Life In 19x19 :: Easy if you know the secret

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Easy if you know the secret http://www.lifein19x19.com/viewtopic.php?f=15&t=14292	Page 1 of 1

Author:	Bill Spight [ Mon Jun 05, 2017 6:19 am ]
Post subject:	Easy if you know the secret
Here is a labyrinthine problem, adapted from Berlekamp and Wolfe's Mathematical Go. (;GM[1]CA[ISO8859-1]AP[GOWrite:2.3.46]SZ[19]FF[4]ST[2]PM[2]AW[pi][oi][ii][fa][qi][ra][ei][fi][gi][ci][di][cd][bd][nf][ed][dd][gd][fd][rl][sl][ql][qd][kl][lq][oq][dq][eg][ig][gg][hg][ok][pk][qk][ps][rd][ms][fs][cs][qn][pn][hb][rn][ib][fb][gb][rb][kn][ln][ob][nn][fn][ge][he][km][nm][de][be][om][mr][rj][kp][lr][or][oj][pr][mj][hj][ij][np][gj][cp][bp][dp][ap][mo][no][ko][ik][fc][jk][kk][lk][oo][mk][po][nk][br][oc][pc][qc][mc][jo][ic][dr][er][ef][hh][bf][cf][ah][ch][bh][eh][rh][sh]PB[ ]GN[ ]C[Problem C-19 in Berlekamp and Wolfe's "Mathematical Go", revised by Bill Spight. White to play and win.]PW[ ]AB[ba][mi][ji][sq][qq][rq][na][il][el][dl][ai][bi][mf][of][jf][id][hd][ld][od][nd][jl][pd][pq][fq][fg][jg][mg][kg][qg][rg][ks][ls][bb][jb][kb][eb][in][jn][nb][mb][bn][dn][en][gn][jm][fm][hm][ie][gm][je][jp][qr][fp][nj][gp][lj][ip][rp][jj][kj][dj][ej][cj][bj][me][pe][oe][re][qe][se][gk][dc][hk][cc][ec][qo][ro][bc][ac][so][ck][ek][bo][kc][jr][kr][do][hr][nc][co][ir][fr][io][gr][hc][ho][go][jh][mh][lh][nh][oh][ph][qh][cm][bm] ) The key, as usual in these problems, is to get the last play to win by 1 pt. One misstep and Black can get jigo (or win by area scoring). Humans can simplify the problem by playing the sente early and by noting the miai, which can be ignored. AlphaGo and other top go programs do not do that, IIUC, and I am not at all sure that they could solve this problem without taking a very long time. OTOH, for humans who know the secret, the solution is almost obvious. Schachus and I have started a discussion about this problem, which I have proposed to move here. I will also take Black against all comers.

