BASIC endgame theory for beginner that actually is useful

Magicwand · #1

Click Here To Show Diagram Code: [go]$$Bcm1 Prisoner Count: B-0 W-0 $$ --------------------------------------- $$ | . . . O X W e O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . . d O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . . c O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . . b O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . a O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

value of a = 1 point gote = 1

value of b = 1 + 1/2 point gote = 1.5
logic: after 1 point is played they both share 1 point endgame so add 1/2
and same logic for below will yield..as below.

value of c = 1 + 1/2 + 1/4 point gote = 1.75

value of d = 1 + 1/2 + 1/4 + 1/8 point gote = 1.825

value of e = 2 point gote

so point e is bigger than a,b,c,d

this is my logic when i play tight endgame.
yes there are more complicated than above example..but i am teaching beginners about basic endgame.
i just came up with above lecture so i might have missed something.
let me know if my logic dont make sense.

Violence · #2

Click Here To Show Diagram Code: [go]$$Wcm1 Prisoner Count: B-0 W-0 $$ --------------------------------------- $$ | . . . O X W 1 O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . . 2 O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . 4 3 O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . 6 5 O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . 7 O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Black has 7 points.

Click Here To Show Diagram Code: [go]$$Wcm1 Prisoner Count: B-0 W-0 $$ --------------------------------------- $$ | . . . O X W 2 O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . 4 3 1 O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . 6 5 O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . 8 7 O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . 9 O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Black has... 7 points...

Magicwand · #3

Violence wrote:

Click Here To Show Diagram Code: [go]$$Wcm1 Prisoner Count: B-0 W-0 $$ --------------------------------------- $$ | . . . O X W 1 O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . . 2 O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . 4 3 O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . 6 5 O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . 7 O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Black has 7 points.

Click Here To Show Diagram Code: [go]$$Wcm1 Prisoner Count: B-0 W-0 $$ --------------------------------------- $$ | . . . O X W 2 O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . 4 3 1 O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . 6 5 O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . 8 7 O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . 9 O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Black has... 7 points...

yes they both have seven points. but my point is that little fraction can addup and can be 1 point difference at the end. but if you follow the value i stated above you will never lose 1 point at the end. that is my point.
during your game you can place value of each endgame and find optimal sequence by that number.

let me know if it is helpful.

Bill Spight · #4

Click Here To Show Diagram Code: [go]$$Bcm1 Prisoner Count: B-0 W-0 $$ --------------------------------------- $$ | . . . O X W e O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . . d O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . . c O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . . b O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . a O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Count:

a: 0.5
b: 1.25
c: 2.125
d: 3.0625
e: 1

Total count: 7.9375

Gain per play:

a: 0.5
b: 0.75
c: 0.875
d: 0.9375
e: 1

White to play can "round down" the count to a local score of 7. Black to play can "round up" to a local score of 8.

Orthodox play, White first.

Click Here To Show Diagram Code: [go]$$Wcm1 Black +7 $$ --------------------------------------- $$ | . . . O X O 1 O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . . 2 O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . 4 3 O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . 6 5 O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . 7 O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

The local score is 7 points for Black.
As advertised.

Click Here To Show Diagram Code: [go]$$Wcm1 White error $$ --------------------------------------- $$ | . . . O X O 2 O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . . 1 O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . . . O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . . . O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . . O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

gains 0.9375 points, to go to a count of 7. However, :b2:

gains 1 point, for a score of 8, which White cannot round down.

Click Here To Show Diagram Code: [go]$$Wcm1 White error, continued. B has 1 W prisoner. $$ --------------------------------------- $$ | . . . O X . X O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . 7 5 3 O O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . . 4 O O . . . . . . . . | $$ | . . . O X X X X X O . . . . . . . . . | $$ | . . . O X . . 6 O O . . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . 8 O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

The local score is 8 for Black.

Magicwand · #5

Bill Spight wrote:

The local score is 8 for Black.

thank you for your help.
but isnt it exactly same as my way of thinking if you multiply by 2?
my way is exactly how korean professionals teach.
i hope i didnt make any mistake.

Bill Spight · #6

Magicwand wrote:

but isnt it exactly same as my way of thinking if you multiply by 2?
my way is exactly how korean professionals teach.
i hope i didnt make any mistake.

Click Here To Show Diagram Code: [go]$$Bcm1 Prisoner Count: B-0 W-0 $$ --------------------------------------- $$ | . . . O X O . O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . . . O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . . . . O . . . . . . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . . O O O . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . . O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Can your way explain why White can get away with making the second largest play in this diagram?

