Source: TSTST-2016-P5
Category: Existence and Construction

Problem:
In the coordinate plane are finitely many *walls*, which are disjoint line segments, none of which are parallel to either axis. A bulldozer starts at an arbitrary point and moves in the $+x$ direction. Every time it hits a wall, it turns at a right angle to its path, away from the wall, and continues moving. (Thus the bulldozer always moves parallel to the axes.) Prove that it is impossible for the bulldozer to hit both sides of every wall.

Solution:
We say a wall $v$ is *above* another wall $w$ if some point on $v$ is directly above a point on $w$. (This relation is antisymmetric, as walls do not intersect.) The critical claim is as follows:

\begin{claim*}
There exists a lowest wall, i.e., a wall not above any other walls.
\end{claim*}

\begin{proof}
Assume not. Then we get a directed cycle of some length $n \ge 3$: it is possible to construct a series of points $P_i$ and $Q_i$ for $i = 1, \dots, n$ (indices modulo $n$), such that the point $Q_i$ is directly above $P_{i+1}$ for each $i$, the segment $\overline{Q_i P_{i+1}}$ does not intersect any wall in its interior, and finally each segment $\overline{P_i Q_i}$ is contained inside a wall. This gives us a broken line on $2n$ vertices which is not self-intersecting. Now consider the leftmost vertical segment $\overline{Q_i P_{i+1}}$ and the rightmost vertical segment $\overline{Q_j P_{j+1}}$. The broken line gives a path from $P_{i+1}$ to $Q_j$, as well as a path from $P_{j+1}$ to $Q_i$. These clearly must intersect, which is a contradiction.
\end{proof}

Thus, if the bulldozer eventually moves upwards indefinitely, it will never hit the bottom side of the lowest wall. Similarly, if the bulldozer eventually moves downwards indefinitely, it will never hit the upper side of the highest wall.


Grading guidelines:
(Partial) 1. Introduced a correct invariant/partial order on walls (e.g. “v is above w if some point of v is vertically above some point of w”), and justified that this relation is antisymmetric because walls are disjoint. 2. Proved (or correctly reduced to proving) that among finitely many walls there exists a “lowest” wall (one not above any other), or dually a “highest” wall, and explained why such an extremal wall would obstruct hitting both of its sides. 3. Established a one-sided trapping statement: if the bulldozer ever starts moving vertically upward (resp. downward) without bound, then it can never hit the bottom side of a lowest wall (resp. the top side of a highest wall). (Almost) 1. Gave the full extremal-wall strategy and concluded the impossibility, but left a localized gap in proving existence of a lowest/highest wall (e.g. the “no directed cycle” argument is sketched but not rigorously justified). 2. Proved existence of a lowest (or highest) wall via the intended cycle/planarity contradiction, and correctly argued that one side of that wall is unhit, but did not fully justify the step that the bulldozer’s motion must eventually go upward indefinitely or downward indefinitely (missing a short case split). 3. Solution is almost complete, but made minor mistakes which are not negligible.