Author:	Bill Spight [ Mon Jun 05, 2017 6:21 am ]
Post subject:	Re: Easy if you know the secret
Continuing discussion with Schachus. (;GM[1]CA[UTF-8]AP[GOWrite:2.3.46]SZ[19]FF[4]ST[2]PM[2]AW[pi][oi][ii][fa][qi][ra][ei][fi][gi][ci][di][cd][bd][nf][ed][dd][gd][fd][rl][sl][ql][qd][kl][lq][oq][dq][eg][ig][gg][hg][ok][pk][qk][ps][rd][ms][fs][cs][qn][pn][hb][ib][rn][fb][gb][rb][kn][ob][ln][nn][fn][ge][he][km][nm][de][be][om][mr][rj][lr][kp][or][pr][oj][mj][hj][ij][np][gj][cp][bp][dp][ap][mo][no][fc][ik][ko][jk][kk][lk][mk][oo][nk][po][br][oc][pc][qc][mc][jo][ic][dr][er][ef][hh][bf][cf][ah][ch][bh][eh][rh][sh]PB[ ]GN[ ]KM[0.00]C[Problem C-19 in Berlekamp and Wolfe's "Mathematical Go", revised by Bill Spight. White to play and win.]PW[ ]FG[259:]RU[Japanese]AB[ba][mi][ji][sq][qq][rq][na][il][el][dl][ai][bi][mf][of][jf][id][hd][ld][od][nd][jl][pd][pq][fq][fg][jg][mg][kg][qg][rg][ks][ls][bb][jb][kb][eb][in][jn][nb][mb][bn][dn][en][gn][jm][fm][ie][hm][je][gm][jp][qr][nj][fp][gp][lj][ip][jj][rp][kj][dj][ej][cj][bj][me][pe][oe][re][qe][se][dc][gk][cc][hk][ec][qo][ro][bc][ac][so][ck][ek][bo][jr][kc][kr][hr][do][ir][nc][co][fr][gr][hc][io][ho][go][jh][mh][lh][nh][oh][ph][qh][cm][bm] ( ;W[ea]C[1pt sente] ;B[da]C[* I reply.] ;W[qs] ;B[rs]C[* Ditto.] ;W[jc]C[* Your current thinking.] ;B[sd]C[* My proposed line follows.] ;W[sc] ;B[oa] ;W[pa] ;B[sn] ;W[jd] ;B[sm] ;W[kd] ;B[ke]C[* Your move, Mr. Phelps?] ) ( ;W[qs]C[1pt sente, but not as sente as the other one. Problably sente enough, though] ;B[rs]C[* I answer.] ) ( ;W[ja]C[2pt gote * Right, using swing counting.] ;B[ka] ;W[ia]C[* Unless you are using the hane as a ko threat or something, best to consider this sequence as a unit.] ) ( ;W[ni]C[2pt gote * Check. ] ) ( ;W[oa]C[1pt reverse sente, but the black sente is only 99% sente * It's 100% local sente. And it is global sente in the current whole board position.] ) ( ;W[sd]C[1pt reverse sente, but the black sente is even less sente then the other one * See next variation.] ) ( ;B[sd]C[* See continuation for threat.] ;B[sb] ;W[sc] ;B[rc]C[* Ko to kill.] ) ( ;W[sg]C[miai count is 2 pt gote(1pt miai value), but I believe it to be slighly better than the "pure" 2pt gote at o11, because it has got a slight "sente touch" to it(it would be 1pt sente if b answers and at some point, b will answer, because) * Eventiually it will carry a threat to gain 1.5 pts., as you show.] ;B[tt] ;W[sf] ;B[tt] ;W[rf] ;B[tt] ;W[qf] ;B[tt] ;W[pf] ;B[tt] ;W[pg] ;B[tt] ;W[og]C[this is now sente because the follow up would be 3pt gote] ) ( ;W[ep]C[also 2 pt gote. hard for me to compare to o11 and t13. the follow up at e5 is sente, because it again threatens a 2pt gote.. *** The follow up is still ambiguous between sente and gote. Before this play the local position is called double up star, written ^^.] ) ( ;W[ao]C[2pt gote miai count. But I believe it to be slightly better than that, because arriving at a7 would be sente. I would rate this a little better than t13, because "the corridor to the sente is shorter" How big is the eventual threat?] ;B[tt] ;W[an] ;B[tt] ;W[am] ;B[al] ) ( ;W[jc]C[2pt gote (surprise...).I think that arriving at l16 will be sente, also at k16, has a minimal sentish touch(maybe getting the cut at m17, is worth some epsilon, though I cant see it) * How big is the eventual threat?] ) ( ;W[sn]C[2pt gote with a slighly "reverse sente"-ish touch because the black move has a sente touch after a very long corridor. * How big is the eventual threat?] ) ( ;W[ad]C[similiar to t6, slighly reverse sente-ish touch, but better than t6, because the black move is a little more sente * How big is the eventual threat?] ) ( ;W[pp]C[2 points gote, plus it makes r1 from "just sente" to absolutely sente. There is also a seki at s1, but I dont think its relevant, because it only works after r4 is filled, but r1 will be taken long before that. It also might give an additional ko threat in the corner after the dame are filled, because the eye is gone, but there seem to be no kos. In total its definitely better than k19/o11, but I dont know how it compares to t13, for example. * Because of the dame at A, there is no seki. So this is just a simple gote.]L[qp] ) ( ;W[fj]C[2 pt gote. THe follow up in the end is gote, so this seems very similar to e4(maybe they are miai?) * Edit. They are similar, but not miai. Both are double up stars, ^^.] ) ( ;W[if]C[2pt gote with a reverse sente touch(better than a16 and t6, in my feeling) How big is the eventual threat?] ) ( ;W[kq]C[2 pt gote. Similar(even the same,imo) to e4 and f10 * Yes. This is a triple up, written ^^^ or ^3.] ) ( ;W[gs]C[Also this move is 1pt miai value as all the others, but it's hardest for me to count. Black g1 would create another 2 point gote at e1. Now if white plays it, then h1 is sente for black (roundabout 1 point), but black can also not take it, because afer white h1, it still leaves a move worth the same miai value(with j1 now really being 1pt sente for black). So no clue, whether withe g1 is better or worse than an ordinary 2pt gote(like o11), I would estimate it worse than the move, who have "a sente touch" for me, like k17, t13... * This is easier than it looks.See continuation and next variation.] ;B[hs]C[* A big threat in this position.] ;W[es]C[* As a rule we consider these three moves as a unit. It gains 1 pt. on average.] ) ( ;B[gs]C[* This move gains 1 pt. on average and threatens to gain 1 more point. The local position before this play is a down.] ) ) Edit: Corrected my error.

