gavinksong

That was really impressive. Good job!

Buffalo. I have no idea.

Actually, since the problem is asking what area the first can obtain for sure, suggesting that the answer should be independent of what the second player picks, my answer must actually be correct.

Not sure what you mean by flat. What am I missing?

BMAD, © confuses me. How do you divide an area into figures that don't share sides or vertices? For example, how can you divide a square into several polygons that don't share any points, sides, or vertices but fully cover the square?

Are you sure about that, bonanova?

Ack. I've made a mistake. :/ To simplify the problem, I assumed that the first player essentially chooses a line, and then the second player chooses a point. I forgot that the second player does not know which point the first player will choose last. Still, at least we know that 1/4 is a lower bound.

would player 2 pick a midpoint? My answer is independent of what the second player picks but...

But there can't be a smallest if you're including integers. Right?

Oh, it probably means "not on the border", doesn't it?

What is the "interior field"?

Close, but TSLF said that there can be only one bottle in each pan. I'm going to have to steal the correct answer from you.

but how do you 'know' these can't be further optimized? Well.... I guess I don't.

To start off the conversation... It's interesting that there are only 20 triangles, and they are not distinct. If my reasoning is correct there should be 24 distinct possible triangles: (4 * 3 * 2) / 3 = 8 triangles with three different numbers (e.g. 012) 4 * 3 = 12 triangles with two different numbers (e.g. 011) 4 triangles with all same numbers (e.g. 000) Those 20 triangles must have been chosen specifically for a reason. I guess two obvious constraints, as bonanova mentioned, is that there must be 20 triangles and 45 dots. However, I don't think they are the only constraints. Either way, the guarantee of a solution must not arise naturally but by design. So I say we should look carefully at the triangles. 012 and 021 are duplicated thrice each. 123 and 132 are duplicated twice each. 112, 113, 221, 223, 331, 332, 013, 031, 023, and 032 are missing.

A point, whether or not it is on a surface or on an edge or a vertex, is illuminated if a straight line can be drawn uninterrupted between it and a light source. Thus, for example, in the three-dimensional case, a cube could be fully illuminated by two lights. Like bonanova said, you may assume that the OP is asking for the fewest number of lights required to guarantee that any convex solid can be fully illuminated by some positioning of the light sources relative to the solid.

I am baffled. If any statement like that existed, why couldn't the UTM output "true" or "undecidable" instead? The "false" output is already wrong. The other two outputs can't be any more wrong, can they? They're just as good as the "false" answer. On the other hand, I can easily think of an answer that fits bonanova's criteria: "The reader of this proposition is human."

I'll have a go.

I made a mistake. 'eight distinct' should be 'six distinct'.