I know it might feel like a step down from that level of psychological cruelty to inflict on players, but I'm curious if 5D Hackenbush would have different combinatorial game arithmetic than regular Hackenbush. I tried playing some 5D tic-tac-toe, and I think we're missing a lot of theory on how these games actually work, because X wins in 8 moves every time, unless the rules allow O to stall the game indefinitely (not technically a tie). Intuitively there should be some way of balancing the game back in O's favor (besides stalling), but how? How can we know that certain restrictions would meaningfully balance the game? More specifically, from any partisan game with known Combinatorics, how, precisely, does the advantage shift with the introduction of time travel in the '5D' style? Can different 5D games be added in the same way as usual? Is it like taking the exponential of it, so you get more of a multiplication? '5D' moves depend on the game history, so you can only perform forwards analysis from the first move (equivalently, the game state includes information non-trivially related to all past game states); Combinatorial Game Theory is already equipped to handle game addition, but thinking about a number which equals the sum of all achievable deviations from all previous numbers, plus the new disadvantage to the "standard game" number, plus that "standard game" number, implies that we can easily get into infinite ordinal advantages. The fact that moving to a past board can alter you opportunities for time travel from the past was seen in Valefisk's game, here. We also saw, with 5D chess, that advantage sometimes doesn't follow from time travel, substantially because the win conditions don't follow the simple addition rules defining Combinatorial Game Theory, and this sublinearity allows the Birthday Property of surreal numbers to come through. Probably. Maybe I'll play some 5D Hackenbush on paper with myself or something, just to get a better grasp on this nonsense.
7 months ago | 4
Oliver Lugg
He's done it. The madman's actually done it.
7 months ago | [YT] | 223