Author:	Schachus [ Mon Jun 05, 2017 7:48 am ]
Post subject:	Re: Easy if you know the secret
so according to my previous plan, it would have to be a5 now? And then continue to push in and once it is filled start with t13?

Author:	Bill Spight [ Mon Jun 05, 2017 8:02 am ]
Post subject:	Re: Easy if you know the secret
Schachus wrote: so according to my previous plan, it would have to be a5 now? And then continue to push in and once it is filled start with t13? If I read you right: (;GM[1]CA[UTF-8]AP[GOWrite:2.3.46]SZ[19]FF[4]ST[2]PM[2]AW[pi][oi][fa][ii][qi][ra][ei][fi][gi][ci][di][cd][bd][nf][ed][dd][gd][fd][rl][sl][ql][qd][kl][lq][oq][dq][eg][ig][gg][hg][ok][pk][qk][ps][rd][ms][fs][cs][qn][pn][hb][rn][ib][fb][gb][rb][kn][ln][ob][nn][fn][ge][he][km][nm][de][be][om][mr][rj][lr][kp][or][pr][oj][mj][hj][ij][np][gj][cp][bp][dp][ap][mo][no][fc][ik][ko][jk][kk][lk][mk][oo][nk][po][br][oc][pc][qc][mc][jo][ic][dr][er][ef][hh][bf][cf][ah][ch][bh][eh][rh][sh]PB[ ]GN[ ]KM[0.00]C[Problem C-19 in Berlekamp and Wolfe's "Mathematical Go", revised by Bill Spight. White to play and win.]PW[ ]FG[259:]RU[Japanese]AB[mi][ba][ji][sq][qq][rq][na][il][el][dl][ai][bi][mf][of][jf][id][hd][ld][od][nd][jl][pd][pq][fq][fg][jg][mg][kg][qg][rg][ks][ls][bb][jb][kb][eb][in][jn][nb][mb][bn][dn][en][gn][jm][fm][hm][ie][je][gm][jp][qr][nj][fp][gp][lj][ip][jj][rp][kj][dj][ej][cj][bj][me][pe][oe][re][qe][se][dc][gk][cc][hk][ec][qo][ro][bc][ac][so][ck][ek][bo][jr][kc][kr][hr][do][ir][nc][co][fr][gr][hc][io][ho][go][jh][mh][lh][nh][oh][ph][qh][cm][bm] ( ;W[ea]C[1pt sente] ;B[da]C[* I reply.] ;W[qs] ;B[rs]C[* Ditto.] ;W[jc]C[* Your current thinking.] ;B[sd]C[* My proposed line follows.] ;W[sc] ;B[oa] ;W[pa] ;B[sn] ;W[jd] ;B[sm] ;W[kd] ;B[ke]C[* Your move, Mr. Phelps?] ;W[ao] ;B[rm] ;W[an] ;B[qm] ;W[am] ;B[al]C[* Right so far?] ;W[sg] ;B[pm] ;W[sf] ;B[pl] ;W[rf] ;B[ol] ;W[qf] ;B[nl] ;W[ml]C[* I assume that you reply.] ;B[if] ;W[pf] ;B[hf] ;W[pg] ;B[gf] ;W[ff]C[* Ditto.] ;B[og]C[* What next?] ) ( ;W[qs]C[1pt sente, but not as sente as the other one. Problably sente enough, though] ;B[rs]C[* I answer.] ) ( ;W[ja]C[2pt gote * Right, using swing counting.] ;B[ka] ;W[ia]C[* Unless you are using the hane as a ko threat or something, best to consider this sequence as a unit.] ) ( ;W[ni]C[2pt gote * Check.] ) ( ;W[oa]C[1pt reverse sente, but the black sente is only 99% sente * It's 100% local sente. And it is global sente in the current whole board position.] ) ( ;W[sd]C[1pt reverse sente, but the black sente is even less sente then the other one * See next variation.] ) ( ;B[sd]C[* See continuation for threat.] ;B[sb] ;W[sc] ;B[rc]C[* Ko to kill.] ) ( ;W[sg]C[miai count is 2 pt gote(1pt miai value), but I believe it to be slighly better than the "pure" 2pt gote at o11, because it has got a slight "sente touch" to it(it would be 1pt sente if b answers and at some point, b will answer, because) * Eventiually it will carry a threat to gain 1.5 pts., as you show.] ;B[tt] ;W[sf] ;B[tt] ;W[rf] ;B[tt] ;W[qf] ;B[tt] ;W[pf] ;B[tt] ;W[pg] ;B[tt] ;W[og]C[this