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Label the positions 1, 2, ..., 1000 so that each one is adjacent to the ones with the next higher and lower numbers, and 1000 is adjacent to 1.

Partial answer for bonus 2:

I think this improves on StephenTG's answer. The first player places Then the second player guesses This requires:

However, So the best we can do is To help look for optimal solutions, here's a spreadsheet showing all the ways to Based on this, one example is

Do you need to select all four pairs of cells before the workers start, then they pick paths that minimize the total number of sections used across all four pipes? Or can you select one pair, they pic …

Partial answer, just an upper bound. Assuming that the carpets can't overlap each other, themselves, or the walls, and that their sizes correspond to all distinct integer solutions of 1 <= x <= y <= 1 …

Assuming that then the best we can do is Let's demonstrate that this is achievable:

I think this covers all optimal solutions:

Solving the puzzle as written: Now if Jaap's piece is added:

These all involve simple deductions along the lines of "if this circle contains X, then this other circle must contain Y", and continuing until exactly one set of choices fails to lead to a contradict …

Let x be the number of lies in the set (0 to 120, inclusive). A computer scan determines that the values of x leading to a non-contradictory situation are so the desired answer is In general, for a …

I think S can be represented as something like