169
money game
Blue
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Problem 1
| 154 169 | White money game Blue |
Chuck Bower: 24/23, 10/5.
This problem reminds me of Kit's "Big Picture" article from last month. If White had both checkers on our 3-point, Kit would say 24/18. If both were on our 1-point he teaches we shouldn't be so bold. But here White has one checker on each of our 1- and 3-points. Hey, Kit, you didn't teach us what todo with this.
Often it's valuable to have a 3rd checker on a point. This is seldom true of the 24-point, however. I think we should start 24/23 and look for a 5.
Hitting could be right, but 8/3* gives up the valuable 8-point simply fora tempo gain. 6/1* gives up less, but starts a less desirable point. Usuallyfor the loose hit inside to be correct, the hitter needs a second motivation like a direct shot at another opponent (homeside) blot or the possibilityof making an advanced anchor by splitting under the gun. Here it merely gainsa tempo, which I don't think is enough.
It looks to me like we can narrow the list of alternatives down to two:the continuance to the opponent's bar (23/18) and safetying the builder on the 10-point (10/5). 23/18 is a confrontational play. It says to White "I dare you!" When it works Blue either gets the 18-point anchor or gains in an exchange of hits. With the two outside blots, one slip (from the bar) could give White a strong cube. This play may be "too big".
If Blue plays conservatively here with 10/5, White is unlikely to dosomething good on BOTH sides of the board. If White anchors up (or hits theflyshot) Blue will be working on an advanced anchor or flyshot of his own.If White builds his 5- or 7-point, the two back blots remain vulnerable,and Blue's new builder on his 5-point will be appreciated. I don't think Blueis sufficiently desperate that he needs to take so many chances with 23/18. 10/5 puts pressure on White's back checkers without much risk. I like the quiet play.
Steve Clark: 24/23, 6/1*.
The first play that comes to mind is 24/23, 10/5. This partly unstacksthe 24 point and gets a spare to the 5 point In most similar positionsthis play would be fairly automatic. Here, however, it leaves me alittle uncomfortable. My opponent seems to have a lot of good rolls andI am not particularly well positioned to improve my inner board. I haveno spares on the 8 point and the spare on the 11 is not very useful.
How about playing 24/18? I am already well behind in the race sogetting hit out there is not too bad. It certainly would be nice tohave a checker there if he hits a blot in my outer board. Nor will I beunhappy with my play if he rolls a number like 3-1 and makes his 5point. This might be the right play.
Other possibilities which come to mind are 24/23 and the hitting him onthe 3 or the ace point. Normally these plays have a feel of desperationabout them which does not seem well placed here. At the same time bothplays have substantial merit. If I hit him on the ace point, he can hitme back but otherwise has few effective rolls. I do misplace mychecker but I will have the next roll, whether I am hit or not, tobring my position together. Hitting on the 3 point leaves a morebalanced position but gives up the 8 point and leaves a lot of blots inhittable positions. This move has attractions but it is too loose and Iwill not do it.
Of the three candidates, I will go with the rather strange play of24/23, 6/1*. Perhaps I have been overly influenced by watching MalcolmDavis make these plays. There has to be some reason he has won allthose events.
Malcolm Davis: 24/23, 6/1*.
Have got away from an early hit on the one-point, but this looks like a case whereit might not be so bad. I am already behind in the race. I would prefer that my opponent notmake another valuable point. I am reluctant to play to the 18 point and let him roll. Thesplit to the 23 seems mandatory, and using the other half of the roll to stack on the 5 pointlooks too stiff.
Ray Fogerlund: 24/23, 6/1*.
I need to split, and I don't want White to have her full roll to develop his position. Also, I do not want to give up the 8 point to hit loose, so I hit from the 6 point and hope for the best.
Steve Hamilton: 24/23, 10/5.
Moving a checker from the 24 point is a must. It overrides all else. Moving10/5 leaves asomewhat inflexible position, but I like it better than 24/18. 24/18 justinvites attack and in my opinion will bring about even more serious problemsfor Blue.
