Dvonn 2 overload

LatexRender and Dvonn-boards

Dvonn 1 mobility

A DaVinci chess problem

Bivalue Sudoku graphs

microsudoku.sty

A 2006 chess puzzle anyone?

Hints for micro-Sudoku

Micro-Sudoku

Sudoku mania (bis)

Sudoku mania

SnortGo 2

Quintominal dodecahedra

Elkies’ puzzles

Fox and geese

SnortGo

ColGo

Antwerp sprouts

Combinatorial game software

I could never have pulled this off in such a short time without Jacques Distler
more or less on constant stand-by (thanks Jacques!). Looking at the
times his emails were send I have no idea in which time zone he lives
(let alone sleeps…). So, here a walk-through the changes :

As
I’m on WP 2.0.5 I’ll start with Frederick’ post. He tells me I have to install first the itex2MML binary as
explained by
Jacques but I find that there is more recent
material and therefore download the most recent imath2MML-package
and follow the readme. There is a Mac OSX binary but it’s not clear
for what processor (PPC/Intel/Binary) but a quick mail to Jacques learns
me that it’s PPC which is fine by me but on the spot he puts a
universal binary online, so whatever your Mac is you can just download
the binary, copy it to /usr/local/bin and make sure its chmodded
755.

Back to Frederick’s post, download and install the plugin itexToMML.php in the usual way
(fortunately I spot just in time that I have to change one line saying
where my itex2MML binary is (in Frederick’s file it is NOT the default
location)). You can verify whether the plugin and itex2MML do what they
are supposed to do by typing a LaTeX-command in a post and save it. The
output will not produce the desired formula but have a look at the
source file and see whether there is some mathML code in it. If so,
fine! If not, go back and check everything.

If this works, it is
“merely” a problem of getting your mathML served. Frederick suggests
to unpack wordpress_mathML.zip in the wp-includes directory (but you
better make sure you have made a copy of the original class.php and
functions-formatting.php files. In the end I decided against this
approach (that is, to replace only the functions-formatting.php but NOT
the class.php file). If you have two or more themes you want to
maintain, it is probably better to change the headers (because this is
what we have to do to get mathML served) only in those themes which are
XML-sound. In my case, the Command Line Interface theme most certainly is NOT!!!).

Go to your
theme-files and look for the header.php (or similar) file and replace
the default header by the code in the addendum to
this post within php-tags. If you can go to your blog-page then you
are in good shape and things should work well (apart possibly from
layout considerations, see below). Of course, in my case i was greeted
by ” XML “yellow screen of death” (as Jacques calls
it) and I was convinced I did something wrong, so I tried out several
useless things for a couple of hours before it dawned on me that the
reason might just be that my blog-files were not valid XHTML (and the
new headers are very demanding on serving only well-form XHTML). I had
to modify all changes I made to sidebars etc. as well as rewrite parts
of my first posts (I used to take a rather liberal view on writing
blog-posts, writing a mixture between Markdown and improvised HTML and
in the process was very lax about closing IMG-tags and the likes).
But after some time and numerous corrections to the files I got the
main-page up and running (and even had the mathML served as a readable
formula) apart from the fact that I barely recognized my own site.

I printed out source files of the page with and without changed
headers and couldn’t find a difference. So, it had to do with the
CSS-style files, but why on earth would the new headers be picky about
CSS? But as a last resort, after narrowing the search down to one
CSS-line, I asked Jacques whether he had an idea what went on. His reply
will be remembered for quite some time :

A fascinating
question. The answer is that it *is* following the CSS directive, but
in XHTML, ‘body’ is not what you think it is. ‘body’ is just big enough
to contain its content. It does not fill the viewport. ‘html’ fills the
viewport. The solution (a solution) is described in
http://golem.ph.utexas.edu/~distler/blog/archives/000203.html

Many hours later, I still haven’t got a clue what
this is all about, but I blindly followed the hint and surely all
problems vanished. In short, another day wasted in front of a
computer-screen.

At the moment I’m back to old headers and
will not be writing mathML for some time as I have the vast job ahead to
validate all my previous posts to XHTML-standards (if not you would see
more yellows screens of death than anything else. So, here’s the
strategy I’ll be taking in the weeks ahead (I’ll sleep on it tonight
so if any of you think there is a better way, reply quickly)

• rewrite each and every post in proper MultiMarkdown using iTeX for
  the most common math and only resorting to LaTeXRender for exotic things
  (such as Sudoku, Chess, Dvonn) and run these posts through Markdown
  (to get basic HTML and all links in place).
• download these
  files to the WP-database (so that in the CLI-interface you will be able
  to follow all links, but will read all iTeX as TeX-commands (as the
  command line intended after all).
• in the process change all
  broken links to the default permalink-structure (with index.php?p=231 or
  so).

