Pushpast

Can the triangles push past the circles?

Allow me to introduce the game I call Pushpast, in which triangles and circles aim to push past each other. The red triangles aim to push past the blue circles to the right, while the circles aim conversely to push past the triangles to the left. As a practical matter of playing the game, one can use colored marbles or X and O marks on paper or whatever.

This is a variation of the Bubble Monster game, which we covered last week, in that the basic rule of Pushpast is both the triangles and the circles act like the children in the Bubble Monster game, during their own turn, but like the bubble monsters on their opponent's turn. Thus, when it is the triangles' turn to move, they select one of the triangles amongst them to push past the circle currently facing it (there must indeed be such a circle), but then having done so, the circle is allowed at its option to reproduce itself any finite number of times behind, adding any number of additional circle duplicates there, just like the bubble monsters do when the children push past. The circles move similarly on their turn by choosing a circle to push past an opposing triangle to the left, which may then add additional duplicates of itself at its discretion behind to the right. The game is over when no more moves are possible, and whichever player made the last move wins. Infinitely long play is a draw.

Here is an example play of a few moves.

We see that the leading red triangle pushed through the opposing blue circle, which duplicated itself with three copies behind. Then, the leading blue circle pushed through its opposing triangle, which duplicated itself twice. And then, the penultimate triangle pushed past its opposing circle, which replaced itself with four copies.

A key difference between Pushpast and the Bubble Monster game is that the triangles and circles don't just want to get through each other, but rather aim to be the team that makes the very last move achieving that state. Can we expect the bubble-monster analysis to carry through? Is this a finite-play game? What can we say about winning strategies?

Interlude

We shall have a full analysis of the game and winning strategy by the end.