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.

Most Asymmetric Tour?

How does one assess how asymmetric a tour is? Or indeed any object? Is there an objective measure of the degree of asymmetry?

Thinking over this question the last two days I have come up with the tour shown which I mantain is the most asymmetric tour possible. There may be better, or alternative ways of measuring asymmetry, but I have concluded that in the case of a tour it is the number of the 21 geometrically distinct moves that occur an odd number of times. In this tour the total is 18, and moreover 12 of these occur only once.

The only moves that occur an even number of times are the corner moves like a1-c2 (where there are eight, which occur in every closed tour) the edge-to edge moves of the type a2-c1 (eight again), and the pair of moves a3-c2 and h3-f2 which are of the same type. Two moves are considered to be of the same type if one can be put in the position of the other by a rotation or reflection of the board.

The 12 moves that occur only once are b1-c3, d1-c3, f1-e3, b2-c4, d2-b3, d2-f3, d3-c5, e3-d5, a4-b6, e4-g5, b5-d6, e5-f7.

Warnsdorf Squares and Diamonds Tour

The first thing I noticed on seeing the figures from the Warnsdorf 1823 book, is that the final figure #96 is a tour of squares and diamonds type! This seems to have been added as an afterthought and is preceded by a diagram in which the squares and diamonds are lettered A, B, C, D using a different type style in each quarter of the board.
This confirms Murray's statement that 'The first composer to give the figure which shows that the cells of the quadrant could be filled by four closed quartes was v. Warnsdorf (1823)'. However the diagram is not in graphic form, and strangely Murray made no mention of the accompanying tour!
Warnsdorf shows tours in numerical form. Here is the tour diagram in graphic form:

This means that Warnsdorf has priority in the showing of a tour of squares and diamonds type. The previous earliest examples I was aware of were the two given by "F.P.H" in 1825 and the near-magic tours in the study by C. R. R. von Schinnern in 1826.

Warnsdorf from the original source

Thanks to the Cleveland Public Library, Ohio USA, I have now been able to study the diagrams in the 1823 book by H. C. von Warnsdorf. The figures relating to his Rule ("Always move the knight to a cell from which it has the fewest exits") are numbers 32 to 39 in Table 5. He begins by applying the rule to the 6x6 board, with four examples, then goes on to the 6x7, 6x8, 7x8 and 8x8 boards, giving one example for each.
Warnsdorf's tours are shown with the cells numbered in the sequence visited by the knight, but here we show them in geometrical form. A black dot indicates the start cell, and a dark circle the end cell. A circle round the initial dot indicates that there are choices of first move, though this may only be because of symmetry. The light circles mark points where the rule provides a choice of two or more moves, so that alternative routes could have been taken.

In the fourth 6x6 example the move e4-c3 (marked by a cross) does not conform to the rule, which requires e4-f6 as in the third 6x6 example. Similarly in Warnsdorf's 8x8 example there is a deviation from the rule at h5. The correct move should be to f4 which has access to only two exits whereas g7 and f6 have three. However these deviations only go to show the robustness of the rule, since in each case it still goes on to complete a tour.