is now sente because the follow up would be 3pt gote] ) ( ;W[ep]C[also 2 pt gote. hard for me to compare to o11 and t13. the follow up at e5 is sente, because it again threatens a 2pt gote.. *** The follow up is still ambiguous between sente and gote. Before this play the local position is called double up star, written ^^.] ) ( ;W[ao]C[2pt gote miai count. But I believe it to be slightly better than that, because arriving at a7 would be sente. I would rate this a little better than t13, because "the corridor to the sente is shorter" How big is the eventual threat?] ;B[tt] ;W[an] ;B[tt] ;W[am] ;B[al] ) ( ;W[jc]C[2pt gote (surprise...).I think that arriving at l16 will be sente, also at k16, has a minimal sentish touch(maybe getting the cut at m17, is worth some epsilon, though I cant see it) * How big is the eventual threat?] ) ( ;W[sn]C[2pt gote with a slighly "reverse sente"-ish touch because the black move has a sente touch after a very long corridor. * How big is the eventual threat?] ) ( ;W[ad]C[similiar to t6, slighly reverse sente-ish touch, but better than t6, because the black move is a little more sente * How big is the eventual threat?] ) ( ;W[pp]C[2 points gote, plus it makes r1 from "just sente" to absolutely sente. There is also a seki at s1, but I dont think its relevant, because it only works after r4 is filled, but r1 will be taken long before that. It also might give an additional ko threat in the corner after the dame are filled, because the eye is gone, but there seem to be no kos. In total its definitely better than k19/o11, but I dont know how it compares to t13, for example. * Because of the dame at A, there is no seki. So this is just a simple gote.]L[qp] ) ( ;W[fj]C[2 pt gote. THe follow up in the end is gote, so this seems very similar to e4(maybe they are miai?) * Edit. They are similar, but not miai. Both of them are double up stars, ^^] ) ( ;W[if]C[2pt gote with a reverse sente touch(better than a16 and t6, in my feeling) How big is the eventual threat?] ) ( ;W[kq]C[2 pt gote. Similar(even the same,imo) to e4 and f10 * Yes. This is a triple up, written ^^^ or ^3.] ) ( ;W[gs]C[Also this move is 1pt miai value as all the others, but it's hardest for me to count. Black g1 would create another 2 point gote at e1. Now if white plays it, then h1 is sente for black (roundabout 1 point), but black can also not take it, because afer white h1, it still leaves a move worth the same miai value(with j1 now really being 1pt sente for black). So no clue, whether withe g1 is better or worse than an ordinary 2pt gote(like o11), I would estimate it worse than the move, who have "a sente touch" for me, like k17, t13... * This is easier than it looks.See continuation and next variation.] ;B[hs]C[* A big threat in this position.] ;W[es]C[* As a rule we consider these three moves as a unit. It gains 1 pt. on average.] ) ( ;B[gs]C[* This move gains 1 pt. on average and threatens to gain 1 more point. The local position before this play is a down.] ) )

