But is this correct terminology? Nobody else uses it apparently. So, let’s try to track down the earliest depiction of this tessellation in the literature…

Stillwell refers to Richard Dedekind’s 1877 paper “Schreiben an Herrn Borchard uber die Theorie der elliptische Modulfunktionen”, which appeared beginning of september 1877 in Crelle’s journal (Journal fur die reine und angewandte Mathematik, Bd. 83, 265-292).

There are a few odd things about this paper. To start, it really is the transcript of a (lengthy) letter to Herrn Borchardt (at first, I misread the recipient as Herrn Borcherds which would be really weird…), written on June 12th 1877, just 2 and a half months before it appeared… Even today in the age of camera-ready-copy it would probably take longer.

There isn’t a single figure in the paper, but, it is almost impossible to follow Dedekind’s arguments without having a mental image of the tessellation. He gives a fundamental domain for the action of the modular group on the hyperbolic upper-half plane (a fact already known to Gauss) and goes on in section 3 to give a one-to-one mapping between this domain and the complex plane using what he calls the ‘valenz’ function (which is our modular function , making an appearance in moonshine, and responsible for the black&white tessellation, the two colours corresponding to pre-images of the upper or lower half-planes).

Then there is this remarkable opening sentence.

Sie haben mich aufgefordert, eine etwas ausfuhrlichere Darstellung der Untersuchungen auszuarbeiten, von welchen ich, durch das Erscheinen der Abhandlung von Fuchs veranlasst, mir neulich erlaubt habe Ihnen eine kurze Ubersicht mitzuteilen; indem ich Ihrer Einladung hiermit Folge leiste, beschranke ich mich im wesentlichen auf den Teil dieser Untersuchungen, welcher mit der eben genannten Abhandlung zusammenhangt, und ich bitte Sie auch, die Ubergehung einiger Nebenpunkte entschuldigen zu wollen, da es mir im Augenblick an Zeit fehlt, alle Einzelheiten auszufuhren.

Well, just try to get a paper (let alone a letter) accepted by Crelle’s Journal with an opening line like : “I’ll restrict to just a few of the things I know, and even then, I cannot be bothered to fill in details as I don’t have the time to do so right now!” But somehow, Dedekind got away with it.

So, who was this guy Borchardt? How could this paper be published so swiftly? And, what might explain this extreme ‘je m’en fous’-opening ?

Carl Borchardt was a Berlin mathematician whose main claim to fame seems to be that he succeeded Crelle in 1856 as main editor of the ‘Journal fur reine und…’ until 1880 (so in 1877 he was still in charge, explaining the swift publication). It seems that during this time the ‘Journal’ was often referred to as “Borchardt’s Journal” or in France as “Journal de M Borchardt”. After Borchardt’s death, the Journal für die Reine und Angewandte Mathematik again became known as Crelle’s Journal.

As to the opening sentence, I have a toy-theory of what was going on. In 1877 a bitter dispute was raging between Kronecker (an editor for the Journal and an important one as he was the one succeeding Borchardt when he died in 1880) and Cantor. Cantor had published most of his papers at Crelle and submitted his latest find : there is a one-to-one correspondence between points in the unit interval [0,1] and points of d-dimensional space! Kronecker did everything in his power to stop that paper to the extend that Cantor wanted to retract it and submit it elsewhere. Dedekind supported Cantor and convinced him not to retract the paper and used his influence to have the paper published in Crelle in 1878. Cantor greatly resented Kronecker’s opposition to his work and never submitted any further papers to Crelle’s Journal.

Clearly, Borchardt was involved in the dispute and it is plausible that he ‘invited’ Dedekind to submit a paper on his old results in the process. As a further peace offering, Dedekind included a few ‘nice’ words for Kronecker

Bei meiner Versuchen, tiefer in diese mir unentbehrliche Theorie einzudringen und mir einen einfachen Weg zu den ausgezeichnet schonen Resultaten von Kronecker zu bahnen, die leider noch immer so schwer zuganglich sind, enkannte ich sogleich…

Probably, Dedekind was referring to Kronecker’s relation between class groups of quadratic imaginary fields and the j-function, see the miracle of 163. As an added bonus, Dedekind was elected to the Berlin academy in 1880…

Anyhow, no visible sign of ‘Dedekind’s’ tessellation in the 1877 Dedekind paper, so, we have to look further. I’m fairly certain to have found the earliest depiction of the black&white tessellation (if you have better info, please drop a line). Here it is

It is figure 7 in Felix Klein’s paper “Uber die Transformation der elliptischen Funktionen und die Auflosung der Gleichungen funften Grades” which appeared in may 1878 in the Mathematische Annalen (Bd. 14 1878/79). He even adds the j-values which make it clear why black triangles should be oriented counter-clockwise and white triangles clockwise. If Klein would still be around today, I’m certain he’d be a metapost-guru.