Hal Heinrich: 24/18.
It is imperative that Blue move off the ace point now!There are two reasonable ways of doing this -- 24/18 and24/23, 10/5. 24/23, 10/5 is the quieter play -- stayingout of harm's way and converting a blot to a builder.Here, however, Blue is well positioned for the more aggressive split to the eighteen point. White cannot hitin the outfield and on the bar as well -- so Blue'soutfield blots are actually well placed. This combinedwith Blue's improved chances for equalizing make themajor split the right choice.
Ron Karr: 24/23, 10/5.
The third checker on the opponent's ace point criesout to be split, if at all possible. This gives more flexibility formaking an advanced anchor or hitting something. Playing 11/10 is nicetoo, making a blocking point and cleaning up some blots; but then the 5would pretty much have to be 6/1*, which doesn't really fit in, andalso doesn't deal with the problem of the back checkers.
After 24/23, hitting is also possible, but I think it's too loose andunnecessary. 24/18 also seems too committal; I don't need to inviteWhite to hit while I have 2 other blots around. 10/5 seems nice andflexible; it preserves the maximum racing equity, and I'll have lots ofoptions depending on what White does next.
George Klitsas: 24/18.
Blue just can't leave those three checkers on the twenty-four point, it's a particularly ugly formation in the early going, one that should be avoided almost at any cost. So, 24/23 is a must with the ace. Let's now think about the five. A serious candidate would be 13/8 if we had a spare on the midpoint, but not now. Either 10/5 or 11/6 bury checkers that should be given the chance to make some blocking points in the outfield (especially the bar point, the nine point and the ten point). We are left with 23/18, which seems to be the best five under the circumstances (the worst is probably 8/3* losing the eight point and leaving blots strewn around and the second worst 6/1* creating a permanent liability in a doubtful tempo gain). Therefore Blue must play 24/18, thematic actually under the name of �action play�, duplicating sixes and fours for White, especially 6-4, 6-3 and 6-1.
Laila Leonhardt: 24/23, 8/3*.
Hitting is important to prevent White from making another point in hisprime. Hitting on the ace point, will most likely render a checker out ofplay and slot a point that Blue has no intention of making. By hitting onthe 3 point, Blue gets an option to get back in the game when White fails tohit back, if White should hit, Blue has with the split from 24 to 23 madeflexibility for getting solid backgame.
Rob Maier: 24/23, 6/1*.
Blue needs to get his back checkers moving, and avoid being hit in hisoutfield. This play does both of these things, and prevents White fromworking on his own offense.
Steve Mellen: 24/23, 6/1*.
In positions with three men on the ace point splitting onechecker as soon as possible is practically automatic. Since there isnothing useful here that can be done with the 5 and 1 in combination I wouldsplit with the ace and look for a five. 8/3 is clearly too loose. 23/18also seems too loose because of the outfield blots. As between theremaining choices I prefer 6/1* over 10/5 because White seems to have a lotof good rolls here to improve his position, and it helps Blue a lot to keephim off balance and gain time to make an advanced anchor or even hit theblot on the 14 point. Getting a fourth checker sent back may not even be abad thing; it certainly is not as bad as letting White make the bar or fivepoint. I like 24/23, 6/1*.
David Montgomery: 24/23, 10/5.
Blue doesn't want three checkers on the 24-point. Blueis further motivated to split because White already hasan advanced anchor slotted.
6/1* with the five is plausible, but after slotting the1-point Blue should be looking to cover it, and Blue'scurrent covers are unattractive.
10/5 duplicates (quadruplicates?) White's 64 and leavesa tight but balanced position that will be easy to improve.
Snowie: 11/10, 6/1*.
The ten point is a very valuable asset. In addition, the loose hit on theace point prevents White from making a point, and he has a lot of point-makingnumbers if I leave him alone. The back checkers can wait.