Magicwand · #7

Bill Spight wrote:

Magicwand wrote:

but isnt it exactly same as my way of thinking if you multiply by 2?
my way is exactly how korean professionals teach.
i hope i didnt make any mistake.

Can your way explain why White can get away with making the second largest play in this diagram?

yes, i see where you are going with that question.

again thank you and appreciate your wisdom.

RobertJasiek · #8

The single most useful endgame advice for "beginners" is:

Play double sente before sente before reverse sente before gote!

Bill Spight · #9

IMO, the most useful thing to tell beginners about yose:

Watch out for shortage of liberties!

jts · **#10**

Magicwand wrote:

Bill Spight wrote:

Magicwand wrote:

but isnt it exactly same as my way of thinking if you multiply by 2?
my way is exactly how korean professionals teach.
i hope i didnt make any mistake.

Can your way explain why White can get away with making the second largest play in this diagram?

yes, i see where you are going with that question.

again thank you and appreciate your wisdom.

Wait, I don't understand. I thought in the second diagram Bill shows, White loses a point for playing the second-largest play.

xed_over · **#11**

Bill Spight wrote:

IMO, the most useful thing to tell beginners about yose:

Watch out for shortage of liberties!

That's the one that always bites me

I don't even know the difference between double, reverse, anti, whatever...

Bill Spight · **#12**

jts wrote:

Wait, I don't understand. I thought in the second diagram Bill shows, White loses a point for playing the second-largest play.

Click Here To Show Diagram Code: [go]$$Wcm1 $$ --------------------------------------- $$ | . . . O X O 1 O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . . 2 O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . . 4 3 O . . . . . . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . 6 5 O O O . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . 7 O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Final local score: B +8.

Click Here To Show Diagram Code: [go]$$Wcm1 $$ --------------------------------------- $$ | . . . O X O 2 O . . . . . . . . . . . | $$ | . . . O X X X O O O O O . . . . . . . | $$ | . . . O X X X X X X X O . . . . . . . | $$ | . . . O X . . . 4 1 O O . . . , . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . . 6 5 3 O . . . . . . . . | $$ | . . . O X X X X X X O . . . . . . . . | $$ | . . . O X . 8 7 O O O . . . . . . . . | $$ | . . . O X X X X O . . . . . . . . . . | $$ | . . . O X . 9 O O , . . . . . , . . . | $$ | . . . O X X X O . . . . . . . . . . . | $$ | . . . O O O O O . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Final local score: B +8.

RobertJasiek · **#13**

Seems like Bill and I are speaking of beginners of different levels:)

xed_over, time is overdue then to learn what those types of endgames are!

http://senseis.xmp.net/?DoubleSente
http://senseis.xmp.net/?Sente
http://senseis.xmp.net/?ReverseSente
http://senseis.xmp.net/?Gote

topazg · **#14**

jts wrote:

Wait, I don't understand. I thought in the second diagram Bill shows, White loses a point for playing the second-largest play.

Bill cheated and changed the diagram

. The B+8 lines are not from the same position as the B+7 lines!

Numsgil · **#15**

MW, I don't actually find this useful, just FYI. I know where the numbers come from from my own understanding of the endgame, but that's based on computer science ideas like decision tree and recursive summing from the leaves to the root. I'd be surprised if that's how you think, though.

So I'm challenging you to take your example and try again, but this time be really really aggressive about saying where the numbers are coming from.

For instance, you say 'a' has a value of 1. Where does that number come from? (I think of it as forming a branch in the decision tree, having W or B occupying 'a' as the decision, determining that it's gote for both sides, then averaging the branches of the tree together, calculating values from the leaves to the root. But maybe you know a simpler way to explain it or calculate it?)

Next you say 'b' is worth 1 + 1/2. Where does 1/2 come from? Especially to beginners, half points (and wilder binary fractions) don't make any sense, since at the end of the game the score (not counting komi) has to be an integer.

As it is, it's like you wrote "x^2 - 8x + 15" on the board, wrote below it "x = 3 or 5", and said "great, everyone knows how to factor now, right?" Rmmm, no, you need to explain a bit more.

topazg · **#16**

Numsgil wrote:

Next you say 'b' is worth 1 + 1/2. Where does 1/2 come from?

When computing a gote move, you have the value of that move, plus half the next move if gote (and full value if sente), plus half of half the third move (if gote) or half the third move (if sente) etc.

Numsgil · **#17**

topazg wrote:

Numsgil wrote:

Next you say 'b' is worth 1 + 1/2. Where does 1/2 come from?

