QUICKIES


Roger Phillips, on a problem by Karl Scherer.  In a chess game, none of white's pieces have been captured, and only two have promoted.  Now, every one of white's pieces has a unique mate in 1.  Possible position.

Three retrograde chess problems by Karl Scherer.

A table about groups I (Ed Pegg) put together ... I want to show pictures 
of all these groups.


SOCCALOT, by David Wilson
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The game is a two-player game based loosely on soccer. It is played on
an 8x8 checker or chessboard. The teams, Red and Black, each have 6
men (checkers). An additional piece (white pawn) represents the ball.
The initial setup is as follows (R = Red, B = Black, O = ball).

+-+-+-+-+-+-+-+-+
| |B|B|B|B|B|B| |
+-+-+-+-+-+-+-+-+
| | | | | | | | |
+-+-+-+-+-+-+-+-+
| | | | | | | | |
+-+-+-+-+-+-+-+-+
| | | | | | | | |
+-+-+-+-+-+-+-+-+
| | | |O| | | | |
+-+-+-+-+-+-+-+-+
| | | | | | | | |
+-+-+-+-+-+-+-+-+
| | | | | | | | |
+-+-+-+-+-+-+-+-+
| |R|R|R|R|R|R| |
+-+-+-+-+-+-+-+-+

Play proceeds in turns, Red plays first. A play is one of the
following:

------------------------------------------------------------
RUN: Move one of your men one space, as a King in chess.
A man may not move to an occupied square (by his own man,
an opponent's man, or the ball).

------------------------------------------------------------
DRIBBLE: If your man is King-adjacent to the ball, you may
move that man and the ball each one space, maintaining
their relative position. Neither the man nor the ball may
move to an occupied square.

For example, note the following position, with Red to play:

+-+-+-+-+
| | | | |
+-+-+-+-+
| |B|O| |
+-+-+-+-+
| |R| | |
+-+-+-+-+
| | |B| |
+-+-+-+-+

Red may make any of five dribbles, resulting in the following
positions:

+-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+
| | | | | | | | |O| | |O| | | | | | | | | | | | |
+-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+
| |B| |O| | |B|R| | |R|B| | | | |B| | | | |B| | |
+-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+
| | |R| | | | | | | | | | | | | |O| | | | | |O| |
+-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+
| | |B| | | | |B| | | | |B| | |R| |B| | | |R|B| |
+-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+ +-+-+-+-+

------------------------------------------------------------
KICK: If your man is King-adjacent to the ball, you may
move the ball any number of unoccupied spaces in the direction
directly away from the man. The ball may move to or across an
occupied square. The man does not move.

For example, in the following position, Red may kick the Ball
to any of the positions marked x.

+-+-+-+-+-+-+-+-+
| | |B| | | |B| |
+-+-+-+-+-+-+-+-+
| | |x| |B|B| | |
+-+-+-+-+-+-+-+-+
| |B|x| |x| | | |
+-+-+-+-+-+-+-+-+
| | |x|x| |B| | |
+-+-+-+-+-+-+-+-+
| | |O| | | | | |
+-+-+-+-+-+-+-+-+
| |R|R| |R|R| | |
+-+-+-+-+-+-+-+-+
| | | | | | | | |
+-+-+-+-+-+-+-+-+
| | | |R| | |R| |
+-+-+-+-+-+-+-+-+

The game is won by getting the ball to the back rank (opponent's
first rank).
--------------------------------------------------------------------
Much of the game is a struggle to gain control of the ball or to
find an opening to the back rank. The course of the game can change
suddenly when the ball is kicked to a remote area of the board.
You should play five or six games before passing judgment, at which
point you will either be totally turned off or hopelessly addicted.
If you like board games, you might want to try this one.  I devised
this game back in 1972, and played a few games with a cousin, who
whipped me soundly. I forgot about the game until 1999. At that time,
the lunchtime chess club was playing around with chess variants, and I
remembered this game (which is not a chess variant). The game caught
on very well, and we played it to the exclusion of chess for several
weeks. Finally, we determined that the game needed a name, and
eventually settled on Soccolot.
----------------------------------------------------------------------

Winning Strategy for Square Removal Game, by Mark Thompson

In the square removal game, a 16x16 square starts the game, and players take turns
removing 1x1, 2x2, or 3x3 squares from it. The last to move loses.

I believe the first player wins the square-removal game. The winning strategy
is to begin by playing a 2x2 in the center of the board, and then play
symmetrically to the second player’s moves by a 180 degree rotation through the
center, until a certain clearly-definable crucial point is reached. Note that if
the first player adopts this strategy, there will never come a time when the
second player can spoil or use up the last place where a 2x2 square could be
removed; if one such place exists at the beginning of the second player's turn,
there will also be another symmetrically opposite. Therefore when the time
comes when there is exactly one place remaining where a 2x2 square can be
removed, it will be player 1’s turn. At this point, player 1 can abandon
symmetric play; he counts the number of 2x2 squares that have been removed thus
far, and if the total is odd, he places the last 2x2 square; if the total is
even, player 1 uses some other size of square to spoil the position so that no
more 2x2 squares can be played.  I think it's clear that this will always
be possible.

After that, player 1 can do anything he likes, because no matter how the game
proceeds, in the end it will have used an even number of 2x2 squares. Note that
the total number of 1x1 plus 3x3 squares used in a game will always be even
(since the entire board has an even number of squares); therefore, if there are
an even number of 2x2 squares used, the game will last an even number of turns,
which means player 2 loses. So as long as player 1 can ensure that an even
number of 2x2 squares are used, he wins; and the strategy described ensures
this.
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Bryce Herdt -- an 8x8 square is filled with the numbers 1-64. Can all sums of orthogonal squares be square
numbers? What is the fewest possible number of differing orthogonal sums?

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Ms.Mami Ishizaki in Japan has found the 20 moves result on 6 by 6.

<Start> <Goal>
+ - - - - - - + + - - - - - - +
| . . # . . # | | . . # . . # | #: Wall
| B . . B A # | | B . . B . # |
| B B B B B . | | B B B B B A |
| B . . B . # | ===> | B . . B . # |
| . # B B . . | | . # B B . . |
| # # . B B . | | # # . B B . |
+ - - - - - - + + - - - - - - + 20 moves

What is the greatest number of moves for a 2-piece sliding block puzzle on
a 7x7 square? What is the greatest number of moves for a 3 piece puzzle in
a 6x6 square? The Sliding Block Puzzle analyzer is *very* nice to play with!
------------------------------------------------------------------------------

All of the solutions below are courtesy of the wonderful Polyforms Group.