Author:	Schachus [ Mon Jun 05, 2017 8:11 am ]
Post subject:	Re: Easy if you know the secret
yes you played like I thought(for white, for black I was surprised). But I havent yet considered, what to do next, so let me see..

Author:	Bill Spight [ Mon Jun 05, 2017 8:41 am ]
Post subject:	Re: Easy if you know the secret
Schachus wrote: yes you played like I thought(for white, for black I was surprised). But I havent yet considered, what to do next, so let me see.. Let me correct a mistake of mine. I got the colors reversed early this morning. E-04 and F-10 are not, repeat, not miai. They are both double up stars.

Author:	Bill Spight [ Wed Jun 07, 2017 1:23 am ]
Post subject:	Re: Easy if you know the secret
Schachus, if you would like, here is a hint. It's already too late. The position after is jigo by territory scoring.

Author:	Schachus [ Wed Jun 07, 2017 4:06 am ]
Post subject:	Re: Easy if you know the secret
Well, I'm not too surprised, but still I dont know what I did wrong/ how I would have had to play. I would assume that you played, what you consider good for black, so I could try and prevent that. The main idea there would be to take a reverse sente first(after taking the 2 sente moves). But it doesnt really seem more promising to me at all. Also at each point there is the option of blocking some of black pushes before one really has to, wich is kind of similar to taking reverse sente points. But I have no clue, how to decide, when exactly that would be the right play. PS: I do know "Nimbers"(https://en.wikipedia.org/wiki/Nimber) and how to add them, does something like this help here?

Author:	Bill Spight [ Wed Jun 07, 2017 5:14 am ]
Post subject:	Re: Easy if you know the secret
Schachus wrote: Well, I'm not too surprised, but still I dont know what I did wrong/ how I would have had to play. The secret is counter-intuitive. Quote: I would assume that you played, what you consider good for black, so I could try and prevent that. The main idea there would be to take a reverse sente first(after taking the 2 sente moves). But it doesnt really seem more promising to me at all. Right. The only reason Black played a reverse sente, , was that it was, in effect, the last play. And Black got to play that reverse sente because was a sente that White had to answer. But White had a sente at that point. What if the White sente had had a threat as large as, or larger than Black's? That sounds like a rhetorical question, but I'll let you draw the conclusion and consider how White might have played differently.

Author: Bill Spight [ Wed Jun 07, 2017 7:10 am ]

Post subject: Re: Easy if you know the secret

Miai

Click Here To Show Diagram Code: [go]$$Wc Miai $$ --------------------------------------- $$ | . X . X O O . . . . . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | . O O O O O O X X O O X . X X X O O X | $$ | . O . O . . O O X X X . X . X X X X X | $$ | . O O . O O X X X X . . X O X O O O O | $$ | . . . . O X O O O X X . X . X O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X . O O O . . | $$ | . X X X X . O O O X X X O X O , . O . | $$ | . . X . X . X X O O O O O O O O O . . | $$ | X . . X X . . . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X . . X X X O O . O O O O X X X | $$ | O O O O . X X . X X O . . O . , . X . | $$ | . . . O . X . . . . . O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . . O . . . . X X O . . O O X . | $$ ---------------------------------------[/go]

I claim, along with Berlekamp and Wolfe, that this position is strict miai. No matter who plays first, the result will be the same. Let me illustrate that with play.

White plays first

Click Here To Show Diagram Code: [go]$$Wc White plays first $$ --------------------------------------- $$ | . X . X O O . . . . . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | 4 O O O O O O X X O O X . X X X O O X | $$ | 6 O . O . . O O X X X . X . X X X X X | $$ | 8 O O . O O X X X X . . X O X O O O O | $$ | 0 . . . O X O O O X X . X . X O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X . O O O . . | $$ | . X X X X 7 O O O X X X O X O , . O . | $$ | . . X . X 9 X X O O O O O O O O O . . | $$ | X . . X X . . . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X . . X X X O O . O O O O X X X | $$ | O O O O . X X . X X O . . O . , . X . | $$ | . . . O . X . . 5 3 1 O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . . O 2 . . . X X O . . O O X . | $$ ---------------------------------------[/go]

Click Here To Show Diagram Code: [go]$$Wcm11 $$ --------------------------------------- $$ | . X . X O O . . a a . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | X O O O O O O X X O O X . X X X O O X | $$ | X O . O . . O O X X X . X . X X X X X | $$ | X O O . O O X X X X . . X O X O O O O | $$ | X 2 4 b O X O O O X X . X . X O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X c O O O . . | $$ | . X X X X O O O O X X X O X O , . O . | $$ | . . X . X O X X O O O O O O O O O . . | $$ | X . . X X d d . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X 3 e X X X O O . O O O O X X X | $$ | O O O O 1 X X . X X O . . O . f . X . | $$ | . . . O . X . g O O O O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . h O X . . . X X O . . O O X . | $$ ---------------------------------------[/go]

After

there are 8 simply gote left that gain 1 pt. for whoever takes it, which Black and White will share equally. Such a gote is called a star, written *.

Now let's look at the result if Black plays first.

