The Chess Variant Pages



New 3D Leapers

By L. Lynn Smith


In 3D Chess, most are quite familiar with the three basic leapers.  These are
the Knight(1x2x3), the Hippogriff(2x2x3) and the Wyvern(2x3x3).  The first being
a planar leaper and the other two cubic.  And they are leapers within the
orthogonal pattern of the playing field.

What if leapers were restricted to the diagonal and triagonal patterns of the
3D playing field?

Let's use the planar leap of the Knight to demonstrate this.

Within a plane, the Knight is able to leap to eight cells and this is its
pattern ('O' represents the Knight and 'X' represents its potential destination):


[ ][X][ ][X][ ]
[X][ ][ ][ ][X]
[ ][ ][O][ ][ ]
[X][ ][ ][ ][X]
[ ][X][ ][X][ ]


This is the orthogonal pattern of this particular leap.

Now if the 1x2x3 pattern was expressed specifically within the diagonal 
pattern of that same plane;

[ ][ ][X][ ][x][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]
[X][ ][ ][ ][ ][ ][x]
[ ][ ][ ][O][ ][ ][ ]
[X][ ][ ][ ][ ][ ][x]
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][x][ ][ ]


Looking at a single leap, comparing an orthogonal pattern to a diagonal:

      orthogonal                  diagonal

[ ][ ][ ][ ][ ][ ][ ]      [ ][ ][X][ ][ ][ ][ ]
[ ][ ][ ][-][X][ ][ ]      [ ][-][ ][-][ ][ ][ ]
[ ][ ][ ][-][-][ ][ ]      [ ][ ][-][ ][-][ ][ ]
[ ][ ][ ][O][-][ ][ ]      [ ][ ][ ][O][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]      [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]      [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]      [ ][ ][ ][ ][ ][ ][ ]



At first glance, this diagonal pattern leap appears to be the leap of the 
2D Camel.  But we are only looking at a single plane.  And in addition to 
two more orthogonal planes, the diagonal pattern has four diagonal plane 
patterns composed of axes from separate orthogonal planes in a 3D playing
field.  So that its potential leap pattern might appear thus:


[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]
[ ][X][ ][X][ ][X][ ]
[ ][ ][X][ ][X][ ][ ]  level 7
[ ][X][ ][X][ ][X][ ]
[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]

[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]
[X][ ][X][ ][X][ ][X]
[ ][ ][ ][ ][ ][ ][ ]  level 6
[X][ ][X][ ][X][ ][X]
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]

[ ][X][ ][X][ ][X][ ]
[X][ ][X][ ][X][ ][X]
[ ][X][ ][ ][ ][X][ ]
[X][ ][ ][ ][ ][ ][X]  level 5
[ ][X][ ][ ][ ][X][ ]
[X][ ][X][ ][X][ ][X]
[ ][X][ ][X][ ][X][ ]

[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]
[x][ ][ ][ ][ ][ ][X]
[ ][ ][ ][O][ ][ ][ ]  level 4
[X][ ][ ][ ][ ][ ][X]
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]

[ ][X][ ][X][ ][X][ ]
[X][ ][X][ ][X][ ][X]
[ ][X][ ][ ][ ][X][ ]
[X][ ][ ][ ][ ][ ][X]  level 3
[ ][X][ ][ ][ ][X][ ]
[X][ ][X][ ][X][ ][X]
[ ][X][ ][X][ ][X][ ]

[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]
[X][ ][X][ ][X][ ][X]
[ ][ ][ ][ ][ ][ ][ ]  level 2
[X][ ][X][ ][X][ ][X]
[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]

[ ][ ][ ][ ][ ][ ][ ]
[ ][ ][X][ ][X][ ][ ]
[ ][X][ ][X][ ][X][ ]
[ ][ ][X][ ][X][ ][ ]  level 1
[ ][X][ ][X][ ][X][ ]
[ ][ ][X][ ][X][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]

Obviously, this piece is much more powerful than the simple Camel.  And
it will always remain within its specific diagonal pattern.  Being a simple
planar leap, it might be the easiest to visualize.

The other two patterns, 2x2x3 and 2x3x3 are cubic and thus involve the
exploitation of three distinct axes.  These will be specific diagonal axes
within the diagonal patterns of the 3D playing field.

The following are single examples of each of these two forms of leaps
within a diagonal pattern.


        2X2X3                             2X3X3

[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][X][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][-][ ][-][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][-][ ][-][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]   level 3   [ ][ ][ ][-][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]


[ ][ ][X][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][-][ ][-][ ][ ][ ]             [ ][ ][-][ ][ ][ ][ ]
[ ][ ][-][ ][-][ ][ ]             [ ][-][ ][-][ ][ ][ ]
[ ][ ][ ][-][ ][ ][ ]   level 2   [ ][ ][-][ ][-][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][-][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]


[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][ ][-][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]
[ ][-][ ][-][ ][ ][ ]             [ ][ ][-][ ][ ][ ][ ]
[ ][ ][-][ ][-][ ][ ]   level 1   [ ][-][ ][-][ ][ ][ ]
[ ][ ][ ][O][ ][ ][ ]             [ ][ ][-][ ][-][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][O][ ][ ][ ]
[ ][ ][ ][ ][ ][ ][ ]             [ ][ ][ ][ ][ ][ ][ ]


Obviously, there are many more potential leaps for each of these
particular patterns.  But they will all remain with the defined
diagonal pattern of the playing field.

These same leaps may be applied to the triagonal patterns of the
3D playing field, creating three additional new 3D leapers.  I
will allow the reader to work out these particular patterns.

But I will give an example of a triagonal pattern Knight leap
to assist in initial visualization:


[ ][X][ ][ ]
[ ][ ][ ][ ]   level 4
[ ][ ][ ][ ]
[ ][ ][ ][ ]

[ ][ ][ ][ ]
[-][ ][-][ ]   level 3
[ ][ ][ ][ ]
[ ][ ][ ][ ]

[ ][ ][ ][ ]
[ ][ ][ ][ ]   level 2
[ ][-][ ][-]
[ ][ ][ ][ ]

[ ][ ][ ][ ]
[ ][ ][ ][ ]   level 1
[ ][ ][ ][ ]
[ ][ ][O][ ]


Now, we arrive at what to call these six new pieces.  A simple
form might be to prefix each piece with its particular pattern.
For example: 'o-' would denote orthogonal, 'd-' for diagonal and
't-' for triagonal.

So that the Knight operating in the orthogonal pattern would be the 
o-Knight, one within the diagonal pattern the d-Knight and the triagonal
as the t-Knight.

And the potential number of composite leapers has now 'jumped'
exponentially.  Although some combinations will be redundant, as 
they are expressed in others. For example, the o-Hippogriff is
present in the pattern of the d-Knight.

Now, I am not advocating the application of each and every one of
these leapers in any one particular game.  But a 3D developer might
consider one or two of them.  

A potential might be to create a 3D game in which the a variety of 
pieces are bound to particular patterns of the playing field.  For
example, if a majority of pieces were bound to the diagonal patterns, 
there exist the potential of two distinct games being played out.
The developer would then have to consider these dynamics when establishing
the goals of the game.