$$W White to play. Which sente? $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O a . W X O . . . $$ | . O b , . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .
[go]$$W White to play. Which sente? $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O a . W X O . . . $$ | . O b , . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .[/go]
There are no kos on the board, nor will there be a future ko. The time has come for White to play sente at "a" or "b". Under these conditions, one play dominates the other. That is, it will sometimes be better, and will never be worse. Which sente should White play?
Enjoy!
The Adkins Principle: At some point, doesn't thinking have to go on?
— Winona Adkins
$$ Position at move 1 $$ --------------------------------------- $$ | . O X . . . . . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O O . O X O X . . X . . . . . . . . | $$ | . O . , . X O . . X X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |
[go]$$ Position at move 1 $$ --------------------------------------- $$ | . O X . . . . . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O O . O X O X . . X . . . . . . . . | $$ | . O . , . X O . . X X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]
If black goes first:
$$Bcm1 Black plays first $$ --------------------------------------- $$ | . O X . . . . . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O O 1 O X O X 2 . X . . . . . . . . | $$ | . O 3 , . X O . 4 X X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |
[go]$$Bcm1 Black plays first $$ --------------------------------------- $$ | . O X . . . . . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O O 1 O X O X 2 . X . . . . . . . . | $$ | . O 3 , . X O . 4 X X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]
Black gets an extra point. But if white goes first:
$$Wcm1 Position at move 1 $$ --------------------------------------- $$ | . O X . . . . . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O O 2 O X O X 1 . X . . . . . . . . | $$ | . O 3 4 . X O . 5 X X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |
[go]$$Wcm1 Position at move 1 $$ --------------------------------------- $$ | . O X . . . . . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O O 2 O X O X 1 . X . . . . . . . . | $$ | . O 3 4 . X O . 5 X X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]
It's a draw. So it seems like the correct play is black's knight approach, because it creates the option for an extra point.
$$W White to play. Which sente? $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O . a W X O . . . $$ | . O 1 , . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .
[go]$$W White to play. Which sente? $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O . a W X O . . . $$ | . O 1 , . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .[/go]
$$W $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O 1 a W X O . . . $$ | . O . , . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .
[go]$$W $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O 1 a W X O . . . $$ | . O . , . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .[/go]
If Black plays a in both diagrams, he is clearly better in the second.
Looked at your solution, emeraldemon. Difference games look complicated.
$$W White to play. Which sente? $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O a . W X O . . . $$ | . O b , . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .
[go]$$W White to play. Which sente? $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O a . W X O . . . $$ | . O b , . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .[/go]
ok i guess this is for kyu level players...but i will give you my answer. b is always better than a.
$$W black will have 3 point in that area $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O 3 4 W X O . . . $$ | . O 1 2 . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .
[go]$$W black will have 3 point in that area $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O 3 4 W X O . . . $$ | . O 1 2 . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .[/go]
$$W black has 3 points same as above. $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O 1 2 W X O . . . $$ | . O 3 4 . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .
[go]$$W black has 3 points same as above. $$ -------------------- $$ | . O X . X O O . . . $$ | . O X X X X O . . . $$ | . O 1 2 W X O . . . $$ | . O 3 4 . X O . O , $$ | . O X X X X X O . . $$ | . O O X O . O X O . $$ | . . X O X X . X O . $$ | . . O O O X . X O . $$ | . . . . . . . . . .[/go]
now second diag black can tenuki and only lose 1 point. first diag black have to answer because he will lose 3 point.
first diag will take away tenuki of black so it is always better.
"The more we think we know about
The greater the unknown" Words by neil peart, music by geddy lee and alex lifeson
Well, everybody got the right answer, but only Magicwand gets full credit.
Difference games give the solutions to problems like this. You mirror the board to make an even positon and then let each player play one of the options. If the result is better for one player, that player's move is dominant. (Kos defeat this procedure, which is the reason for the caveat.)
$$Bcm1 Black plays first $$ --------------------------------------- $$ | . O X . . X O . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O W 1 O X O X 4 3 X . . . . . . . . | $$ | . O 5 , . X O . 2 B X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |
[go]$$Bcm1 Black plays first $$ --------------------------------------- $$ | . O X . . X O . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O W 1 O X O X 4 3 X . . . . . . . . | $$ | . O 5 , . X O . 2 B X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]
emeraldemon got the right answer, but White's reply in the difference game is wrong. This is the correct sequence.
$$Wcm1 White plays first $$ --------------------------------------- $$ | . O X . . X O . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O W 4 O X O X 3 2 X . . . . . . . . | $$ | . O 5 6 . X O . 1 B X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |
[go]$$Wcm1 White plays first $$ --------------------------------------- $$ | . O X . . X O . . O X . . . . . . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . O W 4 O X O X 3 2 X . . . . . . . . | $$ | . O 5 6 . X O . 1 B X . . . . , . . . | $$ | . O X X X X O O O O X . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]
Black to play wins, while White to play can only get jigo. So Black's original play (marked) is dominant.
Last edited by Bill Spight on Fri Dec 10, 2010 1:56 pm, edited 4 times in total.
The Adkins Principle: At some point, doesn't thinking have to go on?
— Winona Adkins
Well, if one play is dominant, it should be easy to construct a board where it wins and the other play does not. I have sort of done that here. (I ask you to imagine parts of the board that are not in play, however. )
(;AP[GOWrite:2.2.21]ST[2]CA[ISO8859-1]FF[4]GM[1]SZ[19]AW[ia][kb][lb][la][sm][rm][rl][rk][bi][ei][ej][eh][ek][qj][pl][ok][sh][sj][ja][qm][qi][pj][pm][oj]AB[ql][jb][ib][hb][ha][qh][ri][rh][ph][sg][rg][bh][ah][ai][aj][ak][bk][ck][dk][ch][dh][pi]CR[bi][ci][di][bj][cj][dj][ia][ja][ka][sh][si][sj][sk][sl][rj][pk][qk][ql]PM[2]GN[ ]FG[259:]C[The unmarked regions of the board are settled, with territory even. White to play and win.]PB[ ]PW[ ] ( ;W[dj] ;B[cj] ;W[di] ;B[ci] ( ;W[si] ;B[rj] ;W[sk] ;B[ka] ;W[ja]C[Black score\: 3 points plus 2 captives\: 5 points Whte score\: 5 points plus 1 captve\: 6 points
White wins by 1 point.] ) ( ;W[ka] ;B[rj] ;W[sk] ;B[si]C[Black score\: 4 points plus 1 captive = 5 points Whte score\: 5 points
Jigo.] )
) ( ;W[di] ;B[ci]C[*** The rest of the play is miai.] ( ;W[dj] ;B[rj] ;W[cj] ;B[ka] ;W[bj] ;B[ja] ;W[sk] ;B[si]C[Black score\: 2 points plus 3 captives = 5 points White score\: 5 points
See next.