If something doesn't seem tractable, try flipping between algebraic and geometric interpretations of a problem. Problems 1 and 3 fell to this approach.
Specific solutions (or suggestive handwaving):
Problem 1:
I thought of it like parity - going left to right, each unichromatic edge doesn't change the color, while each bichromatic edge does. So to have an overall change, we need either 1 bichromatic edge, or 3 (1 and 2 that cancel), or 5 (1 and 4 that cancel)...
Problem 2:
I couldn't understand this one at first. After checking Wikipedia, I think that Rn refers to the space that each point in the sequence lies within. An example of a finite sequence in R2 would then be (1,2),(3,1),(5,3)
Problem 3:
Consider the unit square. We need to draw one continuous line, going from left to right, that covers the entire vertical extent of the square. No matter how you do that, you need to cross the diagonal line from the bottom left to the top right.
Why? Because you need to touch the top and the bottom edges. You can't do that at the bottom-left or top-right corners, since then you'd touch the diagonal line. But then the point where you touch the top edge is entirely within the top triangle, and it cannot touch the bottom edge without entering the bottom triangle. Switching between triangles is identical to crossing the diagonal line.
As for why this isn't true if the set is open rather than closed: if we exclude the edges from our consideration of "does it intersect the diagonal", then it's fairly trivial to construct a curve that stays inside one triangle and has a codomain of (0,1). f(x)=x2 should work.
Strategies that I've found helpful:
If something doesn't seem tractable, try flipping between algebraic and geometric interpretations of a problem. Problems 1 and 3 fell to this approach.
Specific solutions (or suggestive handwaving):
Problem 1:
I thought of it like parity - going left to right, each unichromatic edge doesn't change the color, while each bichromatic edge does. So to have an overall change, we need either 1 bichromatic edge, or 3 (1 and 2 that cancel), or 5 (1 and 4 that cancel)...
Problem 2:
I couldn't understand this one at first. After checking Wikipedia, I think that Rn refers to the space that each point in the sequence lies within. An example of a finite sequence in R2 would then be (1,2),(3,1),(5,3)
Problem 3:
Consider the unit square. We need to draw one continuous line, going from left to right, that covers the entire vertical extent of the square. No matter how you do that, you need to cross the diagonal line from the bottom left to the top right.
Why? Because you need to touch the top and the bottom edges. You can't do that at the bottom-left or top-right corners, since then you'd touch the diagonal line. But then the point where you touch the top edge is entirely within the top triangle, and it cannot touch the bottom edge without entering the bottom triangle. Switching between triangles is identical to crossing the diagonal line.
As for why this isn't true if the set is open rather than closed: if we exclude the edges from our consideration of "does it intersect the diagonal", then it's fairly trivial to construct a curve that stays inside one triangle and has a codomain of (0,1). f(x)=x2 should work.