So, perhaps the tessellation should be called Klein’s tessellation?? Well, not quite. Here’s what Klein writes wrt. figure 7

Diese Figur nun - welche die eigentliche Grundlage fur das Nachfolgende abgibt - ist eben diejenige, von der Dedekind bei seiner Darstellung ausgeht. Er kommt zu ihr durch rein arithmetische Betrachtung.

Case closed : Klein clearly acknowledges that Dedekind did have this picture in mind when writing his 1877 paper!

But then, there are a few odd things about Klein’s paper too, and, I do have a toy-theory about this as well… (tbc)

The essence of such pages is that they need an update, once in a while… I’m a bit embarrassed to admit it, but, I failed to re-install the package, following my own instructions.

Fortunately, there are plenty of good alternatives around, today. After playing a bit with bibtex2html, Ive settled for the bib2html perl script by Patrick Riley.

Included are well documented directions. The most important being that you have to do a

sudo ln -s /usr/bin/perl /usr/local/bin/perl5¹

and change the user-specific entries in the bib2html.conf configuration file (if you have already key.pdf files around, you can just drop them in the output-directory and set . as the paperfiledirlist, and they will be found and linked to automatically. If not, you can always include links in the bib-file). Further, if you want to link to web-pages of your coauthors, remember that the script expects you to use : LastName, FirstName | http://webpage even if you used a different convention in your bib-file.

The result can be seen here. For ebook-download-junks : Ive included again full PDF-files of all versions of my book (starting with version 1 from 1999 to the final published version of last year). They can best be found using the sort-by-type page.

1. thank you Rupert, via the comments

So, what do we mean by this? First some basic rules of combinatorial game theory. To start, we evaluate a position without knowing which player has the move. A zero-game is by definition a position in which neither player has a good move, that is, any move by either player quickly leads to losing the game. Hence, a zero-game is a position in which the second player to move wins.

What is the chess-equivalent of a zero-position game? A position in which neither player has a good move is called a Mutual Zugzwang in chess literature. An example is given by the above position, if we restrict attention only to the 4 pieces in the upper right-hand corner and forget the rest. We don’t know who has the move, but, White cannot move at all and Black cannot move the King or Bishop without losing the Bishop and allowing White to promote the pawn and win quickly. In CGT-parlance, the upper-right position has value where the left options denote the White moves and the right options the Black moves.

All other values are determined by recursion. For example, consider a position in which White has just one move left before the sitution is again a Mutual Zugzwang, and, Black has no good move whatsoever. After white’s move, the position will again be a zero-position and Black has no options, so the value of this position would be denoted by and we call the value of this position to be . Similarly, if white has no options and black has one final move to make, the position would be considered to have value .

Clearly, these are just the three easiest game-values to have and the real kick comes further down the road when one can prove by recursion that some games have non-integer values (such as for a position in which white has one move to get to a mutual zugzwang and black has a move leading to a position of value (defined as before)), or non-number values such as where both white and black’s best move is to get to a mutal zugzwang. Game-values such as are called fuzzy (or confused with zero) and are defined by the property that the first player to move wins.

Similarly, positive game-values are those positions where White wins, independent of who has the move and negatives are those that Black wins. There is a whole menagery of game-values and the WinningWays-booklets give an example based introduction to this fascinating theory.

Brief as this introduction was, it will allow us to determine the exact value of the position in the above diagram. We know already that we can forget about the right-hand upper corner (as this is a zero-position) and concentrate attention to the left-hand side of the board.

It is easy to see that neither Knight can move without loosing quickly, nor can the pawns on a5 and b7. That is, white has just 2 options : either c3-c4 (quickly loosing after d5xc4 2. d3xc4,d4-d3 3. Nc1xd3,Na1-b3) or, and this is the only valid option c3xd4 leading to the position on the left below. Black has only one valid move : d4xc3 leading to the position on the right below.

