A nice quiet cool walk along the prom in the rain with my umbrella this afternoon.

Cool being the most operative word. Hardly anyone about.

## Thursday, 9 August 2018

## Monday, 30 July 2018

### Knight's Tour Notes Update

My efforts to put the information from the website into book form have taken a backward step, to leap forward again I hope. The material has broken apart again to take the form of a series of smaller books of around 80 or 96 pages.

This has quickly resolved into ten parts:

0 - Bibliography with Glossary

1 - Theory of Moves (including leapers)

2 - Odd and Oddly Square Boards (6x6 10x10, etc)

3- Symmetry (including mixed quaternary)

4- Simple Linking of Pseudotours

5 - Oblong Boards

6 - Shaped and Holey Boards

7 - Magic Square Knight Tours

8 - Multimover Magic (King, Queen, etc)

9 - Miscellanea (Figured, Lettered, HexBoards etc)

Updated 4 August.

This has quickly resolved into ten parts:

0 - Bibliography with Glossary

1 - Theory of Moves (including leapers)

2 - Odd and Oddly Square Boards (6x6 10x10, etc)

3- Symmetry (including mixed quaternary)

4- Simple Linking of Pseudotours

5 - Oblong Boards

6 - Shaped and Holey Boards

7 - Magic Square Knight Tours

8 - Multimover Magic (King, Queen, etc)

9 - Miscellanea (Figured, Lettered, HexBoards etc)

Updated 4 August.

## Saturday, 23 June 2018

### disphenocingulum again

It seems my alternative disphenocingulum (J90) has been discovered before.

It is number 25 on this page of "near miss" Johnson solids.

http://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/jsmn.htm

This claims that there is some distortion in the cingulum triangle.

Though in my models I could not detect any distortion. It must be very small.

The author appears to be Jim McNeill as named here:

http://www.orchidpalms.com/polyhedra/

It is number 25 on this page of "near miss" Johnson solids.

http://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/jsmn.htm

This claims that there is some distortion in the cingulum triangle.

Though in my models I could not detect any distortion. It must be very small.

The author appears to be Jim McNeill as named here:

http://www.orchidpalms.com/polyhedra/

## Wednesday, 13 June 2018

### The Disphenocingulum

As a result of contacts on twitter I have become interested in polyhedra, in particular the "Johnson Solids" which are formed of regular polygons but are not the usual suspects.

In particular a drawing was posted of the "disphenocingulum". As a result I decided to make a model in card and ended up with two different versions. The 12 orange triangles form the "cingulum". The "spheno" parts form the roof and keel:

Corresponding flat diagrams of the pieces and their connections are:

Left

Right

The version on the right appears to conform to the patterns shown in the Wikipedia and MathWorld entries for Johnson Solid 90, but the version on the left does not.

So is my new version an alternative ("isotope") of the disphenocingulum? Or is one of them not an authentic Johnson Solid because it has a pair of triangles that are coplanar? It is difficult to tell from the models if some pairs of triangles are at an angle to each other or are flat together. The angle may be very small. In fact the one on the right seems flatter to me than that on the left.

In particular a drawing was posted of the "disphenocingulum". As a result I decided to make a model in card and ended up with two different versions. The 12 orange triangles form the "cingulum". The "spheno" parts form the roof and keel:

Corresponding flat diagrams of the pieces and their connections are:

Left

Right

The version on the right appears to conform to the patterns shown in the Wikipedia and MathWorld entries for Johnson Solid 90, but the version on the left does not.

So is my new version an alternative ("isotope") of the disphenocingulum? Or is one of them not an authentic Johnson Solid because it has a pair of triangles that are coplanar? It is difficult to tell from the models if some pairs of triangles are at an angle to each other or are flat together. The angle may be very small. In fact the one on the right seems flatter to me than that on the left.

## Wednesday, 9 May 2018

### Magic Four-Camel Tour of 1887

Following on from the four Giraffe tour reported previously I have now come across this four Camel tour in my researches in the column "Un Probleme Par Jour" conducted by A. Feisthamel in the French newspaper "Le Siecle". This tour is by "Adsum a Saint-P" (According to H.J.R. Murray this was a pen name of Charles Bouvier). It is mentioned on 13 May 1887 then presented as a problem for solution on 14 May, with the solution appearing on 21 May 1887. The diagonal sums are complementary (i.e. adding to 520). Each rank consists of pairs of complements (adding to 65).