Kit Woolsey: 24/23, 10/5.
I just can't stand the thought of having three checkers on my opponent'sace point. It is the kiss of death. A builder on the five point isnicely placed, so 24/23, 10/5 looks like a good combination.
Chirs Yep: 24/23, 10/5.
24/23 10/5 seems like the right mix between offense and defense. Breaking the midpoint or 8 point (an often underrated point) seems to give up too much. Hitting on the ace point also doesn't feel right. If White had an overwhelming number of threats or if Blue was able to combine the hit with splitting to an advanced anchor then the loose hit might be correct, but that is not the case here.
It is important for White to get off the 24 point -- three checkers stuck on the 24 point is ugly. Thus, I believe the top two moves are 24/18 and 24/23 10/5. I'll go with 24/23 10/5 since it leaves fewer blots, leaves White fewer hitting numbers (in particular, Blue is not vulnerable on both sides of the board), and brings the builder on the 10 point to bear on the 3 point as well as the 4 point.
Summary: Everbody agreed it was vital to get the third checker offthe 24 point. The choice between the loose hit and the purer play looks close,and the vote indicated this.
Play Votes Score24/23, 10/5 6 10024/23, 6/1* 5 9024/18 2 7024/23, 8/3* 1 6011/10, 6/1* 1 6013/8, 11/10 0 4011/5 0 4011/10 8/3* 0 40
Problem 2
| 150 146 | White money game Blue |
Chuck Bower: 24/20*, 20/16, 13/9(2).
I don't see any reason not to start by hitting--24/20* (and neither doesKit since ALL the candidates include this. :) 22//10 is reasonable andpossibly best if the cube were available. 13/9 turns a 4-prime into a 5-prime.Hard to argue with that. 6/2(2) unstacks a heavy point, but puts two checkersbehind White's anchor--usually a bad idea. All the previously mentioned playsexcept 22/10 leave Blue searching for the final 4. That's a strong argumentfor 22/10.
Still, building the 9-point looks awfully strong. Not only does it give White escaping problems, but it will make it easier for Blue to clear the 10-point later. White's position isn't so threatening that Blue needs to be concerned with the two loose back checkers. 20/16 is positive because White may be forced to leave shots immediately, and 20/16 leaves both White's infield and outfield covered (not so after 22/18). As a bonus, 20/16 gives White no good 6, often the right ploy when you've put one opponent checker on the bar.
Steve Clark: 24/20*, 20/16, 13/9(2).
The last time I suggested that the answer to a problem was obvious, itwas obvious, but the obvious answer wasn't my answer. So this time Iwill give the problem with an obvious answer a little more thought, orat least pretend to.
The answer that comes to mind is 24/20*, 20/16, 13/9(2). This creates abusted 5 point prime and makes a nice start on my getaway. 24/20*,22/18, 13/9(2) is a similar play but looks clearly inferior because ouropponent could then hit back with 6's. The preferred play only allowshim to hit back if he rolls 4 or less with both dies.
I do not like making an inner point because it leaves my men all spreadout.. I want to get away at this point in the and get all my mentogether in a nice flexible mass. Making 5 point out of 6 in a row is anice start in that direction. I have tried to take a clear look at thealternatives here, but nothing attracts so I will stick with my obviousplay, at least until another play becomes more obvious.
Malcolm Davis: 24/20*, 22/10.
Am pretty close to winning this game - can't find a better play.
Ray Fogerlund: 24/20*, 22/10.
Hit and run one man to safety. What could be better? Only one problem left to deal with... Don't over-think these things.
Steve Hamilton: 24/20*, 20/16, 13/9(2).
Making a third inner board point with 6/2(2) is attractive, so is runningfor home ahead in the race with 22/10. The downside of these plays isstructure. After either of these plays, how is Blue going to safely clearhis midpoint and his ten point? Clearing these points safely looks to betroublesome after these plays. For this reason I like 24/20*, 20/16, 13/9(2).I want to work with the partial six prime. This blocks most of White'sescaping numbers, but most importantly gives my back checkers a place toland as they come around.