Clearly, the left-diagram has value 0 as it is a mutual Zugzwang. The position on the right takes a moment’s thought : White has one move left d3-d4 leading to a 0-position, whereas black has one move d5-d4 leading to a position of value -1 (as black still has one move left d6-d5, whereas white has none). That is, the CGT-value of the right-hand position is and therefore, the value of the starting position is precisely equal to

(called tiny-one among ONAGers)

It can be shown that has a positive value (that is, White wins independently of who has the first move) but smaller than any positive number-valued games!

Noam Elkies has written a beautiful paper On numbers and endgames: Combinatorial game theory in chess endgames containing many interesting examples (the example above is an adaptation of his diagram9).

Alain’s writings are crystal clear compared to the monstrosity the AMS released to the world : In search of the Riemann zeros - Strings, fractal membranes and noncommutative spacetimes by Michel L. Lapidus.

Here’s a generic half-page from a total of 558 pages (or rather 314, as the remainder consists of appendices, bibliography and indices…). I couldn’t find a single precise, well-defined and proven statement in the entire book.

4.2. Fractal Membranes and the Second Quantization of Fractal Strings
“The first quantization is a mystery while the second quantization is a functor” Edward Nelson (quoted in [Con6,p.515])

We briefly discuss here joint work in preparation with Ryszard Nest [LapNe1]. This work was referred to several times in Chapter 3, and, as we pointed out there, it provides mathematically rigorous construction of fractal membranes (as well as of self-similar membranes), in the spirit of noncommutative geometry and quantum field theory (as well as of string theory). It also enables us to show that the expected properties of fractal (or self-similar) membranes, derived in our semi-heuristic model presented in Sections 3.2 and 3.2. are actually satisfied by the rigorous model in [LapNe1]. In particular, there is a surprisingly good agreement between the author’s original intuition on fractal (or self-similar) membrane, conceived as an (adelic) Riemann surface with infinite genus or as an (adelic) infinite dimensional torus, and properties of the noncommutative geometric model in [LapNe1]. In future joint work, we hope to go beyond [LapNe1] and to give even more (noncommutative) geometric content to this analogy, possibly along the lines suggested in the next section (4.3).
We will merely outline some aspects of the construction, without supplying any technical details, instead referring the interested reader to the forthcoming paper [LapNe1] for a complete exposition of the construction and precise statements of results.

Can the AMS please explain to the interested person buying this book why (s)he will have to await a (possible) forthcoming paper to (hopefully) make some sense of this apparent nonsense?

When someone with a track-record of coming up with surprising mathematical tidbits like John McKay sends me a mystery message claiming to contain “The secret of Monstrous Moonshine and the universe”, I’m happy to spend the remains of the day trying to make sense of the apparent nonsense

Let j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion at oo of the elliptic modular function. Compute sum(c[k]^2,k=1..24) modulo 70

I expected the j-coefficients modulo 70 (or their squares, or their partial sums of squares) to reveal some hidden pattern, like containing the coefficients of Leech vectors or E(8)-roots, or whatever… and spend a day trying things out. But, all I got was noise… I left it there for a week or so, rechecked everything and… gave up

Subject:   Re: mystery message
From:  lieven.lebruyn@ua.ac.be
Date:  Fri 21 Mar 2008 12:37:47 GMT+01:00
To:    mckayj@Math.Princeton.EDU
    
i forced myself to recheck the calculations i did once after receiving your mail.
here are the partial sums of squares of j-coefficients modulo 70 for the first 
100 of them

[0, 46, 26, 16, 32, 62, 38, 3, 53, 13, 63, 39, 29, 59, 45, 10, 60, 40, 30,
 10, 40, 26, 6, 56, 42, 22, 68, 48, 48, 64, 64, 45, 25, 15, 31, 31, 67,
 47, 7, 21, 51, 31, 31, 61, 21, 1, 17, 12, 2, 16, 46, 60, 20, 10, 54, 49,
 63, 63, 53, 29, 29, 23, 13, 13, 27, 27, 17, 7, 67, 43, 43, 52, 42, 42,
 16, 6, 42, 42, 42, 36, 66, 32, 62, 52, 66, 66, 0, 25, 5, 5, 35, 21, 11,
 11, 57, 57, 61, 41, 41]

term 24 is 42…
i still fail to see the significance of it all.
atb :: lieven.

