3 by 3 chessboard

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Faded-Maximus

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Sep 24, 2006
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you have a 3 by 3 chessboard with 3 white pawns and 3 black pawns. given white goes first and both teams play their best what is the outcome?

is the outcome a draw always since pawns can only move forward and kill diagonally. This means that the middle pawn can only kill either the left or right pawn, which means the left or right pawn for each team will always be alive, so it will always be a draw?
 
Hello, Faded-Maximus!

You didn't define a "win".

I would conjecture that the rules are somthing like:

A player wins if:
. . (1) he captures all his opponent's pawns, or
. . (2) one of his pawns reaches the opposite side.

If a player has no legal moves, the game is a draw.

 
Faded-Maximus said:
… both teams play their best …

… it will always be a draw?
Click to expand...


I read the statement about both teams above to mean that a pawn must move to kill anytime it has the possibility to do so.

I'm also guessing that a pawn must reach the opposite side of the board in order to win the game for its team.

Did you draw some pictures and play the game? The potential outcomes will all easily fit on a single piece of paper because there are not many.

Try it. Let us know if you still cannot reach a conclusion.

-
… which means the left or right pawn for each team will always be alive …
Click to expand...


This statement is not true. Draw the possibilities, and you will see why.

 
Many assumptions can be made as playing it's best wasn't established whether its to kill when it can or not. Likewise, a win wasn't specified, but based on chess rules I would think a win is only if one player has killed the three other players..

Assuming kill when you can, I drew out some possibilities and I only end up with draws. Moving the middle white one first: I get a draw where W is white and B is black, X is blank. I figured this can be mirrored a bunch of ways.
XXX
XWX
XBX

Moving a side piece first I get a final board that looks like, which I figured can be mirrored a bunch of ways too.

XXW
WXB
BXX

I don't know how the statement it will be a draw is not true, All the ways I have drawed out, it is impossible to take all 3 of ones pawns.
 
Oh boy...Now you would want to castle - without king or castle...
 
if thinking outside the box, perhaps! But yes, what good is a queen without a king?
 
also, it would be considered a win if a pawn got to the opposite side, for, by chess rules, when a pawn reaches the other side, it can become anything it wants. therefore, if the player is playing to his/her maximum, the pawn would turn into a queen, the queen would massacre the remaining pawns, and the game would be won. btw, it is legal to start a game without a king, for, technicly, the game ends when the king is either dead (suicide) or shure to be dead (checkmate). so, therefore, if there was no king in the first place, there is no suicide or checkmate, so there is no end-game. it is a kingless game, sometimes called a massacre game, because the players no longer have the need to form defence or do trickey manuvers to get at the king. the game ends in either draw or exausion (all peices dead). i will release my findings tomarrow. good-night.
 
i am in the process of writing my response. it should be done by 6:00 Eastern Time Zone. i have to go, so by!
 
never! when i am challenged to a math problem, or see an incomplete one, i will usually stop at nothing to finish it!
 
and this, i just found out, is not one of those times...

i just hit the preveiw button, and the spacing went all cunfusing on my game tree... 8hrs of work down the drain...
 
matt000r000 said:
and this, i just found out, is not one of those times...
i just hit the preveiw button, and the spacing went all cunfusing on my game tree... 8hrs of work down the drain...
Click to expand...
Did you swear?
 
Re: lets wrap this up.

this is the work i did get done:

the results are in!!!!!!!!!!!!!!!!!!!!!
here is the accumulation of 6 hours of tedious work: (WARNING: THIS WILL GET VERY COMPLICATED. NOT FOR THE ILL-MINDED)

so i first decided to settle the debates on the rules. to take the original context, the players are "playing at their best". this most certainly does not mean they must take a pawn whenever available. some of the best plays you can make ignore oppertunities. i, being a programer and a chess fanatic, knew the most efficient way of dealing with chess problems: to use the game tree. for those who do not know what this is, it is an accounting of every possible move that can be taken. every possible outcome is aceived. this is how computers always beat you in chess. checkers, tictacto, and most any other turn-based game, too. then i decided that, if a pawn were to get to the opposite side, it would be a win. for, by the rules of chess, if a pawn got to the opposing side of the chess board, it would turn into any peice not on the board. if the player was truely playing at their best, they would get a queen and massacre the board. finaly, en passants are impossible, because it is impossible for a pawn to move 2 spaces at the start, thus the very thing the en passant refers to is non-existant, thus making it non-existant. so, therefore, the debate is settled: a peice on the opposing side is a win. also, i included any way possible, so stupid mistakes are recorded in the game tree provided. next, i started on my 3-page-long game tree that took most of the 6 hours i spent on this problem.

