The Best of Creative Computing Volume 2 (published 1977)

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Creative Chess (Recreations based on the game of chess)
by Walter Koetke

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Creative Chess
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by Walter Koetke
Lexington High School. Mass.
The game of chess was introduced to Europe
near the middle of the thirteenth century. As one
might expect of a game with over 197,000 ways of
making the first four moves with over 71,000 resulting positions, many people
have always been fascinated with the game but so very few have mastered
it. This article is not, however, about the game of
chess but rather about a few of the recreations it
has spawned.
According to W.W. Ball (1), one of the oldest
known chess related recreatlons was proposed in
the early 1500's. The problem consisted of a 3 by 3
board with two white knights (W) and two black
knights (B) positioned as shown.
BIB
III
WIW
The object is to move the pieces so that the squares
occupied by the white knights are occupied by the
black knights and vice versa.
When one is learning to play chess, some type
of point value for each piece is usually assumed.
For example, I was taught to value a queen as 10,
a castle as 5, and a bishop and knight each as 3.
You may have noticed that different introductory
texts are likely to suggest different point values.
How can this be? How are these point values assigned? If the relative point
values are valid, then
why do the experts disagree?
Suppose we choose to evaluate a piece by computing its "checking power". We can
define the
checking power of a piece as the probability that
the piece will have the king in check if the piece
and the opposing king are placed at random on an
empty board.
***
Computer Glass Box continuedmatics, engineering, ecology, and physical
sciences.
The challenge to educators, then, is to identify such
topics suitable for embodiment as glass box programs, to
search out the kernel concepts to be taught, and to lead
students to better understandings of those concepts using a
programming language.
***
References
***
[1] Papert, S. "Teaching Children Thinking", M.l.T. LOGO
Memo No. 2, Oct. 1971.
[2] Iverson, K. E. "APL in Exposition", IBM Tech. Report
No. 320-3010, Jan. 1972.
[3] Berry, P. et. al. "APL and Insight: The Use of Programs
to Represent Concepts in Teaching", IBM Tech. Report
No. 320-3020, March 1973.
To compute the checking power of a castle, the
castle is placed on an empty board leaving 63 empty
squares. Of these 63 squares. 14 are controlled by
the castle. A king placed at random on the board
would, therefore, have a probability of 14/63=2/9
of being in check. Thus the checking power of a
castle is 2/9.
Computing the checking power of a bishop is a
bit more tedious. A bishop placed on any of the 28
squares in the outer ring of the board commands 7
of the remaining 63 squares. However, if the bishop
is placed on any of the 20 squares in the ring of
squares adjacent to the border ring, then it commands 9 of the remaining 63
squares. Similarly, if
placed on one of the 12 squares of the next inner
ring the bishop commands 11 of those remaining,
and if placed on one of the 4 center squares the
bishop commands 13 of those remaining. Thus the
checking power of a bishop is:
28 7  20 9  12 11 4  13   5
--*--+--*--+--*--+--*-- = -64 63 64 63 64 63 64 63   36
Using similar logic, the checking power of a
knight can be computed as 1/12 and the checking
power of a queen as 13/36. Converting the computed checking power to integers
yields:
***
Piece Checking Power Relative Point Value
Queen      13/36             13
Castle      2/9               8
Bishop      5/36              5
Knight      1/12              3
What do these relative point values suggest?
Perhaps that my own assumed values of 10, 5, 3,
and 3 aren't very good. Perhaps that our definition
of "checking power" should be improved. Perhaps
that the chess masters are supplying relative point
values from their experience rather than from reproducible computation.
Try computing some relative point values on your
own. Can you define "checking power" so you can
produce the relative point values you've assumed?
One alternate definition of checking power begins
as ours did, but then excludes these squares from
the pieces control from which the king could take
the piece. For example, consider the computing
power of a castle using this alternate definition. If
the castle is placed in any one of the 4 corners, it
controls 14 of 63 remaining squares. However, if
the king was placed in either of the 2 squares
adjacent to the castle it could take the castle. Thus
the castle really only controls 12 of the 63 remaining squares. If the castle is
placed in any of the
other 24 outside squares it again appears to control
14 squares, but 3 of these, those adjacent to the
rook, must be excluded because the king could
take the rook from these positions. Similarly, if the

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