A couple of hours later I received his reply and simply couldn’t stop laughing…

From:  mckay@encs.concordia.ca
Subject:   Re: mystery message
Date:  Sat 22 Mar 2008 02:33:19 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

I apologize for wasting your time. It is a joke
depending, it seems, on one’s cultural background.

See the google entry:

Answer to Life, the Universe, and Everything

Best, John McKay

Still confused? Well, do it!

From:  mckayj@Math.Princeton.EDU
Date:  Mon 10 Mar 2008 07:51:16 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

The secret of Monstrous Moonshine and the universe. 


Let  j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion 
at oo of the elliptic modular function.

Compute sum(c[k]^2,k=1..24) modulo 70

Background: w_25 of page x of the preface of Conway/Sloane book SPLAG 

Also in Chapter 27:
The automorphism group of the 26-dimensional Lorentzian lattice
The Weyl vector w_25 of section 2.

Jm

I realize that all of you will feel frustrated by the fact that most university libraries are closed today and possibly tomorrow, hence some help with the background material.

SPLAG of course refers to the cult-book Sphere Packings, Lattices and Groups.

26-dimensional Lorentzian space is 26-dimensional real space equipped with the norm-map

The Weyl vector is the norm-zero vector in

(use the numerical fact that )

The relevance of this special vector is that it gives a one-line description for one of the most mysterious objects around, the 24-dimensional Leech Lattice . In fact

with

where is the unique even unimodular lattice in . These facts amply demonstrate the moonshine nature of the numbers 24 and 70. Apart from this, the previous post may also be of use.

is surprisingly close to the dimension of the smallest non-trivial irreducible representation of the monster group, which is 196883. Note that at that time, the Monster hasn’t been constructed yet, and, the only traces of its possible existence were kept as semi-secret information in a huge ledger (costing 80 pounds…) kept in the Atlas-office at Cambridge. Included were 8 huge pages describing the character table of the monster, the top left fragment, describing the lower dimensional irreducibles and their characters at small order elements, reproduced below

If you look at the dimensions of the smallest irreducible representations (the first column) : 196883, 21296876, 842609326, … you will see that the first, second and third of them are extremely close to the linear, quadratic and cubic coefficient of the j-function. In fact, more is true : one can obtain these actual j-coefficients as simple linear combination of the dimensions of the irrducibles :

Often, only the first relation is attributed to McKay, whereas the second and third were supposedly discovered by John Thompson after MKay showed him the first. Marcus du Sautoy tells a somewhat different sory in Finding Moonshine :

McKay has also gone on to find these extra equations, but is was Thompson who first published them. McKay admits that “I was a bit peeved really, I don’t think Thompson quite knew how much I knew.”

By the work of Richard Borcherds we now know the (partial according to some) explanation behind these numerical facts : there is a graded representation of the Monster-group (actually, it has a lot of extra structure such as being a vertex algebra) such that the dimension of the i-th factor equals the coefficient f in the j-function. The homogeneous components being finite dimensional representations of the monster, they decompose into the 194 irreducibles . For the first three components we have the decompositions

Calculating the dimensions on both sides give the above equations. However, being isomorphisms of monster-representations we are not restricted to just computing the dimensions. We might as well compute the character of any monster-element on both sides (observe that the dimension is just the character of the identity element). Characters are the traces of the matrices describing the action of a monster-element on the representation and these numbers fill the different columns of the character-table above.

Hence, the same integral combinations of the character values of any monster-element give another q-series and these are called the McKay-Thompson series. John Conway discovered them to be classical modular functions known as Hauptmoduln.

In most papers and online material on this only the first few coefficients of these series are documented, which may be just too little information to make new discoveries!

Fortunately, David Madore has compiled the first 3200 coefficients of all the 172 monster-series which are available in a huge 8Mb file. And, if you really need to have more coefficients, you can always use and modify his moonshine python program.