Hal Heinrich: 21/13
Blue retains decent racing chances and some hittingchances by declining the opportunity to hit now. Hittingnow just gives White too many shots at too many blots --though being missed would be sweet! Blue is just notdesperate enough to hit this way. Of the non-hittingplays, 21/13 is best because it escapes a back man --which in turn enables Blue to hold the mid-point longer.
Ron Karr: 24/20*, 22/10.
After hitting, I have a big racing lead, so I want torace. It's tempting to make the 2 point, since it's an efficient useof the stack on the 6 point, and slightly increases White's chances offanning. But it leaves a somewhat stripped position that doesn'treally help me get my other checkers home, and I don't think itsignificantly increases my gammon chances.
White's position is not very good right now, but what are his bestchances of winning? Eventually, he'll have a holding game & I'll justhave to try to get home safely. For now, since he has an awkward2-point board and is behind in the race, he'll be happy to attack ifgiven the chance. I'd rather not have two blots around in the event heis able to attack successfully.
George Klitsas: 24/20*, 20/16, 13/9(2).
For some reason (abstract beauty, perhaps) I am hopelessly attracted by the thought of bringing two down ( 13/9(2) ) making my position (as Blue) more compact, fully aware of the fact that I lose a small amount of outfield control. Another four must be used to hit, of course and for the last one I prefer 20/16 over 22/18 for visible reasons. Another conceivable play is 24/20* 22/10 but it seems to me that Blue will have a hard time bringing home the resulting position (=clearing the ten point and the midpoint). Any play involving making the two point is overextending the position making even more difficult for Blue the task of putting it's pieces together and any play involving 7/3(2) is short-sighted to say the least - for similar reasons. My choice is 24/20* 20/16 13/9(2).
Laila Leonhardt: 24/20*, 20/16, 7/3(2).
Making the 2 point will create bad 2's for White when entering from the bar.Blue should not have a problem getting around the board, so better to make apoint than bringing a checker around.
Rob Maier: 24/20*, 20/16, 13/9(2).
Making the nine point clears the potentially troubling midpoint, and alsocreates a strong prime in case of accidents. For the last four, moving tothe 16 point gives a better chance of getting that checker to safety nextroll.
Steve Mellen: 24/20*, 20/16, 13/9(2).
After hitting Blue is comfortably up in the raceand his game plan is simply to bring the back checkers around safely and winthe race. Because White has no other blots exposed, there is no compellingreason to try and keep him on the bar, and thus making an inside point seemslike a bad idea when the pips could be used more productively to bring thecheckers around. The real question is whether to leave the midpoint now ornot. 22/10 with the last three 4's has the obvious advantage of playingwith only one blot. However, even if Blue manages to safety that blot, boththe 13 and 10 points will be difficult to clear, and in the meantime Whitewill be improving his currently weak board. On the other hand, 13/9(2) notonly breaks a point that will have to be vacated anyway, it is huge in termsof containing White's three back men. Even if White gets a little lucky incontaining Blue's two blots as they come around, the blockade will meanWhite has a lot of work to do before he can extricate those checkers andwin. The final 4, then, is between 22/18 and 20/16. 22/18 means moreaction, which is good in that it increases the chances of sending anotherchecker back. On the bad side, however, Blue's forward position is ratherinflexible after 13/9(2), and more action often means more awkward enteringnumbers down the road. In addition, a fourth checker back may be a benefitto White. On balance I think 20/16, trying to bring the checkers around assmoothly as possible, is the easiest way to win the game.
David Montgomery: 24/20*, 20/16, 13/9(2).
Blue is 40 pips up after playing. His primary goal is tocome home safely. A necessary step towards this goal isclearing the midpoint. This roll lets Blue do that whilemaking an offensive point. 20/16 beats 22/18 because it'sbetter to leave duped 4s and pathetic 3s than diversified6s and reasonable aces to hit.