01 62 13 58 07 52 03 64

49 21 53 02 63 12 44 16

09 39 11 50 15 54 26 56

61 14 59 08 57 06 51 04

20 35 22 41 24 43 30 45

40 10 38 31 34 27 55 25

32 60 28 47 18 37 05 33

48 19 36 23 42 29 46 17

This is the earliest mention of a tour by a {1,3} mover that I am aware of. It seems surprising that later French writers on Mathematical Recreations, such as Lucas and Kraitchik, don't seem to have been aware of these results. Feisthamel indicates that they are only single examples from much more extensive work. Was the work of these composers all lost after their deaths, or is it still hidden away in some obscure French archives?

01 62 13 58 07 52 03 64

49 21 53 02 63 12 44 16

09 39 11 50 15 54 26 56

61 14 59 08 57 06 51 04

20 35 22 41 24 43 30 45

40 10 38 31 34 27 55 25

32 60 28 47 18 37 05 33

48 19 36 23 42 29 46 17

This is the earliest mention of a tour by a {1,3} mover that I am aware of. It seems surprising that later French writers on Mathematical Recreations, such as Lucas and Kraitchik, don't seem to have been aware of these results. Feisthamel indicates that they are only single examples from much more extensive work. Was the work of these composers all lost after their deaths, or is it still hidden away in some obscure French archives?

## Sunday, 6 May 2018

### Sources of Magic Knight Tours, Further Update

I have located another of the tours with missing sources:

(23b) Ligondes (Palamede)

Murray gives the date as 1884, so it may also be in an earlier source,

such as Count Ligondes' own private publication which I have not seen.

This seems to be the last one-chain magic tour published in

Two and four-chain solutions continue to appear up to 30 April 1894,

which is the last date it is signed by A. Feisthamel. From 1 May 1894

the column "Un Probleme Par Jour" has a new editor, Emile Franck.

(23b) Ligondes (Palamede)

*Le Siècle*¶2836 4/11 December 1885.Murray gives the date as 1884, so it may also be in an earlier source,

such as Count Ligondes' own private publication which I have not seen.

This seems to be the last one-chain magic tour published in

*Le Siècle.*Two and four-chain solutions continue to appear up to 30 April 1894,

which is the last date it is signed by A. Feisthamel. From 1 May 1894

the column "Un Probleme Par Jour" has a new editor, Emile Franck.

## Sunday, 22 April 2018

### Magic Four-Giraffe Tour From 1887

On 9 October 2013 I reported here my construction of a Magic Two-Giraffe Tour, i.e. consisting of two sections of {1,4} Giraffe moves joined by two rook moves. In that tour the diagonals add to 272 and 248 which together sum to 520 which is twice the magic constant of the ranks and files.

While studying the cryptotours published in

Diagonally Magic Four-Giraffe Tour by A. E. Reuss of Strasbourg

Problem 3221, Le Siecle 5 March 1887, solution 12 March 1887.

01 25 09 23 42 56 40 64

43 49 39 63 02 26 16 22

03 27 15 21 44 50 38 62

45 51 37 61 04 28 14 20

32 08 24 10 55 41 57 33

54 48 58 34 31 07 17 11

30 06 18 12 53 47 59 35

52 46 60 36 29 05 19 13

The rook moves are 16-17, 32-33, 48-49 and the closure move 64-1. The tour is symmetric about the vertical axis, the ranks consisting of complementary numbers adding to 65, but is not quite symmetric about the horizontal axis.

Naturally I wonder whether Reuss constructed others of this type, or is this just a one-off? Knight tours that he published in the same column were under the pen-name of "X a Belfort".

While studying the cryptotours published in

*Le Siecle*in the column "Un Probleme Du Jour" edited by A. Feisthamel, from 1876 to 1894, I have found this much earlier work on the same subject. This uses four Giraffe paths connected by four rook moves. In this tour the diagonals add to the magic constant 260 as well as the ranks and files.Diagonally Magic Four-Giraffe Tour by A. E. Reuss of Strasbourg

Problem 3221, Le Siecle 5 March 1887, solution 12 March 1887.

01 25 09 23 42 56 40 64

43 49 39 63 02 26 16 22

03 27 15 21 44 50 38 62

45 51 37 61 04 28 14 20

32 08 24 10 55 41 57 33

54 48 58 34 31 07 17 11

30 06 18 12 53 47 59 35

52 46 60 36 29 05 19 13

The rook moves are 16-17, 32-33, 48-49 and the closure move 64-1. The tour is symmetric about the vertical axis, the ranks consisting of complementary numbers adding to 65, but is not quite symmetric about the horizontal axis.

Naturally I wonder whether Reuss constructed others of this type, or is this just a one-off? Knight tours that he published in the same column were under the pen-name of "X a Belfort".

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