In order to reduce bandwidth, here a list containing the first 100 coefficients of the j-function

jfunct=[196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075, 593121772421445058560, 2662842413150775245160, 11459912788444786513920, 47438786801234168813250, 189449976248893390028800, 731811377318137519245696, 2740630712513624654929920, 9971041659937182693533820, 35307453186561427099877376, 121883284330422510433351500, 410789960190307909157638144, 1353563541518646878675077500, 4365689224858876634610401280, 13798375834642999925542288376, 42780782244213262567058227200, 130233693825770295128044873221, 389608006170995911894300098560, 1146329398900810637779611090240, 3319627709139267167263679606784, 9468166135702260431646263438600, 26614365825753796268872151875584, 73773169969725069760801792854360, 201768789947228738648580043776000, 544763881751616630123165410477688, 1452689254439362169794355429376000, 3827767751739363485065598331130120, 9970416600217443268739409968824320, 25683334706395406994774011866319670, 65452367731499268312170283695144960, 165078821568186174782496283155142200, 412189630805216773489544457234333696, 1019253515891576791938652011091437835, 2496774105950716692603315123199672320, 6060574415413720999542378222812650932, 14581598453215019997540391326153984000, 34782974253512490652111111930326416268, 82282309236048637946346570669250805760, 193075525467822574167329529658775261720, 449497224123337477155078537760754122752, 1038483010587949794068925153685932435825, 2381407585309922413499951812839633584128, 5421449889876564723000378957979772088000, 12255365475040820661535516233050165760000, 27513411092859486460692553086168714659374, 61354289505303613617069338272284858777600, 135925092428365503809701809166616289474168, 299210983800076883665074958854523331870720, 654553043491650303064385476041569995365270, 1423197635972716062310802114654243653681152, 3076095473477196763039615540128479523917200, 6610091773782871627445909215080641586954240, 14123583372861184908287080245891873213544410, 30010041497911129625894110839466234009518080, 63419842535335416307760114920603619461313664, 133312625293210235328551896736236879235481600, 278775024890624328476718493296348769305198947, 579989466306862709777897124287027028934656000, 1200647685924154079965706763561795395948173320, 2473342981183106509136265613239678864092991488, 5070711930898997080570078906280842196519646750, 10346906640850426356226316839259822574115946496, 21015945810275143250691058902482079910086459520, 42493520024686459968969327541404178941239869440, 85539981818424975894053769448098796349808643878, 171444843023856632323050507966626554304633241600, 342155525555189176731983869123583942011978493364, 679986843667214052171954098018582522609944965120, 1345823847068981684952596216882155845897900827370, 2652886321384703560252232129659440092172381585408, 5208621342520253933693153488396012720448385783600, 10186635497140956830216811207229975611480797601792, 19845946857715387241695878080425504863628738882125, 38518943830283497365369391336243138882250145792000, 74484518929289017811719989832768142076931259410120, 143507172467283453885515222342782991192353207603200, 275501042616789153749080617893836796951133929783496, 527036058053281764188089220041629201191975505756160, 1004730453440939042843898965365412981690307145827840, 1908864098321310302488604739098618405938938477379584, 3614432179304462681879676809120464684975130836205250, 6821306832689380776546629825653465084003418476904448, 12831568450930566237049157191017104861217433634289960, 24060143444937604997591586090380473418086401696839680, 44972195698011806740150818275177754986409472910549646, 83798831110707476912751950384757452703801918339072000]

This information will come in handy when we will organize our Monstrous Easter Egg Race, starting tomorrow at 6 am (GMT)…

There are almost as many different constructions of as there have been mathematicians interested in that most remarkable of all finite groups.

In the inguanodon post Ive added yet another construction of the Mathieu groups and starting from (half of) the Farey sequences and the associated cuboid tree diagram obtained by demanding that all edges are odd. In this way the Mathieu groups turned out to be part of a (conjecturally) infinite sequence of simple groups, starting as follows :

It is quite easy to show that none of the other sporadics will appear in this sequence via their known permutation representations. Still, several of the sporadic simple groups are generated by an element of order two and one of order three, so they are determined by a finite dimensional permutation representation of the modular group and hence are hiding in a special polygonal region of the Dedekind’s tessellation

