Blue should attack, because White has a checker on the bar and only the 4 point is open. Blue leads by 23 pips, 93 to 116, but Blue’s last checker sits on White’s 3 point behind five White points from the 17 point to the 21 point, so the game is won by closing the board rather than by racing. Blue plays 13/9 11/8, bringing two more builders to bear on the 4 point. If White dances, 22 of Blue’s 36 rolls make the 4 point and close the board. White enters only with a 4, 11 rolls, and the outfield blots face just 5 of them, 4-3, 4-4 and 4-5. Making the bar point with 13/10 11/7 looks tidy, but the bar point blocks nothing an entering checker needs, and only 16 rolls then make the 4 point. It loses 0.119 equity, a BLUNDER.

Blue should run the last back checker, because White owns the cube and a closed board would leave that checker stuck. Blue leads by 28 pips, 92 to 120, White has a checker on the bar, and Blue’s last checker sits on White’s 2 point behind four White points from the 17 point to the 20 point. Blue plays 23/15, jumping the wall with exactly the 2 and the 6. The blots on the 10, 13 and 15 points face 6 shots, all rolls that must first enter with a 4, and Blue wins 79 percent of the games. Closing the board with 10/4 6/4 looks crushing, but a White checker that dances cannot move, so White’s points stay standing while Blue’s own board starts to break. Only 2-5 and 2-6 then free the trapped checker, and Blue wins just 69 percent. It loses 0.219 equity, a BLUNDER.

Same board, but Blue now owns the cube, and the cube makes the closed board the winning plan. Blue leads by 28 pips, 92 to 120, White has a checker on the bar, and Blue’s last checker waits on White’s 2 point behind four White points from the 17 point to the 20 point. Blue plays 10/4 6/4, closing the board, so no roll enters for White. White’s turn is a forced dance, Blue redoubles, and White has to pass. That cash is worth 1.000 equity, and the trapped checker never has to escape. Running with 23/15 wins more games when they are played to the end, 79 percent against 69, but it gives White 6 shots and 11 entering rolls first, and with the cube in Blue’s hands those extra wins are worth less than the sure cash. It loses 0.051 equity, an ERROR.

Blue is priming against White’s ace point anchor, and with the timing on Blue’s side the slot that extends the prime is worth the shots. Blue trails by 7 pips, 136 to 129, holds the 4 through 7 points, and has two checkers on White’s ace point behind White’s 19 through 22 points. Blue plays 10/3, slotting the 3 point. If the slot survives, 35 of Blue’s 36 rolls cover it for a five-point prime from the 3 point to the 7 point, after which the anchor moves only with 1-6. The slot faces 12 shots from the anchor, 14 with the 8 point blot and a hit sends back a checker that adds timing. Stacking with 10/7 10/6 spends two of Blue’s three spares on points already made, so the next awkward roll comes out of the prime. It loses 0.078 equity, an ERROR.

Blue should finish a full prime, because White’s lone checker on Blue’s ace point is trapped and White’s prime must crack first. Blue trails by 16 pips, 122 to 106, holds the 2 through 6 points, and has two back checkers behind White’s six points from the 17 point to the 22 point. Blue plays 24/23 14/12 8/7, anchoring on White’s 2 point and slotting the bar point with builders on the 12 and 13 points. The slot faces 11 shots, any 6 from the ace point, and if it survives 27 of Blue’s 36 rolls cover it for a six-point prime. White’s only spares are then the blot on the 16 point and a third checker on the 18 point, so White soon has to break. Building with 24/23 13/11 8/7 leaves the same slot but only 19 covering rolls. It loses 0.074 equity, an ERROR.

Blue should attack, because a second White checker on the bar freezes White while Blue’s last checker waits for a 6. Blue leads by 44 pips, 80 to 124, and holds the 2 through 6 points. White has one checker on the bar and a blot on Blue’s ace point, and Blue’s last checker sits on White’s 2 point behind White’s 18 through 22 points. Blue plays 7/6 3/1*, hitting a second checker and bringing the bar point checker home. White enters only with an ace, 11 rolls, and must enter both checkers before anything else moves, so White spends whole turns entering while Blue’s gammon chances reach 41 percent. Hitting with 3/2 3/1* wastes a spare on the 2 point and leaves the 7 point checker outside, a blot that 6s hit once White’s checkers enter. It loses 0.049 equity, an ERROR.

Blue should use the time White spends on the bar to build inner points. Blue leads by 48 pips, 89 to 137, holds the 2, 4, 5, 6 and 7 points, and has one checker left on White’s ace point. White has the timing, with five checkers in the outfield, so Blue must attack now. Blue plays 6/3(2) 4/1(2), making the 3 point and the ace point. That gives up the 6 point but leaves six blocking points, a five-point prime from the ace point to the 5 point plus the bar point with a spare. White enters only with a 6, 11 rolls, and Blue’s gammon chances stand at 38 percent. Clearing the bar point with 7/1(2) makes one new point instead of two, leaves five blocking points, and strands a blot on the 7 point that 4-3 hits. It loses 0.045 equity, an ERROR.

Blue is short of timing and should slot now, while a hit costs little, rather than wait and break. Blue trails by 33 pips, 125 to 92, holds the 4 through 8 points against a White blot on Blue’s ace point, and has two checkers on White’s ace point behind a full White prime from the 18 point to the 23 point, so every pip comes from the front. Blue plays 8/2 6/5, slotting the 2 and 3 points. The blots face 20 shots, but a hit checker enters only with an ace, and every dance is a turn in which Blue’s prime stands still. If White misses, 16 rolls make both points and shut the White checker in behind the 2 through 8 points. Making the 2 point with 8/2 3/2 is safe but leaves one spare, so the prime breaks sooner. It loses 0.044 equity, an ERROR.

Blue holds the timing here, so the right play extends the prime even at the cost of shots. Blue trails by 40 pips, 142 to 102, holds the 5 through 8 points against a White blot on Blue’s ace point, and has two checkers on White’s ace point behind six White points from the 17 point to the 22 point. White’s prime is under strain, since 16 of White’s 36 rolls force a break next turn. Blue plays 13/8 6/4, slotting the 4 point. The slot faces 14 shots, but if it survives 35 of Blue’s 36 rolls cover it for a five-point prime, and a hit only sends a checker back behind a prime about to crack. Slotting with 10/5 6/4 uses the wrong spare, since the 10 point checker is a builder for the 4 point and the midpoint checker is not. It loses 0.040 equity, an ERROR.

Blue is far ahead in the race but short of timing, so the hit should make a point rather than spend a spare, while White’s three midpoint checkers can wait. Blue leads by 52 pips, 77 to 129, holds the ace point and the 3 through 6 points, and White has a blot on Blue’s 2 point. Blue’s last checker sits on White’s 3 point behind White’s 17 through 21 points. Blue plays 6/2* 4/2, hitting and making the 2 point for five inner points. White enters only with a 6, 11 rolls, each hitting the 6 point blot. If White dances, 35 of Blue’s 36 rolls play without breaking a point. Hitting loose with 7/3 4/2* allows the same 11 hitting rolls but spends the 7 point checker, and 15 of Blue’s rolls would then break a point. It loses 0.037 equity, an ERROR.