When computing a gote move, you have the value of that move, plus half the next move if gote (and full value if sente), plus half of half the third move (if gote) or half the third move (if sente) etc.

"Half" isn't a great way of thinking about it. I prefer "averaging" my best move and his best move. But yeah, that's what I do for positions. But you can't just take it for granted that "beginners" understand that. Especially on non-trivial problems with a decision tree that's 3 or 4 moves deep.

Also, how do you actually calculate it in a game? I can do accurate end game calculations only by drawing out the decision tree and each position on graph paper, and even then it's a bit of a battle with the numbers and double checking the work. Doing it properly in your head for non trivial positions seems really hard. You're basically doing a depth first traversal of a tree in your head! So unless there's some clever mnemonic tricks I don't know, you're required to store in your head the entire tree, plus information about whether you've already visited a node or not, plus the values of the nodes all the way up to the root for the current node you're evaluating.

My guess is that there's a clever way to traverse the tree without requiring a great deal of storage for the current state of the traversal, but if there is I haven't figured it out.

And actually, thinking about it, I have a question I dont' actually know the answer for even on paper: how do you count a position where you can make a series of 1 point gote moves that end in a 30 point sente move for you? Unless it's very early in the game, your opponent isn't going to let you make that 30 point sente exchange. Which means that the "gote" move before it is actually a sente move your opponent will respond to. But now imagine decreasing that 30 point value to something like 5 points. Do you count or not count that 5 points? It's actual value depends on the largest move your opponent has available at the time you make the 1 point gote move that has the 5 point sente followup. So do you remember two different values for the count, with a threshold for when it switches from one to the other? What if the position is more complicated and there's many such "false sente" moves that may or may not make the gote move above them sente depending on the global situation?

Bill Spight · **#18**

Numsgil wrote:

Also, how do you actually calculate it in a game?

Memory and experience help. Also, you seldom have to be precise.

Quote:

I can do accurate end game calculations only by drawing out the decision tree and each position on graph paper, and even then it's a bit of a battle with the numbers and double checking the work. Doing it properly in your head for non trivial positions seems really hard. You're basically doing a depth first traversal of a tree in your head! So unless there's some clever mnemonic tricks I don't know, you're required to store in your head the entire tree,

Once you have evaluated a node, you can (usually) forget the tree below it. Also, if there are no kos below a node, its value lies between the results where Black plays first and White plays first. While that seldom gives you the answer, those numbers help keep you oriented, or let you make a quick estimate.

Quote:

And actually, thinking about it, I have a question I dont' actually know the answer for even on paper: how do you count a position where you can make a series of 1 point gote moves that end in a 30 point sente move for you?

There is a trick. You can evaluate it like a sente. Assume that you make a play and your opponent replies.

Quote:

But now imagine decreasing that 30 point value to something like 5 points. Do you count or not count that 5 points? It's actual value depends on the largest move your opponent has available at the time you make the 1 point gote move that has the 5 point sente followup.

You are making it too hard for yourself. The point of the theory is to provide heuristics that usually work, since it is impossible to read out the whole tree in time. (It can also guide you in reading the whole tree when that is possible, by giving you a line that is correct or nearly so.)

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So do you remember two different values for the count, with a threshold for when it switches from one to the other?

The count will be the same.

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What if the position is more complicated and there's many such "false sente" moves that may or may not make the gote move above them sente depending on the global situation?

That is a separate topic from evaluation. It is dealt with in some detail on SL: http://senseis.xmp.net/?GoInfinitesimals

jts · **#19**

topazg wrote:

jts wrote:

Wait, I don't understand. I thought in the second diagram Bill shows, White loses a point for playing the second-largest play.

Bill cheated and changed the diagram

. The B+8 lines are not from the same position as the B+7 lines!

Yes, that's what got me. I actually had to open two windows so that I could look at both sets of diagrams at once... I couldn't figure out what he was doing differently! :oops:

Thanks for the examples, Bill.

Bill Spight · **#20**

topazg wrote:

Bill cheated and changed the diagram

. The B+8 lines are not from the same position as the B+7 lines!

What I did:

I lengthened the C corridor, adding 15/16 of a point to Black. That made the count 8 7/8. White to play can "round down" to 8.

The orthodox play is to save the White stone, gaining 1 point to make the count 7 7/8, which Black can "round up" to 8. But playing in one of the long corridors would gain 15/16 of a point to make the count 7 15/16. That is enough so that Black can only "round up" to 8, as well.

BASIC endgame theory for beginner that actually is useful

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