Beta (East) to play
This is a very difficult puzzle. If you think that Beta can win in two or three turns, you've missed something and should keep looking.
Beta can't quite win and needs to play for a draw.
Many players in Alpha's position would fail to see the correct play on move 3 (a move that loses four ships to destroy one is likely to be dismissed out of hand). So besides being the unique line where Beta can force a draw, Beta's plan above stands a good chance of winning if Alpha makes a mistake.
Alpha's other defenses are no better, such as
and Beta will win on their next turn.
Beta's other lines of attack end up losing. For example, if Beta tries to invade with their second r2 instead of their r3:
Or if Beta goes for the yellow fleet catastrophe:
Or if Beta starts the invasion with r3:
This position is from a ladder game on SDG where Beta was challenging Alpha for position #1. SDG user Draw5PlayAll helped me with the analysis.
At this point in the actual game, Beta played
and probably intended to continue with
but Alpha won quickly after
Note that the SDG website does not allow Homeworlds games to end in a draw for any reason, so even if Beta recognized that playing for a draw was best, the site could not have recorded such a result. The players would have simply alternated captures until one of them resigned.