Snowie: 24/20*, 22/10.
The value of the nine point is an illusion. I'm not priming anybody with Whitehaving five checkers on the midpoint. My plan is simply to bring them aroundand win, and my play brings one checker around safely.
Kit Woolsey: 24/20*, 20/16, 13/9(2).
The nine point looks big. That point will be staring White in the face forthe rest of the game as he tries to extricate his checkers. For the finalfour, 20/16 avoids unnecessary complications.
Chirs Yep: 24/20*, 22/10.
After hitting and moving the rest of the roll, Blue will be up 40 pips. It seems clear that Blue should want to disengage contact and race. Making the 9 point creates a nice block, but leaves a stripped position and strands the back men. If Blue's back men get stuck (or are hit and reenter badly from the bar), he will have to awkwardly dump checkers from his 6 point deep into his board. 6/2(2) is reasonable since it makes another inner board point and makes it more likely that White will enter on Blue's 4 point rather than deeper in Blue's board. On the other hand it dumps two useful spares behind White's anchor, which is often not right when one still has two back men to safety. 7/3(2) also dumps two spares behind White's anchor as well as making it more difficult for Blue to play safely in the future since the 7 point is a valuable landing spot for the checkers on the 10 point and midpoint.
I prefer 24/20* 22/10, the only move which runs one back checker to complete safety. Blue will then try to run the other checker to safety, after which he will play against White's holding game. When up by 40 pips, it's usually right to try to race.
Summary: The panel made it very clear that this is an outfield game, andthat it is not correct to make the a point behind White's anchor. The ninepoint is appealing, but as Snowie points out maybe that is an illusion.
Play Votes Score24/20*, 20/16, 13/9(2) 9 10024/20*, 22/10 5 8024/20*, 20/16, 7/3(2) 1 6024/20*, 22/18, 13/9(2) 0 5024/20*, 20/16, 22/14 0 5024/20*, 20/16, 6/2(2) 0 4024/20*, 22/18, 7/3(2) 0 4024/20*, 22/18, 6/2(2) 0 40
Problem 3
| 134 120 | White money game Blue |
Chuck Bower: 18/13, 18/15.
Unless you expect White to play like a dufus, Blue is probably going to need his board, so the only safe candidate (6/1, 4/1) looks like a reject. Both remaining plays leave 3 and 2-1 as hitting numbers, and 18/15 also gives White the bonus 5-4 roll. But this difference is small enough that it should only be used as a tiebreaker if the two plays look about equal otherwise (and they don't).
One apparent advantage to keeping the 18-point is that if the blot on the8-point is hit, the pseudo-anchor will provide some security. But unless Blue's entering number either hits back or lands upright on the 18-point, Blueis going to be left with the unenviable choice of breaking the home board or laying open more blots. For example, White rolls 3-2 playing 20/17*/15. Now 16 rolls fan, 4 rolls (3-1 and 3-2) hit back (but leave a homeboard shot), 6 rolls (4-1, 4-2, 5-1) force Blue to open a new blot, 6-2 bounces out leaving ablot, 2-1 enters and shifts in the homeboard. 2-2 gives Blue the distasteful decision of opening a new blot or burning an homeboard checker. 6-1 and 5-2 hop the prime safely. 1-1 is a very good roll, but not a crusher.
With the race lead (and thus the corresponding timing disadvantage), Blue is likely to be forced to pull up anchor first, and this looks like the time. If missed, Blue's distribution of builders has the added advantage of point-making rolls next time with 64, 53, and 42. If White were to miss after Blueplayed 13/8, 6/3, White will not be under huge pressure leaving a blot becauseof Blue's homeboard blot, and Blue's position will be sufficiently strippedthat more decisions like this one are likely to be coming in the near future.As with the alternate (13/8) shot, Blue will have some problems if he is hit.However, the spare left high in the homeboard (on the 6-point) when the 18-point is broken is in much better position to help with awkward entering rolls.
