In some ways, this axiom is just a corollary of the first. It also doubles as a melancholy metaphor for the challenges of interpersonal relationships.

Axiom #2: If two cars cannot merge at first, they can never merge.

To borrow a term from physics, any two cars in Trainyard are either in phase or out of phase, and no amount of track can change this state. If they are in phase, they can merge with each other, and if they aren't, they can't.

This follows from the first axiom. In the example below, the two cars begin one square apart from each other. If we wanted them to merge, we could delay the first car (as shown), but since we can only delay it by multiples of two, we can never bridge that one-square difference.

If the yellow car begins one square farther away, then the cars are two squares apart, which means they can merge.

This pattern continues, with the cars either in or out of phase as you move them farther apart.

We can also change phase by shifting the cars in time rather than space. If two cars emerge from the same home square, the second one will be delayed by one square from the first.

Thus, any two cars that directly follow one another out of a home square will always be out of phase. If more than two cars emerge from the same home square, then all even-numbered cars will be in phase with every other even-numbered car, and likewise for the odd-numbered ones. In the example below, all of the red cars are in phase, and all of the yellow cars are in phase. The solution proves this, since each car merges with all others of the same color. (Notice that in the end, the final red and yellow cars are still one square out of phase.)

This property of phase has important ramifications for solving many puzzles, even when you are not trying to merge cars at all. The quickest way to check whether two cars are in or out of phase is to simply connect them directly. If they meet in the middle of a square (the first example below), they're in phase; if they meet on a line (second example), they're out of phase.

The last important thing to know about phase is that cars out of phase with each other will never collide if their tracks cross each other at right angles. With this in mind, we can see an easy solution to the second example puzzle above, and we know that we'll have to delay one car from the first example in order to avoid a collision in the center.

In the next section, we'll look at some of the most useful forms for Trainyard solutions and see what makes them tick.

Next: The Molehill