is only one knight move away from cell 64!

This tour was constructed by a Mr. William Beverley and published in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, in 1848. {[4], page 324; [5], pages 191-192.}

Knight Tours on other square and rectangle boards.

George Jelliss shows various knight tours on various size boards on his Knight Tour Notes page. [6]

to the right is one of four 12x12 magic knight tours (MKT) by Awani Kumar discovered in 2003. The 12 rows, 12 columns and 2 diagonals all sum to 870.

The following three early reentry knight tours on
5 x 6 boards go back in history, but are not magic.

Knight Tours on Cubes and Tesseracts

10	19	32	5
31	6	9	20
8	11	18	29
17	30	7	12

1	14	23	28
24	27	2	13
15	22	25	4
26	3	16	21

Awani Kumar has published his investigations into knight tours of magic cubes and cuboids on G. P. Jelliss’s Games and Puzzles Journal. [7] Here is Figure 14 from that Journal

On this almost MKT of a 2x4x4 cuboid, all rows and columns sum to 66. However, only 8 (one-half) of the pillars sum to 66/2. All 2x2 horizontal squares also sum to 66.

Among many other examples, Kumar shows an order-4 magic knight tour cube (rows, columns, pillars and triagonals) in figure 41. However, I show an earlier one by Guenter Stertenbrink. [8]

Kumar reports that a magic knight tour of the order 8 cube has yet to be found. So far, his best effort has one-third of the orthogonal lines summing correctly to 2052.

Addendum: April 28, 2007. Awani Kumar constructed the first order-8 Magic Knight Tour.
Then on June 19, 2007 he announced the first order-12 Magic Knight Tour cube.
In both cases, the knight tour was confirmed correct by Guenter Stertenbrink.
Both cubes are shown on my Cube_Update-5 page.

Dan Thomasson illustrates his method of attempting to make an 8x8x8 magic knight tour. [9]
Follow the links at the end of his page to see his Latin knight squares and Latin Cube based on the previous 3-D cube. All rows, columns, main diagonals, and pillars each add up to 260, while each number is used only once per row, column, pillar (pile), or level (plane).

Erich Friedman has a page where he discusses MKT but shows a large variety of other types of knight and other chess piece tours. [10]
Here as an example of what he calls a knight realization. This is a reentrant tour.
This figure is credited to Tino Jonker.

Order-8 Magic Knight Tour

News Flash! May 22, 2007.
Awani Kumar announced by email the first two solutions to an order- 8 Magic Knight Tour. All rows, columns, pillars, and the 4 triagonals
of this cube sum to 2052. This cube is simple magic with no included magic squares. The knight tour is re-entrant.

Guenter Stertenbrink confirmed (also by email) the same day that all knight moves are correct.

On June 19, 2007 Kumar announced an order-12 magic knight tour cube.

This is the listing of the first of his two order-8 cubes.

A                                                     B 
 19   482   509    16   461   480    35    50         510    15    20   481   472   453    58    43
490    27     8   501    36    49   462   479           7   502   489    28    57    44   471   454
511    14    17   484   465   452    63    46          18   483   512    13   460   473    38    55
  6   503   492    25    64    45   466   451         491    26     5   504    37    56   459   474
117   100   411   398   429   448    67    82         414   395   116   101   440   421    90    75
410   399   120    97    68    81   430   447         113   104   415   394    89    76   439   422
387   118   109   412   433   420    95    78         108   413   390   115   428   441    70    87
112   409   386   119    96    77   434   419         391   114   105   416    69    88   427   442

C                                                     D
495    30     1   500    33    52   463   478           2   499   496    29    60    41   470   455
 22   487   508     9   464   477    34    51         507    10    21   488   469   456    59    42
  3   498   493    32    61    48   467   450         494    31     4   497    40    53   458   475
506    11    24   485   468   449    62    47          23   486   505    12   457   476    39    54
 99   406   397   124    65    84   431   446         396   125   102   403    92    73   438   423
400   121    98   407   432   445    66    83         103   402   393   128   437   424    91    74
405   388   123   110    93    80   435   418         126   107   404   389    72    85   426   443
122   111   408   385   436   417    94    79         401   392   127   106   425   444    71    86

E                                                     F
259   274   237   256   213   318   195   300         282   267   248   229   314   209   304   199
238   255   260   273   292   203   214   317         247   230   281   268   207   296   313   210
287   270   241   228   315   212   301   198         262   279   236   249   216   319   194   297
242   227   288   269   206   293   316   211         235   250   261   280   289   202   215   320
369   136   159   362   165   334   339   188         154   367   376   129   330   161   192   343
158   363   372   133   340   187   326   173         373   132   155   366   191   344   169   322
135   146   361   384   331   164   189   342         368   377   130   151   168   335   338   185
364   381   134   147   190   341   172   323         131   150   365   380   337   186   327   176

G                                                     H 
239   254   257   276   291   204   309   222         246   231   284   265   208   295   218   305
258   275   240   253   310   221   196   299         283   266   245   232   217   306   303   200
243   226   285   272   205   294   219   308         234   251   264   277   290   201   312   223
286   271   244   225   220   307   302   197         263   278   233   252   311   224   193   298
359   370   137   160   179   348   325   174         144   153   354   375   352   183   170   321
140   157   358   371   166   333   180   347         355   374   141   156   329   162   351   184
145   360   383   138   349   182   171   324         378   143   152   353   178   345   328   175
382   139   148   357   332   163   350   181         149   356   379   142   167   336   177   346

References

[1] Guenter Stertenbrink, Computing Magic Knight Tours at http://magictour.free.fr/
[2] Eric Weisstein’s Mathworld has this news item http://mathworld.wolfram.com/news/2003-08-06/magictours/
[3] Iver Peterson’s Trek Talk also reported this news http://www.maa.org/mathland/mathtrek_10_06_03.html
[3] Thomasson’s 4x4 tour diagrams are at http://www.borderschess.org/SMKT2.htm
[4] Edward Falkener, Games Ancient and Oriental and How to Play Them, Dover Publ. 1961, 486-20739-0. This book was first published in 1892
[5] Martin Gardner, Mathematical Magic Show, Knopf, 1977, 0-394-40822-5
[6] George Jellis’s tours of rectangles at http://www.ktn.freeuk.com
[7] Awani Kumar’s 3-D knight tours are at http://www.gpj.connectfree.co.uk/gpj43.htm
[8] Stertenbrinks 4x4x4 magic knight tour is shown on my Unusual Cubes page.
[9] Thomasson’s 8x8x8 attempt is at http://www.borderschess.org/KTMS.htm
[10] Erich Friedman’s page on chess piece tours http://www.stetson.edu/~efriedma/mathmagic/0805.html