4-16-10 Match Play/Answer

The best way to understand the right play here is to ask yourself what's the best thing that can happen to Red right now.  If you think about it, you are no longer trying to hold White in your inner board, you'd love for him to start leaving while you can still hit him and put some of his checkers on the bar while you bring this baby home. 

Sure, you can play this game "quietly" and hold White some more, and you'll win plenty of games and plenty of gammons.  That's a good move.  But "good" is the enemy of "best," so if you see that "normal" good play and just make it, you would be passing up a play that actually wins you more games and more gammons.

Make your board and "force" White to come out from your inner board while you've still got "friends" on the other side of the board to serve as a "welcoming committee."