View from my Window

The photo is a typical view from one of the windows of my flat at present.

There is scaffolding all round the building as the landlord is having the upper walls cleaned and repainted.

Sorry I've not been keeping this diary up to date.
I've been concentrating on trying to finish my book on Knight's Tours.
More on this soon.

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.

Bright Sea at Sunset

After sunset tonight the sea was strangely brighter that the sky.

But I'm afraid the photo is inadequate to show it.
Earlier the setting sun was very red but I didn't have the camera with me.

Magic Rectangles 5 by 7

01 25 17 09 33 06 35
20 16 28 04 12 15 31
26 07 29 18 10 34 02
21 23 13 32 05 24 08
22 19 03 27 30 11 14

11 31 16 06 15 26 21
17 27 07 22 32 12 09
13 03 28 18 08 23 33
29 24 04 14 34 19 02
20 05 35 30 01 10 25

13 09 05 35 33 07 24
23 10 30 02 31 08 22
15 16 17 18 19 20 21
14 27 06 34 04 29 12
25 28 32 01 03 26 11

05 27 15 12 25 19 23
32 03 16 29 14 28 04
06 35 26 18 10 01 30
34 08 22 07 20 33 02
13 17 11 24 21 09 31

The first two magic rectangles above were constructed on 7 November
and are rather irregular. The magic constants are 90 and 126 (i.e.
5x18 and 7x18).

The third is slightly more regular, having the sequence 15 to 21 along
the middle rank, similar to the method used for the 3x7 magic
rectangles reported here on 13 October.

The fourth is the most regular having diametrally opposite cells
complementary (i.e. adding to 36) except for two cases 32-04 and 34-
02. It may be that a completely symmetric magic tour is impossible,
but I've not been able to prove this so far. (It is possible on the
3x5 board as I showed in Chessics #26, 1986.)

Obviously every rank and file contains an even number of odd numbers,
and hence an odd number of even numbers. There are 8 numbers of the
form 4n, occurring in pairs of complements, and 9 of the form 4n + 2
consisting of the middle number 18 and four pairs of complements.
There remain 9 each of the forms 4n +1 and 4n + 3, which are
complements of each other.

The method I use for constructing these is to first put the numbers
into the array in some regular pattern, and then to adjust the rank
and file sums to give the magic values by making transpositions of
pairs or groups of numbers.

The ranks and files of these magic rectangles can be permuted without
affecting the magic property. So each generates 5!x7!/4 = 151200
(oriented with the long lines horizontal). The division by 4 is to
avoid counting rotations and reflections separately.

I've been trying to find an arrangement that uses the least number of
types of move but so far have not managed to reduce the number to less
than nine.

=====
Addendum 15 November:
A symmetric solution is possible. One is given in W.S.Andrews
Magic Squares and Cubes Fig.458 due to C.Planck.
Planck also seems to have anticipated my work on the 3x5 case,
since the 39 solutions are mentioned but not diagrammed.

Crane working on Hastings Pier

I meant to post this photo of the crane working on Hastings Pier a while ago.
It was taken 28 September. The crane has since gone.

How it was brought there and taken away I didn't see.
I don't think they were drilling for oil!
Impressive engineering work.

Magic Rectangle Tours

I have been doing a search for magic rectangle tours
on 3 by 7 board (and others) by pieces with limited moves,
and have found only these three so far.

They all contain the rank 8, 9, 10, 11, 12, 13, 14 (in some order)
adding to magic constant 77. The files add to 33.

Each tour is presented in forward and reverse numbering,
and oriented with the 1 (or 2) in the top left.

------
Amazon (Queen + Knight) magic tours:

15 01 19 02 21 03 16 == 06 19 01 20 03 21 07
12 14 09 11 08 13 10 == 12 09 14 11 13 08 10
06 18 05 20 04 17 07 == 15 05 18 02 17 04 16

This uses seven types of move
Rook 02, 03, 04, 05, 06, Bishop 11, Knight 12

01 19 18 04 03 15 17 == 02 17 20 01 06 15 16
12 09 13 08 14 11 10 == 10 13 09 14 08 11 12
20 05 02 21 16 07 06 == 21 03 04 18 19 07 05

This uses nine types of move
Rook 01, 02, 03, 04, 05, Bishop 11, 22, Knight 12

------
Raven (Rook + Nightrider) magic tour:

01 17 15 05 21 02 16 == 06 20 01 17 07 05 21
14 13 12 09 08 11 10 == 12 11 14 13 10 09 08
18 03 06 19 04 20 07 == 15 02 18 03 16 19 04

This uses seven types of move:
Rook 01, 02, 03, 04, 05, Nightrider 12, 24

------
Does anyone know of previous work on this subject?
I published some results on the 3 by 5 board
in Chessics #26 (1986) including a symmetric tour
using only four types of move.

I can only find an article by Marian Trenkler of Slovakia
http://math.ku.sk/~trenkler/
published in Mathematical Gazette 1999.

A Figured King Tour

The wazir tour with squares in a row on boards 2x2, 6x6, 10x10 and so on was published in Chessics #21 (1985). It occurred to me yesterday to look at the same problem on the 8x8 board but using the king as the touring piece. It appears that a solution with the square numbers in order of magnitude is just beyond the realm of possibility (with a knight move in place of one of the king moves it can probably be done). However I did find a solution with the numbers slightly out of order:

I set this as a puzzle on twitter, but haven't had any claims of anyone solving it yet. Of course the tour is not completely determinate. Some of the parallel pairs of moves can be replaced by crossing diagonal moves. But with the condition "minimum crossovers" it is probably unique. It includes a 6x6 solution with the numbers in correct sequence in the central area.