Tinies and Minies

  Difficulty: Intermediate   Keywords: EndGame, Theory

Tinies and Minies

What infinitesimals correspond to these corridors?

[Diagram]
Some tinies  



The top one corresponds to TINY-1, also written as “``+_1``”. The second one corresponds to TINY-2 (``+_2``).[1] The third corresponds to TINY-3 (``+_3``), and the fourth to TINY-4 (``+_4``).

Table of contents Table of diagrams
Some tinies
``uarr - +_1``
``uarr - +_1``, Black first
``uarr - +_1``, White first
``uarr - { 0 | +_1 }``
``uarr - { 0 | +_1 }``, Black first
``uarr - { 0 | +_1 }``, White first

The opposite of a TINY is a MINY.

Tinies are White's sente, of course. For getting the last play, they favor Black, since Black plays last regardless of who plays first. By convention, we say that they are positive. (Positions favoring White are negative.)

In general, in a fight for the last play, White should play the tinies first, starting with the one with the largest threat.

Tinies are called tinies because they are positive, but less than ↑ (UP).

Comparing a tiny with ↑

We can confirm `` +_1 < uarr `` with a difference game, showing ``uarr - +_1 = uarr + -_1 > 0``.

[Diagram]
``uarr - +_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]
``uarr - +_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]
``uarr - +_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 ``uarr > 2 * +_1``. In fact, UP is greater than any number of tinies. Furthermore, ``+_1`` is greater than any number of ``+_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 | +_1 }`` ? Let's compare it with ↑ via a difference game.

[Diagram]
``uarr - { 0 | +_1 }``  

White has two captured Black stones.

[Diagram]
``uarr - { 0 | +_1 }``, Black first  

Black gets tedomari and wins the chilled game.

[Diagram]
``uarr - { 0 | +_1 }``, White first  

White gets tedomari and wins the chilled game.


Since the player with sente wins the difference game, ``{ 0 | +_1 } " || " uarr``[2], and it has an atomic weight of 1.[3] (You may verify that it is greater than ``ast``.)

Similarly, ``{ 0 " || " 0 | +_1 }`` has atomic weight ``2``. Etc. We may write such games (using MathJax) as ``0^n +_x``, where ``n`` is the number of zeroes, and also the atomic weight.

See also


Notes and discussion

[1] Derivation of the chilled values:

The top corridor has the following game tree:

``{ 4 " || " 3 | ast }``

(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``, also written as “``+_G``” (using MathJax: code as “``+_G``”); similarly MINY-``G`` may be written as “``-_G``”.

The bottom (PJT: Ώsecond?) one has the game tree:

``{ 5 " || " 4 | 0 }``

which chills to

``{ 4 " || " 4 | 2 } = 4 + +_2``

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

(back to footnote mark in text: [1])


[2] Read “``"||"``” as “is confused with”; do not confuse confusion with the notation “``"||"``” for compressing game forms!

[3] This is not a proof. There are other games that are confused with ↑ that do not have an atomic weight of ``1``. Atomic weights are not necessarily numbers. –BillSpight


Tinies and Minies last edited by PJTraill on January 21, 2019 - 19:28
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