Finally, Blue's cube ownership argues for breaking the anchor now. Ifafter breaking, the expected future position is Blue holding the 13-point and White having to give up his own midpoint, an efficient cubeturn will be in theworks. The alternative of sitting back on the 18-point means Blue will probably have to hit a blot first, or run with doubles before shipping it over--both potential big market losers. Putting all factors together, I thinkit's time to break the 18-point.
Steve Clark: 13/8, 6/3.
Our nasty opponent has built his inner board in order so that we wouldreally rather not be hit. At the same time the safe play of 6/1, 4/1has a nasty drawback: our board will be a wreck and our opponent willno longer be afraid of being hit Unless we are lucky to remake ourboard, he will be able to stay back reasonably safely and win with anyhit on one of our blots. The risk of being hit now versus the risk ofbeing hit later is always difficult to judge but here I will take mychances now (and probably later too, el yucho). After all he only has a4 point board. I might be more frightened if he had made 5 points.
I normally am not attracted to 18/13, 18/15 in this type of position andhere the play does not look any better than usual. It does start thegetaway but there are substantial risks. First I am giving up mydefense if I am hit. I would rather retain his bar point also to givemy plot a place of safety in case it is hit. Furthermore I take awaymuch of the opponents flexibility by retaining the bar. He will havemany more awkward rolls if I stay back.
So that leave 13/8, 6/3. If I get hit, I will still be a favorite tocome in and I will still have a defensible position. I start the 3point which might be useful in the near future. This is the play I wouldmake.
Malcolm Davis: 13/8, 6/3.
Hate to volunteer the 3 shot, but don't like breaking the 18 point so soon as it seems pretty hard to get home, even if I get away with leaving the extra shots. It seems too earlyto start breaking my board. If I get hit on the 8 point, at least I still have sone defense.
Ray Fogerlund: 13/8, 6/3.
Not a happy roll, but, oh well. Slot and bring down a builder. If White hits, I still have the bar anchor and some flexibility. I can stay out for a few turns and still have a threatening position. I may lose this game, and maybe even one other, this year.
Steve Hamilton: 13/8, 6/3.
If I'm going to get hit I DON'T want it to be after moving 18/13,18/15. Thatmove plays right into White's hands. Chances are very slim that blot willmake it around safely. 13/8,6/3 forces White to break his anchor to hit, andalso keeps a lifeline on White's barpoint in case the blot is hit.
Hal Heinrich: 13/8, 6/3.
Playing safe and stacking the ace point is premature inthis position -- getting hit is survivable as White'sboard won't reach five points for a couple/few rolls. Inaddition, it makes White's position very easy to play.Breaking the eighteen point would be fine if Blue wereable to claim or reach a comfortable position afterbeing missed. As it is, Blue will be happy to have theeighteen point upon being hit. Which brings us to 13/8,6/3 -- this plays starts two good points, retains Blue'sassets, and is a bit safer than breaking the eighteen point.
Ron Karr: 13/8, 6/3.
Ugh. I'm ahead in the race, but it's going to be hardto bring this home. I can avoid leaving a shot now, but my board getsseriously damaged, and what are the chances of playing safely in thefuture? I could make a break for it by clearing the 18 point, but, Iget hit almost half the time, which is very bad, having given up theanchor, and if I'm missed there are still several hurdles to clear.
I think it's best to operate on the assumption that I'm very likely tobe hit sooner or later, and preserve my board as well as the securityof the anchor. If I do get hit, at least White can't prime me, andI'll have several rolls to enter before getting closed out. If I'mmissed, it's true I'm still likely to have future problems, but I havesome chance to make the 8 point, which would be nice.
George Klitsas: 13/8, 6/3.
