## Thursday, 18 December 2014

### Magic Rectangle 7 by 9

The first rectangle here shows a King Tour in which the ranks sum to all the successive values from 285 to 291 and the files to the successive values from 220 to 228.

01 56 57 15 14 42 43 29 28
55 02 16 58 41 13 30 43 27
54 17 03 40 59 31 12 26 45
18 53 39 04 32 60 25 11 46
19 38 52 33 05 24 61 47 10
37 20 34 51 23 06 48 62 09
36 35 21 22 50 49 07 08 63

The middle rank and file are naturally magic, consisting of pairs of complements (adding to 64) plus the middle average number 32, giving the required totals of 288 and 224.

This type of King Tour with consecutive rank and file totals seems to be possible on any odd-sided oblong where the sides have no common factor. I've not seen this result published anywhere before. The moves are completely regular, being diagonal except where they meet a board edge when the king takes a lateral step along the edge and then resumes its diagonal moves as if reflected from the edge.

I used this regular numbering of the cells to construct the following magic rectangle by a series of interchanges of pairs of entries.

01 59 57 14 15 42 43 29 28
55 02 18 58 41 13 30 44 27
60 17 03 40 54 33 11 25 45
16 52 38 08 32 56 26 12 48
19 39 53 31 10 24 61 47 04
57 20 34 51 23 06 46 62 09
36 35 21 22 49 50 07 05 63

This began with the interchange of 56 with 59 and 5 with 8 which fixed the top and bottom ranks and the second and eighth files (without altering the total of the middle file). Then the interchange of 16 with 18 and 46 with 48 fixed the second and sixth ranks and the third and seventh files(without altering the total of the middle rank). After that it became a bit more difficult to find suitable changes that did not disrupt the previous ones. The resulting tour uses ten different types of move instead of just two. Can it be done with less disruption?

Here is an earlier example I found based on a knight tour.

03 14 57 56 05 63 31 17 42
38 55 04 41 48 23 40 09 30
58 02 27 18 45 21 29 39 49
54 28 44 52 32 12 20 36 10
15 25 35 43 19 46 37 62 06
34 53 24 13 16 51 60 11 26
22 47 33 01 59 08 07 50 61

This uses 16 different types of move, so can hardly be called a "tour" at all. It is also not symmetric (or "associated" as magic square devotees term it) since the pairs of complements 55-9, 41-23, 53-11 and 13-51 lie along the second and sixth ranks instead of being diametral.