Singer

Judging the sincerity of the coordinates was Singer's joy

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

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@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

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@NoahZinsmeister ceil(sqrt(x)) = R

implies sqrt(x) <= R (only true because R is integer)

implies x <= R^2

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@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 https://t.co/6FHKpVThB1 relies on integer properties of ceil in many places. Doesn't prove it's right. But more likely.

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@NoahZinsmeister Proof by contradiction

First assume ceil(sqrt(ceil(x))) > ceil(sqrt(x)) for some x

Let ceil(xqrt(x)) = R (R for “right” as it’s on the right side of the inequality)

R is an integer since ceil is always an integer

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@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

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@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

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Or, Dark Forest: Hope You Get Lucky And Find Big Planets 🙃

Hope players enjoy the final day of the game!

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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.

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If Dark Forest games were movies, 0.4 would be Dark Forest: Deep Space Exploration

https://t.co/jNoksrasqu

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If Dark Forest games were movies, 0.3 would be Dark Forest: Silver Supply Chain.