Let us try to figure out where the sporadic with the next simplest permutation representation is hiding : the second Janko group , via its 100-dimensional permutation representation. The Atlas tells us that the order two and three generators act as

e:= (1,84)(2,20)(3,48)(4,56)(5,82)(6,67)(7,55)(8,41)(9,35)(10,40)(11,78)(12, 100)(13,49)(14,37)(15,94)(16,76)(17,19)(18,44)(21,34)(22,85)(23,92)(24, 57)(25,75)(26,28)(27,64)(29,90)(30,97)(31,38)(32,68)(33,69)(36,53)(39,61) (42,73)(43,91)(45,86)(46,81)(47,89)(50,93)(51,96)(52,72)(54,74)(58,99) (59,95)(60,63)(62,83)(65,70)(66,88)(71,87)(77,98)(79,80);

v:= (1,80,22)(2,9,11)(3,53,87)(4,23,78)(5,51,18)(6,37,24)(8,27,60)(10,62,47) (12,65,31)(13,64,19)(14,61,52)(15,98,25)(16,73,32)(17,39,33)(20,97,58) (21,96,67)(26,93,99)(28,57,35)(29,71,55)(30,69,45)(34,86,82)(38,59,94) (40,43,91)(42,68,44)(46,85,89)(48,76,90)(49,92,77)(50,66,88)(54,95,56) (63,74,72)(70,81,75)(79,100,83);

But as the kfarey.sage package written by Chris Kurth calculates the Farey symbol using the L-R generators, we use GAP to find those

L = e*v^-1  and  R=e*v^-2 so

L=(1,84,22,46,70,12,79)(2,58,93,88,50,26,35)(3,90,55,7,71,53,36)(4,95,38,65,75,98,92)(5,86,69,39,14,6,96)(8,41,60,72,61,17, 64)(9,57,37,52,74,56,78)(10,91,40,47,85,80,83)(11,23,49,19,33,30,20)(13,77,15,59,54,63,27)(16,48,87,29,76,32,42)(18,68, 73,44,51,21,82)(24,28,99,97,45,34,67)(25,81,89,62,100,31,94)

R=(1,84,80,100,65,81,85)(2,97,69,17,13,92,78)(3,76,73,68,16,90,71)(4,54,72,14,24,35,11)(5,34,96,18,42,32,44)(6,21,86,30,58, 26,57)(7,29,48,53,36,87,55)(8,41,27,19,39,52,63)(9,28,93,66,50,99,20)(10,43,40,62,79,22,89)(12,83,47,46,75,15,38)(23,77, 25,70,31,59,56)(33,45,82,51,67,37,61)(49,64,60,74,95,94,98)

Defining these permutations in sage and using kfarey, this gives us the Farey-symbol of the associated permutation representation

L=SymmetricGroup(Integer(100))("(1,84,22,46,70,12,79)(2,58,93,88,50,26,35)(3,90,55,7,71,53,36)(4,95,38,65,75,98,92)(5,86,69,39,14,6,96)(8,41,60,72,61,17, 64)(9,57,37,52,74,56,78)(10,91,40,47,85,80,83)(11,23,49,19,33,30,20)(13,77,15,59,54,63,27)(16,48,87,29,76,32,42)(18,68, 73,44,51,21,82)(24,28,99,97,45,34,67)(25,81,89,62,100,31,94)")

R=SymmetricGroup(Integer(100))("(1,84,80,100,65,81,85)(2,97,69,17,13,92,78)(3,76,73,68,16,90,71)(4,54,72,14,24,35,11)(5,34,96,18,42,32,44)(6,21,86,30,58, 26,57)(7,29,48,53,36,87,55)(8,41,27,19,39,52,63)(9,28,93,66,50,99,20)(10,43,40,62,79,22,89)(12,83,47,46,75,15,38)(23,77, 25,70,31,59,56)(33,45,82,51,67,37,61)(49,64,60,74,95,94,98)")

sage: FareySymbol("Perm",[L,R])

[[0, 1, 4, 3, 2, 5, 18, 13, 21, 71, 121, 413, 292, 463, 171, 50, 29, 8, 27, 46, 65, 19, 30, 11, 3, 10, 37, 64, 27, 17, 7, 4, 5], [1, 1, 3, 2, 1, 2, 7, 5, 8, 27, 46, 157, 111, 176, 65, 19, 11, 3, 10, 17, 24, 7, 11, 4, 1, 3, 11, 19, 8, 5, 2, 1, 1], [-3, 1, 4, 4, 2, 3, 6, -3, 7, 13, 14, 15, -3, -3, 15, 14, 11, 8, 8, 10, 12, 12, 10, 9, 5, 5, 9, 11, 13, 7, 6, 3, 2, 1]]