First of all, 6/1 4/1 should be rejected. Blue is almost guaranteed to leave a shot anyway (actually, in all probability, more than one, at different times), so weakening his inner board at such an extend (not to mention the damage to his race chances) does not seem a wise course here. We are left with play a (18/13 18/15) and play b (13/8 6/3). Both are conceivable and should be close in equity terms. Still, I give the nod to play b, mainly because, if hit, Blue remains semi-alive with the eighteen point (White's bar point) kept, serving as a crucial link and a landing point for his back checker - White has neither the ammunition not the overall structure to attack it in that case. After play a, instead, if hit, Blue is almost dead as White should be able to slot his bar point while Blue is on the bar and make a five (or a six) point prime and win a number of easy gammons. Play b is better from other aspects as well, namely fewer immediate hits and the fact that White loses his anchor in order to do so. Theonly drawback I see in play b is that it does not make a move towards clearing things up, but such a tactic is often best taking into account all the parameters of a position. I vote for 13/8 6/3.
Laila Leonhardt: 18/13, 18/15
Blue is highly unlikely to be able to clear without leaving at least 1direct shot. Blue owns the cube, and will have a chance to get back in thegame even if hit. If White had less checkers in the outfield than Blue, thenBlue would be better off waiting for White to break first.
Rob Maier: 13/8, 6/3.
Breaking the bar point is pretty bad if we get hit, and nowhere near ginif we don't, so I have to reject it. 6/1, 4/1 is safe for the moment, butnext roll, or the roll after that could be just as bad, and we would beleaving shots with a weaker board. 13/8, 6/3 seems to be the rightcompromise. The bar point will be some measure of protection if we arehit, we maintain a four point board, and we have checkers to play safelynext roll if we aren't hit.
Steve Mellen: 18/13, 18/15.
Butchering the home board is terrible. Anyone who thinksthey can bring this home without leaving a shot is dreaming (even though ouropponents always seem to do it effortlessly). 13/8 6/3 certainly has itsbenefits; some outfield structure would definitely help Blue bring thishome, and if the blot is hit, the 18 point will give Blue reasonable chancesto recirculate the checker and stay in the game. What troubles me aboutthis play is that Blue's position is so stripped that there is little chanceto do anything useful with the checker on the 8 point. The most likelyscenario is that even if Blue is missed, he will end up having to bury thischecker next turn and be back in the same awkward situation. I am notthrilled with breaking the 18 point, because even if Blue gets away with itthe position is far from gin, but it seems like the 18 point will have to bebroken very soon and this may well be the best roll Blue will get to do it.I would play 18/13, 18/15.
David Montgomery: 13/8, 6/3.
I can't stand 6/1 4/1. Blue will never put his positionback together after this play.
18/13 18/15 is reasonable. Blue is ahead and would certainlylike to clear the 18-point. After Blue safeties the blot he canclose in on an efficient double.
But the clean up may not be so easy. 13/8 6/3 makes it easierto handle next turn, leaves three fewer shots this turn, andslots the valuable 8-point. Since Blue's racing lead isn'toverwhelming, this is my choice.
Snowie: 13/8, 6/3.
This is the only play which leaves me with a tenable position. There is now wayI will break up my board, and breaking White's bar point only makes his lifeeasier while I still have problems even if I don't get hit.
Kit Woolsey: 13/8, 6/3.
Busting up my board is sick, and will leave me hopelessly places for the restof the game. Running the back checkers would be fine if it were a claimwhen it worked, but I would still have plenty to do. My play leaves fewerand less appealing shots, and gives me some chance to improve next roll.White is going to have problems also, and I want to be ready when he does.
Chirs Yep: 13/8, 6/3.
6/1 4/1 is very ugly. With two checkers out of play, Blue will have a very difficult time rebuilding his board. It will often be too dangerous to hit White, should the opportunity arise. Blue's race lead means that he will likely have to give up the first shot; if White hits this shot, White's stronger board usually allows him to aggressively play for the closeout. I believe that Blue can do better with one of the other two plays.
