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MatPlus.Net  Forum Promenade In Quantum Chess, is there a 100 percent chance that a game will end? |
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| | | (1) Posted by Sarah Hornecker [Sunday, Mar 17, 2019 10:38]; edited by Sarah Hornecker [19-03-17] | In Quantum Chess, is there a 100 percent chance that a game will end?
This is more something for Hauke "Schroedinger's Plachutta" Reddmann.
http://www.youtube.com/watch?v=N3zFEpEPVF8
Quantum Chess was invented a while ago. A piece can have a quantum state which collapses if another piece with a 100 percent chance moves to the square where that piece could be. This sounds complicated, and it is.
If the first move is Sb1-c3, the knight moves from b1-c3 with a 100 percent chance.
If the first move is Sb1^c3, the knight has a 50 percent chance to be on b1 and a 50 percent chance to be on c3.
So since one can always make a quantum move, meaning there is a 50 percent chance to be on square x or square y - with chances being different also depending on what is played, the exact maths behind it are complicated, but for this let us make a simplified version of a 50 percent chance.
Let us simplify it even further. In practice White might be able to cast a net around Black to ensure the win. But let us assume the following.
So let us assume at each move there is a 50 percent chance that Black loses, as White captures the king (this is possible as win condition there). Given that the chance to win at each turn if 50 percent and there is no losing chance as Black can't reach White's king. So is White certain to win?
Simplified:
Let us assume there is no 3-fold repetition or 50 move rule, i.e. a game can really go on forever.
If the winning chance is 50 percent after each move: Will White eventually win, or is there an infinitesimal chance that the game goes on infinitely?
(Similar problem: https://www.youtube.com/watch?v=A5-Q2GdD5xw - is there an infinitesimal chance of extinction that becomes 100 percent in an infinite time period, or not?)
| | | (2) Posted by François Labelle [Monday, Mar 18, 2019 13:17] |
Is White certain to win? No.
Does White have a 100 percent chance of winning? Yes.
To a mathematician, "certain" and "100 percent chance" are not exactly the same thing, and your experiment is one example where they differ. This circumstance is called "almost surely" ( https://en.wikipedia.org/wiki/Almost_surely ). The Wikipedia page gives the example of throwing a dart randomly at a square and hitting exactly a diagonal (an example of "almost never"). It also gives your example of tossing a coin repeatedly.
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