Here, the first string gives the numerators of the cusps, the second the denominators and the third gives the pairing information (where [tex[-2[/tex] denotes an even edge and an odd edge. Fortunately, kfarey also allows us to draw the special polygonal region determined by a Farey-symbol. So, here it is (without the pairing data) :

the hiding place of …

It would be nice to have (a) other Farey-symbols associated to the second Janko group, hopefully showing a pattern that one can extend into an infinite family as in the inguanodon series and (b) to determine Farey-symbols of more sporadic groups.

At first this seems wrong as there is an obvious mate in two : 1. Qe2-f1, Kh1xh2 2. Rg3-h3 The ingenious point being that black claims a draw after 1. Qe2-f1 invoking the 50 moves rule which states

The game is drawn, upon a correct claim by the player having the move, if
(a) he writes on his scoresheet, and declares to the arbiter his intention to make a move which shall result in the last 50 moves having been made by each player without the movement of any pawn and without the capture of any piece, or
(b) the last 50 consecutive moves have been made by each player without the movement of any pawn and without the capture of any piece.

I’d love to have the ‘official’ solution to this problem. Here’s what I’ve come up with, spending the better part of the afternoon… It will not be optimal but hopefully isn’t too far off. The crucial part is a maneuver unlocking the lower left-hand cluster of pieces (in particular the ordering of white and black rook). All captures were done with pawns and the final pawn move was b2-b3. Immediately before it the situation might look something like the situation on the left (essential is that the white king should not be too far from its home-square as he will be needed later to block the white rook)

after b2-b3 the bisshop on c1 travels to f4 and the black rook squeezes in to block the white rook so that also the black king can come in and position himself at e2. Then the two rooks evacuate the first row, allowing the black king to go to h1 and then the white king comes in to block the white rook from checking the black king (situation on the left below).

Finally, the white rook comes in and positions itself at e2, afterwards the white king evacuates the first row via b2 and travels to the right-hand upper corner entering via g7. Meanwhile, the black rook comes to g1, the white rook then travels to a3 and the black rook to a2. Then, the white king goes to b7 allowing the bishop to unlock the rook on a8 going to g7, allowing finally the king to go to d8…

Perhaps there is a much simpler and more elegant solution, so if you know, please comment. Oh, btw. how is the original problem solved. Well white first cancels the 50-move rule by 1. Kd8xd7 to continue for example with 2. Rg3-g4, 3. Qe1

Wrong! There exist concrete positions, provable mate in two in which it is NOT possible to determine the winning first move for white! So, what’s wrong with the argument above? We did assume that, given the position, it is possible to determine all legal moves for the two players. So?

Well, some moves are legal only depending on the history of the game. For example, you are only allowed to do a castling if your king nor your rook made a prior move. Further, you can only make an en-passant-capture on the next move.

But surely all this is just theoretical? No-one ever constructed a provable 2-mate with impossible winning move. Wrong again. The logician Raymond Smullyan did precisely that in his retro-chess puzzle book Chess mysteries of Sherlock Holmes. Here’s the position :

Presumably every chess player goes for the mate : 1. Kf5-e6 2. g7-g8D But what if black counters your first move with a castling 1. … 0-0-0 ? Surely he isnt allowed to do this. Why not, is there any clue in the position to prove that either the king or the rook must have moved before?

Well, what was black previous move then? It cannot be the pawn move e6-e5 as before that move the white King would be in check, so what was it? Just one possibility left : it must have been e7-e5.

This offers then another winning strategy for white, as white can capture en-passant. 1. d5xe6 e.p. and then if black castles 1. … 0.0.0, 2. b6-b7 or is black does any other move : 2. g7-g8.

Hence, whatever the games’ history, white has a mate in two! However, looking ONLY at the given position, it is impossible for him to judge whether Kf5-e6 will do the trick!

Anyone seen similar constructions?

UPDATE

According to the wikipedia page on Retrograde chess analysis, the Smullyan-idea is an adaptation of a much older problem due to W. Langstaff in the Chess Amateur of 1922. Here’s the situation (the solution is the same as above)