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.

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?

Sources of Magic Knight Tours, Further Update

I have located another of the tours with missing sources:
(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.

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 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".

Sources of Magic Knight Tours: Update

Two minor updates to the list of sources of magic knight tours of the standard chessboard have come to light from my researches in Le Siecle.

Tour 00i by an Unknown composer appears as problem 772 on 25 April 1979 with solution on 2 May 1879. The word puzzle used with it is given as by M. Jacquemin-Molez but the composer of the design is not identified. The column is signed as by A. Feisthamel as usual, but if by him the designer would probably be cited as M. A. F. so I think we have to classify it as still Unknown and from an earlier source. The 1879 date is one or two years earlier than the 1880/81 date given by Murray.

Tour 00f by Palamede (the pseudonym of Count Ligondes) appears as problem 2134 on 31 August 1883 with solution on 7 September 1883. There does not seem to be any mention of its cyclic properties.

National Chess Library

Here is a new image of the layout of the bookcases in the ECF Offices at Battle,
with the bookcases numbered 1 to 49.

And a key to the present subject arrangement.

READING ROOM AND OFFICE
1-2 History, 3-6 Game Collections, 7-10 Openings
11-14 English Periodicals 15 Office Use
16-17 Middle Game 18 Endgame 19 Problems
20 Variants 21-22 Archive

ROOM 8
23-25 Introductory Books 26 Office Use
27 Russian 28-30 East European 31 Multilingual
32-34 German 35 Braille 36 Library Catalogues
37-38 West European 39 French
40 Chess Handbooks 41-42 Reference
43 World Championship 44 UK Regional
45 Year Books 46-49 Tournaments.

Magic Square in Figured Tour

In my booklet on Figured Tours published in 1997, there is a knight's tour, A5 on the 8x8 board in which the first nine even numbers form a magic square. The tour is by T. R. Dawson and originally published in Vie Rennaise 19 November 1932. It led me to wonder if the same task could be done on a smaller board of minimum size.

The 3x5 board with a cell added at the middle of each side is of 19 cells, that can be chequered to show 9 dark cells in the form of a diagonal square and 10 light cells. Unfortunately this board has no knight tour, for the same reason as the 3x5 board, formation of a closed path firmed by the knight moves available at b3 and f3.

However if the extra cells are added on the long sides to form a rotary symmetric board as shown below the board becomes tourable. In fact there are 62 symmetric tours. Among these there are just six in which the square formed by the even numbers is magic. As shown here:

I wonder if this result can be new?