Check out Janggi (Korean Chess), our featured variant for December, 2024.

Enter Your Reply

The Comment You're Replying To
Jörg Knappen wrote on Thu, Nov 21, 2013 08:54 AM UTC:
I think a got a proof for the hex geometry.

We orient the hexes such that there is a horizontal line of rook movement,
and denote that direction by 1. The other directions of rook movement are
denoted by \omega and (\omega-1) [the use of the letter \omega is
inspired by Eisenstein numbers]. The centre of a hex is given by a+b\omega
with a,b integer numbers.

First step is a drawing: When we go horizontally firs and vertically as a
hex bishop second, we can reach only one half of the hexes (a+2b\omega).
We repeat this for the other rook directions and mark the hexes
accordingly. They fall in two classes: (i) hexes which can be reached in
one way only (ii) hexes that can be reached in all three way.

The second class forms a grid described by 2a+2b\omega (both coordinates
must be even.

Finally we map these to rook and bishop moves. The path to a three-way
reachable hex (2a+2b\omega) using horizontal and vertical moves
(elementary vertical bishop step: (2\omega -1)) consists of b bishop steps
and b+2a rook steps. Therefore the number of rook and bishop steps are both
odd or both even, giving an even SOLL.

The other direction: Take r rook steps and s bishop steps and demand that
r+s is even. Then we go to r+s*(2\omega -1) = (r-s) +2s\omega. This is a
three-way reachable square again, because (r+s) even implies (r-s) even.

Edit Form
Conduct Guidelines
This is a Chess variants website, not a general forum.
Please limit your comments to Chess variants or the operation of this site.
Keep this website a safe space for Chess variant hobbyists of all stripes.
Because we want people to feel comfortable here no matter what their political or religious beliefs might be, we ask you to avoid discussing politics, religion, or other controversial subjects here. No matter how passionately you feel about any of these subjects, just take it someplace else.
Avoid Inflammatory Comments
If you are feeling anger, keep it to yourself until you calm down. Avoid insulting, blaming, or attacking someone you are angry with. Focus criticisms on ideas rather than people, and understand that criticisms of your ideas are not personal attacks and do not justify an inflammatory response.
Quick Markdown Guide

By default, new comments may be entered as Markdown, simple markup syntax designed to be readable and not look like markup. Comments stored as Markdown will be converted to HTML by Parsedown before displaying them. This follows the Github Flavored Markdown Spec with support for Markdown Extra. For a good overview of Markdown in general, check out the Markdown Guide. Here is a quick comparison of some commonly used Markdown with the rendered result:

Top level header: <H1>

Block quote

Second paragraph in block quote

First Paragraph of response. Italics, bold, and bold italics.

Second Paragraph after blank line. Here is some HTML code mixed in with the Markdown, and here is the same <U>HTML code</U> enclosed by backticks.

Secondary Header: <H2>

  • Unordered list item
  • Second unordered list item
  • New unordered list
    • Nested list item

Third Level header <H3>

  1. An ordered list item.
  2. A second ordered list item with the same number.
  3. A third ordered list item.
Here is some preformatted text.
  This line begins with some indentation.
    This begins with even more indentation.
And this line has no indentation.

Alt text for a graphic image

A definition list
A list of terms, each with one or more definitions following it.
An HTML construct using the tags <DL>, <DT> and <DD>.
A term
Its definition after a colon.
A second definition.
A third definition.
Another term following a blank line
The definition of that term.