Clearly, this is a work that will take a couple of
weeks but it may be fun to reread these old posts and possibly add new
information about the subjects. When I’m making these changes, I’ll
use the new headers so if you are using a smart browser look out for the
yellow screens. When they happen, either use a dumb browser (such as
Safari) or go into CLI-interface mode where everything should still
work. I plan to start with the oldest posts as this seems more fun to
me.

$\xymatrix@=.3cm @!C
@R=.7cm{.& & & & & & & & & & & & & \\ & & & \SBlack \connS & &
\bull{d}{5} \conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & &
\bull{g}{5} \conn & & \bull{h}{5} \conn & & \SWhite \connS & & \SWhite
\connS & & \SWhite \conneS & & & \\ & & \SBlack \connS & & \SBlack
\connS & & \Black{6} \connS & & \bull{e}{4} \conn& & \bull{f}{4} \conn &
& \bull{g}{4} \conn & & \bull{h}{4} \conn & & \SWhite \connS & &
\SWhite \connS & & \SWhite \conneS & & \\ & \SBlack \connbeginS & &
\SBlack \connS & & \BDvonn{2} \connS & & \bull{d}{3} \conn & & \SBlack
\connS & & \BDvonn{3} \connS & & \White{4} \connS & & \SWhite \connS &
& \Dvonn \connS & & \SWhite \connS & & \SWhite \connendS & . \\ & &
\Black{5} \connbeginS & & \SBlack \connS & & \SBlack \connS & &
\bull{d}{2} \conn & & \SBlack \connS & & \bull{f}{2} \conn & &
\bull{g}{2} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite
\connendS & & \\ & & & \bull{a}{1} \con & & \bull{b}{1} \con & &
\Black{5} \conS & & \bull{d}{1} \con & & \bull{e}{1} \con & &
\bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & & \White{2}
& & & \\ .& & & & & & & & & & & & & } $

Probably, Black lost the
game with its last move d1-f3 thereby disconnecting its pieces into two
clusters. White (the human player) must already have realized at this
moment he had a good chance of winning (as indicated in the previous
post) by letting Black run out of moves by building large stacks on the
third row, White building a stack of the appropriate size which then
jumps on the largest Black stack on the final move. Btw. this technique
is called *sharpshooting* in Dvonn-parlance

The concept
of manipulating the height of a stack so that it can land precisely on a
critical space. It’s a matter of counting and one-digit addition. Notice
that this doesn’t necessarily mean putting your own stacks atop one
another – the best sharpshooting moves are moves which also neutralize.
To counter a sharpshooting move is called “spoiling”.

But
for this strategy to have a chance, White must keep the Black stacks
containing the Dvonn pieces on the third row. At the moment the stack on
c3 can move to c1 or to c5 and with his next move White counters this
by *overloading* the stack, that is

To spoil a move or
prevent a lifting move by moving atop the enemy stack. Even if the
opponent has enough control to retake the stack, he cannot move it
because it has become taller.

So, White sacrifies his
height 4 stack on g3 with the move g3-c3. Black must take back
immediately (if not, White moves c3-i3 and all Black’s material in the
farmost right cluster is lost) but now the previously mobile Black
height 2 stack at c3 has become an immobile (or *old stack*) height 7
stack which has no option but to stay on c3 (clearly Black will never
move it to j3…). Next, White performs a similar startegy to
neutralize the *young* height 3 Black stack on f3 by overloading it by 2
and hence after the forced recapture it becomes a height 6 Black stack
which must remain on f3 forever. Here are the actual moves 1) g3-c3
b2-c3 2) h2-h3 b4-c5 3) h3-f3 e2-f3 and we end up with the
situation we analyzed last time, that is

$\xymatrix@=.3cm @!C
@R=.7cm{.& & & & & & & & & & & & & \\ & & & \Black{2} \connS & &
\bull{d}{5} \conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & &
\bull{g}{5} \conn & & \bull{h}{5} \conn & & \SWhite \connS & & \SWhite
\connS & & \SWhite \conneS & & & \\ & & \bull{b}{4} \conn & & \SBlack
\connS & & \Black{6} \connS & & \bull{e}{4} \conn& & \bull{f}{4} \conn &
& \bull{g}{4} \conn & & \bull{h}{4} \conn & & \SWhite \connS & &
\SWhite \connS & & \SWhite \conneS & & \\ & \SBlack \connbeginS & &
\SBlack \connS & & \BDvonn{7} \connS & & \bull{d}{3} \conn & & \SBlack
\connS & & \BDvonn{6} \connS & & \bull{g}{3} \conn & & \bull{h}{3}
\conn & & \Dvonn \connS & & \SWhite \connS & & \SWhite \connendS & . \\
& & \Black{5} \connbeginS & & \bull{b}{2} \conn & & \SBlack \connS & &
\bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & &
\bull{g}{2} \conn & & \bull{h}{2} \conn & & \SWhite \connS & & \SWhite
\connendS & & \\ & & & \bull{a}{1} \con & & \bull{b}{1} \con & &
\Black{5} \conS & & \bull{d}{1} \con & & \bull{e}{1} \con & &
\bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & & \White{2}
& & & \\ . & & & & & & & & & & & & & } $

