Can do 1 kingdom card a 2 landscapes:
KC-KC (way of the rat, way of the horse to draw new KC + treasure, play new KC, repeat)
Donald says we should avoid using 2 Ways. But sounds nice.
And, for hat problem number 1:
Decide ahead of time to always guess odd or always even. Then everyone is guessing the same hat distribution. This produces a 50% chance of victory, because there's an equal number of even and odd distributions.
I think this is the right idea, but there isn’t the same number of odd and even distributions. So we would actually have to check what are the odds of having an odd number of white hats, total, and compare with the odds of having an even number. Then everybody agrees to guess the hat color that leads to a total number of white hats that is odd/even, depending on their previous result.
2. Similar to the previous problem, but there are 127 players. Furthermore, each player simultaneously guesses their hat color OR abstains. The players win if all guesses are correct and there was at least one guess, and otherwise, they all lose. What is the maximum probability of victory that they can achieve?
This one... I think this is the answer.
50%.
With the distribution being completely random, there's no way any single person can have more than a 50% chance to guess their own hat colour. No amount of planning a way a person might decide to guess or not guess will give them a better than 50% chance when they actually make their guess. Therefore, one person is designated to guess at random, while everyone else abstains. The person making a guess then has a 50% chance to guess correctly, making everyone win.
People were able to compute plenty of derivatives perfectly well far before any of the modern rigorous notions of limits, which was primarily due to Cauchy. They did so mostly with "handwaving."
The key issue here is more subtle, and it's more about being circular than being handwavy. The approximation e^h = 1+h when h is small is only valid for the number e. It's equivalent to the fact that the tangent line to the graph of e^x at x=0 is y=1+x, which can't be established without knowing that the derivative of e^x is e^x in the first place.
Still not quite sure I understand. Doesn't x^h approach 1+h as h approaches 0 for any positive x?
The "handwaving" is just a less formal way of talking about limits, right?
This isn't a great reason, but if you multiply 72 or 288 by 2.5 you get 180 mod 360. It's not a great reason because if you multiply (72 + 360) by 2.5 you get 360 again.
It's also not true: 288*2.5 = 720, while 216 * 2.5 = 240.
I'm somewhat confused about complex numbers right now. Or rather about how WolframAlpha deals with them, which is presumably the correct way.
If I ask for solutions to the equation x^5 = 1, it gives me all the complex numbers with angles n * 72°, n = 0,1,2,3,4. (Left.) If I ask for solutions to the equation x^{5/2} = 1, it gives me the ones with n = 0, 2, 3. (Right)
So what exactly is the property that the points with angles 72° and 288° don't have?
Tournament + Herald
Used Herald overpay to topdeck province and tournament. I bought an early province and got 3 prizes with it on my next shuffle.
Something tells me that would have taken multiple shuffles...
Fortune could basically be reworded to have a buy restriction if you want... "You can't gain or buy this unless 5 Gladiators have been gained this game". (Yes it's not the same in all cases, but it is in most cases). To me, having 5 cards sitting on top of the card is absolutely a buy restriction.
That's besides the point. If my opponent gains 5 Gladiators, I can buy Fortune for $8D8 without ever gaining any Gladiators.
Instead of comparing Ducat to Candlestick Maker, we should really be comparing it to Market Square.
Isn't that like comparing Gold to Smithy? How does that work?
