It's conceivable that the maximum number of plies (half-moves) you need is 218. The best known lower bound on needed number of plies is 185 for "Harry Goldsteen's furthest position" https://timkr.home.xs4all.nl/chess2/diary.htm
So perhaps the hardest-to-reach position manages to improve on that by an additional 33 plies.
Meanwhile, for the game of Go, as played on a standard 19x19 board, we have:
The maximum number of possible next moves is 361, which happens only in the initial empty position.
The 361 hardest-to-reach positions (assuming logical rules like [2]) are all the positions with 360 white stones and 1 empty point. these take 2*361 = 722 ply to reach, with black passing all their turns.
And these answers were found without checking all 208168199381979984699478633344862770286522453884530548425639456820927419612738015378525648451698519643907259916015628128546089888314427129715319317557736620397247064840935 legal positions :-) [1]
White cannot play on the last empty point as this would be suicide, which is prevented by the (assumed) superko rule forbidding repetition of the empty position.
A game of Go can be legally infinite due to recaptures. (player passes 360 times, then eats the entire board and it starts over).
It's also a natural infinite game due to Kos which can be the best move to play. This requires a set of extra rules to prevent. (Ko, superKo, triple kos, etc)
The black b2 pawn has no moves in the position with white to move.
If it were black's move, it would still have no moves since it is pinned by the white queen on e5. If it were not pinned it would have 4 moves, as it can also underpromote.
The phrasing "Legal but non-reachable " makes clear that they use some notion of legal that differs from the normal one of reachability.
It's hard to imagine what sensible notion that could be though.
Something like: each side having only one king, no pawns on first/last row, at most one king in check, etc ?!
That's ceil(log2(~4.8x10^44)) = 149 bits. But to make it efficiently decodable you'd use the ceil(log2(8726713169886222032347729969256422370854716254)) = 153 bit representation of the ChessPositionRanking project linked to in my other comment. The ChessCounter project does not provide an efficiently decodable code.
One is "legal" the other is "total problem space"
From a computing point of view, the total problem space is what matter because you still have to "compute" if it's legal or not before moving on. There isn't a straightforward way to only iterate over legal positions.
> Pascal contends that a rational person should adopt a lifestyle consistent with the existence of God and should strive to believe in God. The reasoning for this stance involves the potential outcomes: if God does not exist, the believer incurs only finite losses, potentially sacrificing certain pleasures and luxuries; if God does exist, the believer stands to gain immeasurably, as represented for example by an eternity in Heaven in Abrahamic tradition, while simultaneously avoiding boundless losses associated with an eternity in Hell.[2]
Should a rational person play in the lottery? If they lose they incur only small losses. If they win, they stand to gain immensely.
No, the expected pay-off is still negative.
Pascal tries to make the expected value positive by promising infinite rewards, but God or any supposed behaviours to please that God are not really needed in there.
There might be another God who provides the same rewards to those who refrain from the supposed behaviours.
Perhaps a rational person should simply believe in an infinitely rewarding afterlife for those who believe in one.
Yes, Pascal's Wager never made any logical sense whatsoever. In addition, it's nonsensical in a spiritual sense, a kind of dishonest bit of psychological judo.
Bear in mind that the goal of Pascal's wager was to convert people to Christianity. That the wager implicitly assumes the Abrahamic God is the only one that can possibly exist is intentional.
This kind of "dishonest psychological judo" is common to Christian apologists to this day. If you've ever been in a conversation with someone trying to convert you, it becomes obvious that they're just trying to find the one crazy logic trick or rhetorical flourish that will flip you.
This is a great introduction to that most ancient and elegant model of computation. But I was a little sad to read:
> One can represent natural numbers in a simple way, as follows:
> Definition (ordered tuples, natural numbers) The ordered tuple ⟨a0 , … an⟩ of λ-terms is defined as λx [x a0 … an]. One then defines the λ -term ┌ n ┐ corresponding to the natural number n as: ┌ 0 ┐ = I and, for every k, ┌ k + 1 ┐ = ⟨F, ┌ k ┐⟩
which is a rather convoluted and impractical way to represent natural numbers compared to the Church numerals Cn which iterate a given function n times on a giving argument.
There are cases where it makes sense to use tuples to represent natural numbers, namely to simplify the definition of predecessor, subtraction, and comparison. But that uses 1-tuples rather than 2-tuples [1].
Replacing Church numerals by tuple numerals allowed a googologically VERY large number (based on the so-called Bashicu Matrix System) to be expressed in only 350 bits rather than 404 bits. [2]
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