Black plays first

Click Here To Show Diagram Code: [go]$$Bc Black plays first $$ --------------------------------------- $$ | . X . X O O . . . . . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | 3 O O O O O O X X O O X . X X X O O X | $$ | 5 O . O . . O O X X X . X . X X X X X | $$ | 7 O O . O O X X X X . . X O X O O O O | $$ | 9 . . . O X O O O X X . X . X O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X . O O O . . | $$ | . X X X X 8 O O O X X X O X O , . O . | $$ | . . X . X 0 X X O O O O O O O O O . . | $$ | X . . X X . . . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X . . X X X O O . O O O O X X X | $$ | O O O O . X X . X X O . . O . , . X . | $$ | . . . O . X . . 6 4 2 O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . . O 1 . . . X X O . . O O X . | $$ ---------------------------------------[/go]

Click Here To Show Diagram Code: [go]$$Bcm11 $$ --------------------------------------- $$ | . X . X O O . . . . . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | X O O O O O O X X O O X . X X X O O X | $$ | X O . O . . O O X X X . X . X X X X X | $$ | X O O . O O X X X X . . X O X O O O O | $$ | X 1 3 . O X O O O X X . X . X O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X . O O O . . | $$ | . X X X X O O O O X X X O X O , . O . | $$ | . . X . X O X X O O O O O O O O O . . | $$ | X . . X X . . . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X 4 . X X X O O . O O O O X X X | $$ | O O O O 2 X X . X X O . . O . , . X . | $$ | . . . O . X . . O O O O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . . O X . . . X X O . . O O X . | $$ ---------------------------------------[/go]

After 14 plays we reach the same position. In these sequences each play by the second player is necessary to preserve the miai.

(Although not necessarily unique.)

Except for the play on the bottom edge, each play is in a corridor that leads to a star. In the top left, Black has 6 plays before reaching the star, plus 1 play on the bottom side before doing so there. In the center there are 3 corridors, where White has 2 + 2 + 3 plays before reaching the stars. Each player has 7 plays before reaching the stars.

This problem is a fight for the last play that gains 1 pt., and in the top left corridor White has a local advantage in that fight. White can get the last local play unless Black makes 6 plays to reach the star, and then takes the star. The corridor is called a sextuple down star, written v6*. (Yes, it's a peculiar notation. Take your complaints elsewhere.

) The point is that before reaching the star White has 6 chances to take the last local play, which might be the last whole board play. By moving into the corridor Black takes away a potential last (whole board) move for White. You can see the similarity to filling outside liberties in a semeai.

Black does not necessarily take the star, because, while it is a potential last move for White on the whole board, it is also a potential last move for Black on the whole board. For why we say that the corridor is 6 v's plus *, see this page: http://senseis.xmp.net/?CorridorInfinitesimals , which includes a discussion of ^^*.

In this problem White has a v6* + v = v7*. Black has ^^* + ^^* + ^3 = ^7**. Up(^) and down(v) are opposites, so v7 + ^7 cancel out. * + ** = ***. Two stars are miai, as we know, so they cancel out, leaving *. (The similarity to Nim heaps is not accidental.) As we see, when the ups and downs are all played out, there are actually 5 stars left. But 4 of them are miai, leaving one star, as advertised.

(BTW, you don't have to do the strange arithmetic. There are four corridors plus the down on the bottom edge, so playing them out will leave five stars.

) There are three stars elsewhere on the board, so everything is miai.

Note that if there had been, in effect, an odd number of stars on the board after :w35:

would have been a mistake, as it would have allowed White to take the last star. Instead, Black should have taken the star, allowing White to play the sente.

Edit: To underscore the miai, the second player can make a mistake by not preserving it, as in the next two diagrams.

Black mistake

Click Here To Show Diagram Code: [go]$$Wc Black mistake $$ --------------------------------------- $$ | . X . X O O . . . . . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | . O O O O O O X X O O X . X X X O O X | $$ | . O . O . . O O X X X . X . X X X X X | $$ | . O O . O O X X X X . . X O X O O O O | $$ | . . . . O X O O O X X . X . X O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X . O O O . . | $$ | . X X X X 1 O O O X X X O X O , . O . | $$ | . . X . X 3 X X O O O O O O O O O . . | $$ | X . . X X 4 . . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X . . X X X O O . O O O O X X X | $$ | O O O O 5 X X . X X O . . O . , . X . | $$ | . . . O . X . . . . . O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . . O 2 . . . X X O . . O O X . | $$ ---------------------------------------[/go]

is a mistake. Maybe Black thought :w3:

was sente.

Click Here To Show Diagram Code: [go]$$Bcm6 $$ --------------------------------------- $$ | . X . X O O . . . . . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | 1 O O O O O O X X O O X . X X X O O X | $$ | 3 O . O . . O O X X X . X . X X X X X | $$ | 5 O O . O O X X X X . . X O X O O O O | $$ | 7 9 0 . O X O O O X X . X . X O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X . O O O . . | $$ | . X X X X O O O O X X X O X O , . O . | $$ | . . X . X O X X O O O O O O O O O . . | $$ | X . . X X X . . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X 2 . X X X O O . O O O O X X X | $$ | O O O O O X X . X X O . . O . , . X . | $$ | . . . O . X . . 8 6 4 O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . . O X . . . X X O . . O O X . | $$ ---------------------------------------[/go]

Black's mistake allows White to get the last play with :w15:

. There are six stars left on the board.

