What is Smoothing?
During the bear in, when you bring your last checkers home into your inner board, you do best to spread them evenly among your six, five, and four points. Likewise, while bearing off, you will do best to use the numbers which miss (that is, do not allow you to remove a checker) to smooth by filling empty and thin points.
It is more important to fill gaps and shortages on higher points than lower. It is seldom right to use a number for smoothing when you can bear a man off with that number directly. It is never right when that number is an ace.
||Black to play 4-2.|
Although black can bring both checkers home with 9/5, 8/6, this play stacks the six and five points. It is better to use the 4 to fill the gap on the four point. Of all possible gaps, the 4-gap is the most harmful because it forces a player to move 6/2 or 5/1 with any 4, piling men on the two and one points from where they may be borne off later with large numbers, wasting pips.
After moving 8/4 with his 4, black can bring his last checker closer by moving 9/7, but does better to move 6/4 with his 2, further smoothing his distribution.
||Black to play 1-1.|
It is imperative for black to dump two checkers from his heavy six point onto his thin five point. Therefore he cannot afford to use three of his aces to bring his last checker home. 6/5(2), 4/3(2) and 6/5(2), 4/3, 9/8 are about equally good.
Note that it is not imperative to fatten the thin three point (a lower point), which can be resupplied from the six point later by the 3’s which miss.
||Black to play 6-3.|
No beginner would do so, but many experts might play 6/3, 4/off, leaving an even number of checkers with no gaps. Yet the beginner’s play, 6/off, 3/off, is best. The Sconyers program gives black’s winning chances as 55.9% after 6/off, 3/off; 53.5% after 6/off, 4/1; and only 52.3% after 6/3, 4/off.
A little thought reveals why. After 6/off, 3/off, the 3-gap costs a roll when black rolls two 3’s, a hazard which disappears as soon as black rolls four numbers higher than 3. But after 6/3, 4/off, black can lose a roll by rolling two 2’s, a hazard which disappears only when black rolls five numbers higher than 2. Meanwhile, black has wasted 2 pips.
6/off, 4/1 avoids the pip wastage and leaves only four men beyond the thin two point, so it is better than 6/3, 4/off. But 6/off, 4/1 risks losing a roll if black rolls two 3’s before three numbers higher than 3 or two 2’s before four numbers higher than 2, so it is not as good as 6/off, 3/off.
||Black to play 1-1.|
Bearing off a man from the one point is always right (as Paul Magriel was first to note), so black’s first three aces are automatic: 1/off(3). With his fourth ace, black should move 6/5 to fill the higher point, rather than 3/2 to fill the lower.
The Sconyers program confirms this. 1/off(3), 6/5 yields 58.5 winning chances against 57.7% for 1/off(3), 3/2.