now, before i get to the game tree, i would like to demonstrate a few things. first, the syntax used. "p", when refering to a move, means a pawn moved. "W" or "B" denotes the color of the pawn. the subsequent number refers to the pawn's number (1, 2, or 3). the pawns are numbered as folows:
Code: 
--------------------------
| pB1 |  pB2 | pB3  |
--------------------------
|        |        |         |
--------------------------
| pW1 | pW2 | pW3|
--------------------------
after specifying what peice i am refering to, i write the movements of the peice. the movements are shown by how many spaces up the unit moved, and how many spaces right the unit moves. and, to prevent future questions, yes, that does mean a downward move is a negetive number, and likewise with a left move. they are written similar to an ordered pair, exept the "y" coordinate, or the movement up or down, is first. also, if a pawn takes another pawn, it is refered to after the move is stated. so, the end product looks something like this:
Code: 
pW1(1,-1)pB2
pB3(-1,1)pW3[B]
pW2(1)D
the first one reads: "white pawn 1 moves up 1 and left one, taking black pawn 2"
the seccond one reads: "black pawn 3 moves down 1 and right 1, taking white pawn 2 and winning the game for black"
the third one reads: "white pawn 2 moves up 1, causeing a draw"
so, yes, when the pawn only moves up or down, the seccond move coordinate is excluded, since it would be "0".