$~\xymatrix@=.3cm @!C @R=.7cm{ & & \Black{2} \connS & &
\bull{d}{5} \conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & &
\bull{g}{5} \conn & & \bull{h}{5} \conn & & \SWhite \connS & & \SWhite
\connS & & \SWhite \conneS & & \\ & \bull{b}{4} \conn & & \SBlack
\connS & & \Black{6} \connS & & \bull{e}{4} \conn& & \bull{f}{4} \conn &
& \bull{g}{4} \conn & & \bull{h}{4} \conn & & \SWhite \connS & &
\SWhite \connS & & \SWhite \conneS & \\ \SBlack \connbeginS & &
\SBlack \connS & & \BDvonn{7} \connS & & \bull{d}{3} \conn & & \SBlack
\connS & & \BDvonn{6} \connS & & \bull{g}{3} \conn & & \bull{h}{3}
\conn & & \Dvonn \connS & & \SWhite \connS & & \SWhite \connendS \\ &
\Black{5} \connbeginS & & \bull{b}{2} \conn & & \SBlack \connS & &
\bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & &
\bull{g}{2} \conn & & \bull{h}{2} \conn & & \SWhite \connS & & \SWhite
\connendS & \\ & & \bull{a}{1} \con & & \bull{b}{1} \con & & \Black{5}
\conS & & \bull{d}{1} \con & & \bull{e}{1} \con & & \bull{f}{1} \con & &
\bull{g}{1} \con & & \bull{h}{1} \con & & \White{2} & &} $

The
reason behind this unwanted clipping is that Latexrender uses
**convert** to take the relevant part of a ps-page containing only the
TeXed formula on an empty page by performing clipping and then converts
it into a GIF-file (or any other format you desire). The obvious way
round this is to enlarge my template by adding two additional rows and
columns and putting visible nonsense there (such as dots) to enlarge the
relevant part so that no clipping is done of essential info. But then
(1) the picture generated becomes even larger than that above and (2) I
don’t want you to see the extra nonsensical dots… The essential line
in the **class.latexrender.php** file is

$command =
$this->_convert_path." -density ".$this->_formula_density.
" -trim -transparent \"#FFFFFF\" ".$this->_tmp_filename.".ps ".
$this->_tmp_filename.".".$this->_image_format;

So
I needed to delve into the [manual pages for the convert command](http://amath.colorado.edu/computing/software/man/convert.html)
of the ImageMagick-package. To my surprise, the *-trim* option (which I
thought to adjust somewhat by adding parameters) doesn’t exist! Still, I
got around my second problem using the *crop* option and around the
first by using the very useful *geometry* option. The latter is also
useful if you find that the size of the output of Latexrender is not
compatible with the size of your regular text. Of course you can amend
this somewhat by using the *extarticle* documentclass (as suggested) but
if you want to further adjust it, use for example

-geometry
86%

to size the output to exactly 86% (or whatever you need).
So, whenever I want to do some Dvonn-blogging from now on I’ll change my
class.latexrender.php file as follows

$command =
$this->_convert_path." -crop 0x0-10% -crop 0x0+10% -density
".$this->_formula_density. " -geometry 80%
-transparent \"#FFFFFF\" ".$this->_tmp_filename.".ps ".
$this->_tmp_filename.".".$this->_image_format;

