Singer

If I destroy you, what business is it of yours?

@NoahZinsmeister I should clarify ceil(sqrt(x)) = R implies sqrt(x) <= R is always true because that's how the ceil function works *not* "only because R is integer" Besides, R already must be an integer

@NoahZinsmeister ceil(sqrt(x)) = R implies sqrt(x) <= R (only true because R is integer) implies x <= R^2

@VitalikButerin @matthuang @NoahZinsmeister The original statement relies on integer properties of ceil function. A proof that does not utilize those properties can't be right. For example twitter.com/hideandcleanse… relies on integer properties of ceil in many places. Doesn't prove it's right. But more likely.

@matthuang @NoahZinsmeister I don't think this proof is valid. If we change the exponent from 0.5 to 0.49, the "changing the output by 1 changes the required input by >=1" property still holds, but there are counterexamples at `i^(1/0.49) - epsilon` for integer i

@NoahZinsmeister @VitalikButerin So both ceil(sqrt(ceil(x))) > ceil(sqrt(x)) and ceil(sqrt(ceil(x))) < ceil(sqrt(x)) lead to contradictions Thus, when x > 0, ceil(sqrt(ceil(x))) == ceil(sqrt(x)) is always true

@NoahZinsmeister @VitalikButerin Other direction: assume ceil(sqrt(ceil(x))) < ceil(sqrt(x)) for some x This implies sqrt(ceil(x)) < sqrt(x) since if ceil(M) < ceil(N) then M < N This implies ceil(x) < x. Contradiction

Or, Dark Forest: Hope You Get Lucky And Find Big Planets 🙃 Hope players enjoy the final day of the game!

If Dark Forest games were movies, 0.4 would be Dark Forest: Deep Space Exploration twitter.com/hideandcleanse…

Intermission: Scoring differences in Dark Forest 0.3 and 0.4 With Dark Forest 0.4 nearing an end, now seems like a good time to share observations on what most top players have surely already realized out about scoring changes.