Author:	Schachus [ Wed Jun 07, 2017 11:07 am ]
Post subject:	Re: Easy if you know the secret
Ok. Let me try to understand why the ^^* from your example is indeed a ^^. By definition, we have ^^= {^,^^\|^,^^}, right?(if black plays in one of the ups, then we get ^, if he plays in the star, we get ^^, if white plays in one of the ups, we get ^*=^, if he plays the star, we get ^^). Now ^^ is alwas better for black then ^, no matter what is there otherwise, right? so this means, white would never play to ^^, but always to ^, so we have ^^={^,^^\|^}. On the other hand , the position we are given is {0\|^}. How do I know, this is the same? Now in a way similar to the argument given in the link, I could try to show that {^,^^\|^}+{v\|0}=0 (Showing that the inverted position together with a ^^* is miai). The left hand side should be {^+{v,0},^^+{v,0},,^\|^+{v\|0},^}. I see that ^+{v\|0}={{v\|0},0\|+{v\|0},^}={0\|+{v\|0},^}. Also I have +{v\|0}={{v\|0},v\|{v\|0},}. I cant make progress with this miai argument either, because I always fail to simplify {v\|0} inside, so it doesnt seem to make things easier. The way you obtain the miai in the link is by saying "B1 - W4 is correct play for both sides. The result is obviously miai, with a score of -2." I'm exactly not sure about the "correct play" statement, because I'm still new to which infinitesimals are really always better than others(in fact even when saying ^^ is better than ^ for black above(or similarly ^is better than 0 for black, how to prove that? I see that 0=={\|}, so now comparing with ^={0\|}, why is this always better for black? (ok, if this is the only play, I can see that leaving 0 is better than leaving *, but if it isnt the only play!?))), and thats why I fail to simplify the above. Can you eloborate on this? Edit: Some of my concerns are gone, once I realized the following: if I have a game A, that one player can win, no matter, whose turn it is, then the game X+A is always better for him, then the game X, for any finite game X(meaning, if he has a win a winning startegx for X, then also for X+A), which I can simply see by induction on the maximal number of moves X can take(this dereases with every move). This exmplains, for example, why ^ is always better than 0 for black. Edit2: This argument actually onlx needs that I have a winning startegy for A on the other players turn. This explains to me also the miai concept a little better. A game A in which the starting player loses always loses is miai, right? Cause I can apply the above to show that X is at least as good as X+A for either player, hence, they are just equal.

Author:	Bill Spight [ Wed Jun 07, 2017 12:24 pm ]
Post subject:	Re: Easy if you know the secret
Truth to say, I find it easier for go infinitesimals to deal with go positions. Like so: (;GM[1]CA[ISO8859-1]AP[GOWrite:2.3.46]SZ[9]FF[4]ST[2]PM[2]AW[ba][bb][cb][db][eb][gb][hb][ib][gc][gd][ge][gf][ih][cc][bd][ad][cd][dd][ed][cf][gf][gg][hg][ig]PB[ ]GN[ ]C[The board has a Black corridor with two White moves before the star. It also has a star and two downs. To prove: These go infinitesimals are equal. I.e., the second player gets the last play.]PW[ ]FG[259:]AB[ca][fa][fb][fc][ia][dc][fd][fe][gh][ae][ce][be][de][ee][bg][bf][hh][ii][cg][dg][eg][fg][fh][hi] ( ;B[ea] ;W[ff]C[* White gets the last play.] ) ( ;B[ec] ;W[ff] ;B[ga] ;W[ha]C[* White gets the last play.] ) ( ;B[ff] ;W[ec] ;B[ea] ;W[da] ;B[ga] ;W[ha]C[* White gets the last play.] ) ( ;W[ff] ;B[ga]C[* Black gets the last play.] ) ( ;W[ec] ;B[ga] ( ;W[ff] ;B[ha]C[* Black gets the last play.] ) ( ;W[ha] ;B[ea] ( ;W[ff] ;B[da] ;W[ef] ;B[df]C[Black gets the last play.] ) ( ;W[da] ;B[ff]C[ Black gets the last play.] ) ) ( ;W[ea] ;B[ha] ;W[ff] ;B[ef]C[* Black gets the last play.] ) ) ( ;W[ga] ;B[ea] ;W[ff] ;B[ef]C[*** Black gets the last play.] ) ) ¿Está claro?