which
produces the output

$\xymatrix@=.3cm @R=.7cm{.& & & & & & & & & &
& & & \\ & & & \Black{2} \connS & & \bull{d}{5} \conn & & \bull{e}{5}
\conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn
& & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & & \\ & &
\bull{b}{4} \conn & & \SBlack \connS & & \Black{6} \connS & &
\bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & &
\bull{h}{4} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite
\conneS & & \\ & \SBlack \connbeginS & & \SBlack \connS & &
\BDvonn{7} \connS & & \bull{d}{3} \conn & & \SBlack \connS & &
\BDvonn{6} \connS & & \bull{g}{3} \conn & & \bull{h}{3} \conn & &
\Dvonn \connS & & \SWhite \connS & & \SWhite \connendS & . \\ & &
\Black{5} \connbeginS & & \bull{b}{2} \conn & & \SBlack \connS & &
\bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & &
\bull{g}{2} \conn & & \bull{h}{2} \conn & & \SWhite \connS & & \SWhite
\connendS & & \\ & & & \bull{a}{1} \con & & \bull{b}{1} \con & &
\Black{5} \conS & & \bull{d}{1} \con & & \bull{e}{1} \con & &
\bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & & \White{2}
& & & \\ . & & & & & & & & & & & & & } $

which (I hope) you will
find slightly better…

DVONN is a stacking game. It is played
on an elongated hexagonal board, with 23 white, 23 black and 3 red
DVONN-pieces. In the beginning the board is empty. The players first
place the DVONN-pieces on the board and next their own pieces. Then they
start stacking pieces on top of each other. A single piece may be moved
1 space in any direction, a stack of two pieces may be moved two spaces,
etc. A stack must always be moved as a whole and a move must always end
on top of another piece or stack. If pieces or stacks lose contact with
the DVONN pieces, they must be removed from the board. The game ends
when no more moves can be made. The players put the stacks they control
on top of each other and the one with the highest stack is the winner.
That’s all!

All this will become clearer once we fix a
specific end-game, for example

$\xymatrix@=.3cm @!C @R=.7cm{ & &
\Black{2} \connS & & \bull{d}{5} \conn & & \bull{e}{5} \conn & &
\bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn & &
\SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & \\ &
\bull{b}{4} \conn & & \SBlack \connS & & \Black{6} \connS & &
\bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & &
\bull{h}{4} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite
\conneS & \\ \SBlack \connbeginS & & \SBlack \connS & & \BDvonn{7}
\connS & & \bull{d}{3} \conn & & \SBlack \connS & & \BDvonn{6} \connS &
& \bull{g}{3} \conn & & \bull{h}{3} \conn & & \Dvonn \connS & & \SWhite
\connS & & \SWhite \connendS \\ & \Black{5} \connbeginS & &
\bull{b}{2} \conn & & \SBlack \connS & & \bull{d}{2} \conn & &
\bull{e}{2} \conn & & \bull{f}{2} \conn & & \bull{g}{2} \conn & &
\bull{h}{2} \conn & & \SWhite \connS & & \SWhite \connendS & \\ & &
\bull{a}{1} \con & & \bull{b}{1} \con & & \Black{5} \conS & &
\bull{d}{1} \con & & \bull{e}{1} \con & & \bull{f}{1} \con & &
\bull{g}{1} \con & & \bull{h}{1} \con & & \White{2} & &} $

with
White to move. Some comments about notation : the left-slanted columns
are denoted by letters from a (left) to k (right) and the rows are
labeled 1 to 5 from bottom to top (surprisingly this ‘standard’
webgame-notation differs from the numbering on my Dvonn-board where the
rows are labeled from top to bottom…). So, for example, the three
spots on the upper right are k3,k4 and k5 (there are no k1 or k2 spots).
The three Dvonn pieces are colored red and in the course of the game a
stack may land on a Dvonn piece and so stacks containing a Dvonn piece
are denoted with a red halo. For example, the symbol on spot f3 stands
for for a stack of 6 pieces, one of which is a red Dvonn piece, under
the control of Black (that is, the top-piece is Black). Further note
that a piece or stack can only move if it is not surrounded by 6 other
pieces or stacks (so the White pieces on j3 and j4 cannot (yet) move). A
piece can only move by one step in either line-direction provided there
is another piece or stack on that position. The same applies for stacks
: an height 3 stack for example can move in each lin-direction by
exactly 3 steps provided there is a piece or stack to jump onto. For
example, the height 6 stack on d4 can only move to j4 whereas the height
6 stack on f3 cannot move at all! Similarly, the two black height 5
stacks are immobile. At the moment black has all its stacks defended,
that is, if White should be able to jump onto one of them (which White
cannot at the moment), Black can use one of its neighbouring pieces to
take the stack back under its control. So, any computer program would
‘evaluate’ the position as favourable for Black : Black has stacks of
total height 34 safely under control (there are no immediate threats to
be seen : the [horizon effect](http://www.comp.lancs.ac.uk/computing/research/aai-aied/people/paulb/old243prolog/subsection3_7_5.html) in such programs) whereas White
can only claim potential stacks of total height 13… Still, Black
has already lost the game. White has more pieces which are quite mobile
as opposed to the immobile black stacks, so Black will soon run out of
moves to make and his end position will have some large stacks on the
third row. All white has to do is to let Black run out of moves and then
continue (Dvonn forces each player to make a move if they still can and
to pass the move otherwise, so the most mobile player can still continue
long after the other player was forced to stop) to build a White stack
of the appropriate height on the third row to jump on the highest Black
stack with its last move! Here is how the play continued : 1) j2-k3 ;
a3-b3 2) i1-k3 ; c5-c3 3) i2-i3 ; c2-c3 4) i3-k3 ; d4-j4 5)
j3-j4 ; e3-f3 6) i4-j4 ; c4-b3 to arrive at the position where
Black is no longer able to make any moves at all