on to the game trees:
(by the way, i recomend having a chessboard of some kind in front of you. i drew one and used 3 quarters and 3 nickels.)
Code: 
first moves
--------------
   |
   |---------------|----------------------------------------|
   |                  |                                                |
pW1(1)         pW2(1)                                        pW3(1)
   |                  |                                                 |
|-| |------------ |                      |---------------------|-----------------|
|    |                                       |                                               pB2(-1)
|    |                                   pB1(-1)                                           |
|    |                                       |                                                |-----------------|-------------------------------|
|    |        |-------------------------|----------------------------------|                           |                                     |
|    |    pW2(1,-1)pB1             pW3(1,-1)pB2[W]                 pW2(1)               pW1(1)[D]                   pW1(1,1)pB2
|    |        |                                                                         |                                                                  |
|    |        |--------------------------|------------------------|          |------------|------------------------|                   |-------------|------------------------|
|    |        |                               |                             |                        pB(-1,1)pW3         pB(-1,-1)pW2               pB1(-1,1)pW1[D]              |
|    |    pB2(-1)                     pB2(-1,1)pW3[D]       pB2(-1,-1)pW1          |                             |                                                                   |
|    |         |                                                             |                     |---|--------|                 |----------|-----------|-------|                            |
|    |         |---------------|                                      pW2(1)[W]          |               |                               |             |        |                            |
|    |         |                  |                                                          pW2(1)[W]    pW2(1,1)pB3[W]     pW3(1)[W]     |     pW3(1,-1)pB3[W]       |
|    |     pW2(1)[W]     pW1(1,1)pB2                                                                                                            pW1(1,1)pB3                           |
|    |                             |                                                                                                                                  |                                     |
|    |                          pB3(-1,-1)pW2        |-------------------------|------------------------------------------------------------|                    |------------|
|    |                             |                    pB2(-1,1)pW3           pB1(-1)[B]                                                                                      |
|    |        |----------------|                         |                                                             |---------------------------------------------|------|
|    |    pW1(1)[W]     pW3(1)[W]          pW2(1)[W]                                               pB1(-1)                                             pB3(-1,-1)pW1
|    |                                                                                                                       |                                                      |
|    |             |---------------------------|----------------------------------------------------------|                                                   pW3(1)[W]
|    |        pW2(1,-1)pB1                  pW1(1,1)pB3[W]                                           pW(1)[W]
|    |             |
|    |        pB3(-1,-1)pW2
|    |             |
|    |             |-----------------|
|    |             |                    |
|    |        pW3(1)[W]     pW1(1)[W]
|    |
|    |----------------------------------|---------------------------|-------------------------------------|------------------------------------------------------------|
|                                          pB1(-1)                       pB1(-1,1)pW2                        pB3(-1,-1)pW2
|                                              |                                 |                                             |
|                   |---------------------|                                 |-----------------|
|                   |                         |                                 |                     |
|               pW3(1)                 pW2(1,1)pB3[W]      pW1(1,1)pB1          |
|                   |                                                            |                    |------------------------------------------------------------------|
|       |---------|------------------|                                    |-----------------------------------------|                                                  |
|       |                                  |                                    |                                                 |                                                  |
|      pB2(-1,1)pW3              pB3(-1,-1)pW2              pB3(-1)[D]                                   pB3(-1,-1)pW1                                  |
|       |                                   |                                                                                       |                                                |
|       |---------------------|        |------------------------|----------------------|---------------------| |---------------------|                     |
|       |                         |                                       |                          |                         |                            |                     |
|      pW2(1)[W]       pW2(1,1)pB3[W]               pW1(1,1)pB3           pW3(1)[W]          pW3(1,-1)pB2[W]       |                      |
|                                                                           |                                                                                  |                      |
|                                           |-------------------------|                                    |-------------------------------------|                      |
|                                           |                              |                                    |                                             |                     |
|                                       pB2(-1,1)pW3            pB1(-1)[B]                    pW3(1)                                  pW3(-1,-1)pB3[D]    |
|                                           |                                                                   |                                                                    |
|                                       pW2(1)[W]                    |-----------------------------|                          |---------------------------------|
|                                                                       pB2(-1,-1)[B]                 pB3(-1)[W]           pW1(1)                                   |
|                                                                                                                                            |                                       |
|                               |-------------------------------|-----------------------------|--------------------------|                                       |
|                          pB3(-1)                       pB1(-1)[B]                    pB2(-1,-1)pW1              pB1(-1,1)pW3[B]                        |
|                               |                                                                        |                                                                          |
|             |-------------|--------------|---------------------------|                    |-------------------|                                                 |
|             |                                  |                                |                pW3(1)              pW3(1,-1)pB1                                   |
|   pW1(1,1)pB2[W]             pW1(1)[W]                         |                      |                      |------------|--------|                     |
|                                                                                  |                     |-------------|                      |         |                      |
|             |--------------------------------------------------------|             pB1(-1)[B]     pB2(-1)[B]             |       pB2(-1)[B]           |
|             |                                                                                                                      pB3(-1,-1)pW3[B]                      |
|   pW3(1,-1)pB1                                                                                                                                                              |
|             |                                                             |---------------------------------------------------------------------------------------|
|             |------------------|                                 pW3(1)                                                                                            pW3(1,-1)pB1
|             |                      |                                    |                                                                                |--------------------|
|   pB3(-1)[B]            pB2(-1,-1)pW1                       |----------------------|--------------------|                     pB3(-1)               pB3(-1,-1)
|                                    |                            pB(-1)[B]            pB1(-1,-1)pW1[B]      pB2(-1,1)pW3            |
|                               pW3(1)[W]                                                                                  |                     pW1(1)
|                                                                            |-------------------------------------------|                          |
|                                                                       pW1(1)                                       pW1(1,1)pB1                 |--------------|
|                                       |------------------------------|                                                   |                    pB3(-1)[B]        pB3(-1,-1)pW1
|                                   pB1(-1)[B]                        pB2(-1)[B]          |-----------------------|                                             |
|                                                                                               pB2(-1)[B]             pB3(-1,-1)pW1[W]        |-------------|
 
Sure a lotta work...wish I could follow your diagram...

Can you not use photobucket or something similar?
 
it was perfect... and then "code" destroyed it! and i couldn't do it normal, because the multiple spaces would becoume one...

whats photobucket?
 
Huh?

matt000r000 said:
I seem to be the only one in life that acctualy (sic) likes math for fun...
Click to expand...


If you visit math sites on the Internet, then you're not paying attention.

 
matt000r000 said:
whats photobucket?
Click to expand...
Google the darn thing!

Not sure what you're doing, but your "flow diagram?" sure seems long,
since you're starting with a small 3by3 array, and only 3 empty spots.
 
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