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Tinies and Minies
    Keywords: EndGame

Tinies and Minies

What do these games chill to?

[Diagram]
Tinies


The top one has the following game tree:

                    { 4 || 3 | * }

(An unchilled * is a dame.) It chills to

                    { 3 || 3 | 2 } = 3 + { 0 || 0 | -1}

{ 0 || 0 | -1} is an infinitesimal called TINY-1. Its negative is MINY-1. A game with the canonical form { 0 || 0 | -G}, where G is a game greater than some positive number, is called TINY-G. You can also write it as a plus sign with a subscript, G, but SL does not have subscripts. You can write a miny with a minus sign plus subscript.

The bottom one has the game tree:

                    { 5 || 4 | 0 }

which chills to

                    { 4 || 4 | 2 } = 4 + TINY-2

As usual, we ignore the integer and just call these TINY-1 and TINY-2.

Tinies and minies are, of course, sente. Tinies are positive, as you may confirm. Tinies are called tinies because they are positive, but less than ^.

We can confirm this with the difference game, ^ - TINY-1 = ^ + MINY-1, which is greater than 0.

[Diagram]
^ - TINY-1


To make the value infinitesimal, let's make the mean value 0 by saying that Black has captured 1 White stone. We are mainly interested in the tedomari at temperature 1.

[Diagram]
^ - TINY-1 Black first

Black wins. The board score is the same, but Black has the captured stone. However, Black has made one more play than White, which costs one point in the chilled game, so the score is 0. Black wins because he gets tedomari.


[Diagram]
^ - TINY-1 White first

Black wins. The score is even, but Black got tedomari.

Since Black wins no matter who goes first, the difference is greater than 0, and ^ is larger.



You may verify that ^ > 2 x TINY-1. In fact, UP is greater than any number of tinies. Furthermore, TINY-1 is greater than any number of TINY-2's, etc.

Atomic Weight of Tinies and Minies

Tinies are less than ^, so their atomic weight is 0, as is the atomic weight of minies.

But what about { 0 | tiny-1 } ? Let's compare it with ^ via a difference game.

[Diagram]
^ - {0|tiny-1}

White has two captured Black stones.


[Diagram]
^ - {0|tiny-1} Black first

Black gets tedomari and wins the chilled game.


[Diagram]
^ - {0|tiny-1} White first

White gets tedomari and wins the chilled game.



Since the player with sente wins the difference game, { 0 | tiny-1 } is confused with ^, and it has an atomic weight of 1.[1] (You may verify that it is greater than *.)

Similarly, { 0 || 0 | tiny-1 } has atomic weight, 2. Etc. We may write such games as 0(n)tiny-x, where n is the number of zeroes, and also the atomic weight. (We should really use superscripts, but that's not easy on SL.)

See also Infinitesimals, Playing Infinitesimals, Other Infinitesimals, and More Infinitesimals.



This is a copy of the living page "Tinies and Minies" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.