$\xymatrix@=.3cm
@!C @R=.7cm{ & & \bull{c}{5} \conn & & \bull{d}{5} \conn & & \bull{e}{5}
\conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn
& & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & \\ &
\bull{b}{4} \conn & & \bull{c}{4} \conn & & \bull{d}{4} \conn & &
\bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & &
\bull{h}{4} \conn & & \bull{i}{4} \connS & & \White{9} \connS & &
\SWhite \conneS & \\ \bull{a}{3} \connbegin & & \Black{3} \connS & &
\BDvonn{10} \connS & & \bull{d}{3} \conn & & \bull{e}{3} \conn & &
\BDvonn{7} \connS & & \bull{g}{3} \conn & & \bull{h}{3} \conn & &
\bull{i}{3} \conn & & \bull{j}{3} \conn & & \WDvonn{6} \connendS \\ &
\Black{5} \connbeginS & & \bull{b}{2} \conn & & \bull{c}{2} \conn & &
\bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & &
\bull{g}{2} \conn & & \bull{h}{2} \conn & & \bull{i}{2} \conn & &
\bull{j}{2} \connend & \\ & & \bull{a}{1} \con & & \bull{b}{1} \con & &
\bull{c}{1} \con & & \bull{d}{1} \con & & \bull{e}{1} \con & &
\bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & &
\bull{i}{1} & &} $

Note that all pieces and stacks no longer
connected to a Dvonn piece must be removed. So, for example, after the
third move by Black, the Black height 5 stacks on c1 was removed. All
white now has to do is to built an height 8 stack on k3 and jump onto
the height 10 Black stack on c3 to win the game. The (only) way to do
this is by 7. j5-k5 and 8. k5-k3 to finish with 9. k3-c3 with final
position (note again that the White right-hand pieces and stacks are no
longer connected to a Dvonn piece and are hence removed)

$\xymatrix@=.3cm @!C @R=.7cm{ & & \bull{c}{5} \conn & & \bull{d}{5}
\conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn
& & \bull{h}{5} \conn & & \bull{i}{5} \conn & & \bull{j}{5} \conn & &
\bull{k}{5} \conne & & \\\ & \bull{b}{4} \conn & & \bull{c}{4} \conn &
& \bull{d}{4} \conn & & \bull{e}{4} \conn& & \bull{f}{4} \conn & &
\bull{g}{4} \conn & & \bull{h}{4} \conn & & \bull{i}{4} \conn & &
\bull{j}{4} \conn & & \bull{k}{4} \conne & \\\ \bull{a}{3} \connbegin
& & \Black{3} \connS & & \WDvonn{18} \connS & & \bull{d}{3} \conn & &
\bull{e}{3} \conn & & \BDvonn{7} \connS & & \bull{g}{3} \conn & &
\bull{h}{3} \conn & & \bull{i}{3} \conn & & \bull{j}{3} \conn & &
\bull{k}{3} \connend \\\ & \Black{5} \connbeginS & & \bull{b}{2} \conn
& & \bull{c}{2} \conn & & \bull{d}{2} \conn & & \bull{e}{2} \conn & &
\bull{f}{2} \conn & & \bull{g}{2} \conn & & \bull{h}{2} \conn & &
\bull{i}{2} \conn & & \bull{j}{2} \connend & \\\ & & \bull{a}{1} \con &
& \bull{b}{1} \con & & \bull{c}{1} \con & & \bull{d}{1} \con & &
\bull{e}{1} \con & & \bull{f}{1} \con & & \bull{g}{1} \con & &
\bull{h}{1} \con & & \bull{i}{1} & & } $

So White wins with 18 to
Black’s 15. This shows that it is important to maintain mobility and
also that it is possible to win a Dvonn-game from computers. In fact,
the above end-game was played against a computer-program (Black). The
entire game can be found
[here](http://www.littlegolem.net/jsp/game/game.jsp?gid=426457&nmove=91)
.