Author:	Schachus [ Sun Jun 11, 2017 10:00 am ]
Post subject:	Re: Easy if you know the secret
Yes, I think that is pretty much clear to me:)

Author: Bill Spight [ Sun Jun 11, 2017 4:57 pm ]

Post subject: Re: Easy if you know the secret

OK, back to the original problem.

Black takes sente

Click Here To Show Diagram Code: [go]$$Bcm34 Black takes sente $$ --------------------------------------- $$ | . X . X O O . . . . . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | . O O O O O O X X O O X . X X X O O X | $$ | . O . O . . O O X X X . X . X X X X X | $$ | . O O . O 2 1 X X X . . X O X O O O O | $$ | . . . . O X O O O X X . X . 3 O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X . O O O . . | $$ | . X X X X . O O O X X X O X O , . O . | $$ | . . X . X . X X O O O O O O O O O . . | $$ | X . . X X . . . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X . . X X X O O . O O O O X X X | $$ | O O O O . X X . X X O . . O . , . X . | $$ | . . . O . X . . . . . O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . . O . . . . X X O . . O O X . | $$ ---------------------------------------[/go]

After 33 moves we reach a point where each player has a sente. :b34:

takes her sente before playing the reverse sente, which is the last play, as the rest of the board is strict miai.

But what if it were White's turn?

Black's Zwischenzug

Click Here To Show Diagram Code: [go]$$Wc Black's Zwischenzug $$ --------------------------------------- $$ | . X . X O O . . . . . . . X X O . O . | $$ | . X . . X O O O O X X . X X O . . O . | $$ | X X X X X O . X O O X . O X O O O . O | $$ | . O O O O O O X X O O X . X X X O O X | $$ | . O . O . . O O X X X . X . X X X X X | $$ | . O O . O 3 2 X X X . . X O X O O O O | $$ | . . . . O X O O O X X . X 4 1 O X X O | $$ | O O O . O . . O . X . X X X X X X O O | $$ | X X O O O O O . O X . . X . O O O . . | $$ | . X X X X . O O O X X X O X O , . O . | $$ | . . X . X . X X O O O O O O O O O . . | $$ | X . . X X . . . X X O . O X X X O O O | $$ | O X X . . X X X . X O . . O O X X X X | $$ | O X . X X O X . X X O O . O . O O O X | $$ | O X X X . . X X X O O . O O O O X X X | $$ | O O O O . X X . X X O . . O . , . X . | $$ | . . . O . X . . . . . O . . O X X X X | $$ | . O . O O X X X X X X O O . O O X . . | $$ | . . O . . O . . . . X X O . . O O X . | $$ ---------------------------------------[/go]

is sente, but before answering it, Black plays her sente, which carries a larger threat, as an intervening move (Zwischenzug in chess parlance). That way Black still gets the last play.

Back to the original problem. If White only had a sente with a larger threat than Black's (or at least as large) when we reach the point where each player has a sente, then White could play it as a Zwischenzug, get the last play, and win the game.

How to do that? Start in the corridor with the smaller sente threat at the end, saving the larger sente threat for later, when it is needed.

See the main line.

Author:	Schachus [ Mon Jun 12, 2017 10:49 am ]
Post subject:	Re: Easy if you know the secret
Nice:) Can you now tell me, why you think the fist non-sente play would be obvious!? I could have thought for hours, I would not have come up with the thinking that the larger threat at the end of the corridor is decisive, even if its much farther away

Author:	Bill Spight [ Thu Jun 15, 2017 8:02 pm ]
Post subject:	Re: Easy if you know the secret
Schachus wrote: Nice:) Can you now tell me, why you think the fist non-sente play would be obvious!? I could have thought for hours, I would not have come up with the thinking that the larger threat at the end of the corridor is decisive, even if its much farther away Yes, for those who don't know the secret, the first play is counterintuitive. I remember being surprised, myself. After all, given a choice of sente of the same size with no kos around, it is obviously right to choose the sente with the largest threat. (At least for simple threats. ) So with a choice of corridors (and other such situations) with distant sente, it also makes sense to choose the one with the largest threat. But if you have read Mathematical Go, you know the secret that this problem illustrates, and you know that you choose the one with the smallest threat. A quick glance reveals that the T-13 corridor has the smallest threat at the end. Quote: I don't always play in corridors with distant sente, but when I do, I start in the one with the smallest threat.

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