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**Puzzles and Challenges / Re: Best Asymptotic Point Scoring**

« **on:**November 23, 2018, 02:15:19 am »

Very nice! This is some good progress. It got me thinking about the Busy Beaver amount of Coin thread again, where we also achieved triple-up-arrow. The question there is, how many coins can be generated by an n-card deck which can't generate arbitrarily many coins? Coins and VP are mostly synonymous, and with f(n) coins you can pretty much get whatever f(n)-size deck you want. This suggests that if there is a solution for the coin problem achieving the growth rate, then there should be a solution to this VP problem which achieves a f

This suggests that there should be a quadruple-up-arrow solution to the VP problem, arising from the triple-up-arrow solution to the coin problem. However, we don't quite yet have this: the current triple-up-arrow solution for the coin problem involves gaining arbitrarily many cards costing <6, which isn't an issue for that problem because the number of actions (and so the amount of draw) is bounded. But for this problem, we could gain arbitrarily many estates for unbounded VP. The converse seems to be true in this case, though: bitwise's triple-up-arrow solution does give a double-up-arrow solution for the coin problem too.

That said, maybe we can still use this. Suppose we add the landmark wall to the kingdom, so that the estates aren't worth a point anymore. Then as long as we have another way of getting points (I think obelisk on an expensive card or palace both work), it may be possible to use that solution. I'll go see how well that works...

EDIT: the way we gained arbitrarily many cards in the coin solution was to inherit estate as catacombs, trash them with watchtower gaining hunting grounds (reduced cost to zero), trash the hunting grounds for 3x estate, repeat. However, we could gain arbitrarily many duchies in this way, so wall doesn't fix that. Still, maybe we can find another way to gain arbitrarily many cards in a way that excludes duchy and province (and at least one of gold, a potion-cost, or a debt-cost)

^{n}(2) growth rate, where f^{n}() is f() iterated n times and 2 is a constant.This suggests that there should be a quadruple-up-arrow solution to the VP problem, arising from the triple-up-arrow solution to the coin problem. However, we don't quite yet have this: the current triple-up-arrow solution for the coin problem involves gaining arbitrarily many cards costing <6, which isn't an issue for that problem because the number of actions (and so the amount of draw) is bounded. But for this problem, we could gain arbitrarily many estates for unbounded VP. The converse seems to be true in this case, though: bitwise's triple-up-arrow solution does give a double-up-arrow solution for the coin problem too.

That said, maybe we can still use this. Suppose we add the landmark wall to the kingdom, so that the estates aren't worth a point anymore. Then as long as we have another way of getting points (I think obelisk on an expensive card or palace both work), it may be possible to use that solution. I'll go see how well that works...

EDIT: the way we gained arbitrarily many cards in the coin solution was to inherit estate as catacombs, trash them with watchtower gaining hunting grounds (reduced cost to zero), trash the hunting grounds for 3x estate, repeat. However, we could gain arbitrarily many duchies in this way, so wall doesn't fix that. Still, maybe we can find another way to gain arbitrarily many cards in a way that excludes duchy and province (and at least one of gold, a potion-cost, or a debt-cost)