18/13 18/15 gives up the back anchor in an attempt to swing the timing back to Blue. If Blue can then run the last back checker to safety, the timing will usually swing to Blue's favor -- White will then usually have to break either the 20-anchor or the midpoint before Blue has to break his midpoint. Unfortunately, the remaining back checker is usually in jeopardy for two rolls instead of one. If White hits, Blue will have no return shots and his back checker will be stranded in a lot of trouble. White will often be able to form a 5 or 6 prime, especially if Blue stays on the bar for any length of time. I believe that the best move is 13/8 6/3 which leaves White fewer shots over the near-future (including over the next roll) compared to 18/13 18/15. This is important, since White currently has a strong board. Also, in order to hit, White has to give up his anchor and give Blue return shots. This is significant since Blue also currently has a strong board.
Summary: The decision to pay now was unanimous, and the large majorityof the panel decided that running doesn't really work even when it does. Thislooks like a sensible conclusion.
Play Votes Score13/8, 6/3 12 10018/13, 18/15 3 706/1, 4/1 0 40
Problem 4
| 147 157 | White money game Blue |
Chuck Bower: 13/11, 8/7.
This positions has the markings of prime-vs-prime and Blue seems to havethe better timing. Splitting the back checkers doesn't look like the rightapproach, so we can concentrate our efforts on the homeside. Note that Blue'sownership of the cube gives him the freedom to improve his position without theconcern of a drop/take decision.
8/5 is safe, but Blue is trying to build the 7- and 8-points. The safeplay leaves all of Blue's builders for the outfield camped on one point. Thetime to build is now. Between 13/10 and 13/11, 8/7, I prefer the latter primarilly because White is likely to have more trouble playing a 6 next turn than a 5, so better to give him a 5 to hit than a 6. In addition, the second builder being farther back (11-point instead of 10-point) serves better if White trys to jump a back checker out.
Steve Clark: 24/23, 8/6.
Where is my rollout program when I really need it? 12 (or so) differentplays and I have no idea. All of the answers look ugly. The safetyplays of 24/23, 8/6 and 8/5 hold few attractions because I seem to havefew ways to win after either play. On second thought maybe 24/23, 8/6is not so bad. He won't particularly want to hit me unless he rollssome big double. 24/23 is not a bold step forward but any advancedanchor would be helpful here. This play does leave a lot of checkersstacked up on the 13 and the 6 points.
I do not like the bolder steps forward because all of them leavemultiple blots in positions where White would love to hit them, or pointon them, or have a good roll even if he missed them. The combination ofmy blot on the 8 point and a forward blot in his inner board is too uglyfor words.
For a while I was attracted to the dubious merits of 24/23, 13/11,duplicating 6's or something. This play does have some attractivefeatures. I makes a modest try for an advanced anchor and does unstackthe 13 point. This play is great assuming that he misses me and Ifollow up with a reasonable roll. That, however, is a lot ofassumptions. This play has a lot of things that could go wrong.
Another try for a prime which is less bold is 13/11, 8/7. With thisplay if I get lucky, I will have a better chance of making my point. Hewill have fewer shots at me. But I will still be stuck back on his acepoint.
I would like to think that I have offered something useful in mydiscussion of this problem but I am doubtful. Nor am I confident of myplay but I will try 24/23, 8/6.
Malcolm Davis: 24/23, 13/11.
Is probably a little loose - 4 blots. However, seems like I need to split, and playingto the 11 point with the 2 unstacks the 13 point. Not splitting could be right, in which case 13/10 would be my choice, but I will stick with 24/23, 13/11.
Ray Fogerlund: 13/11, 8/7.
My anchor gives me chances and limits some of White's options, so I am going to keep that. Otherwise, I would like to make a better prime, so I slot the bar and bring a builder down. I own the cube, so I just try to make incremental gains, (like making the bar point), while the game evolves into whatever it is going to become.
Steve Hamilton: 24/22, 24/23.
Those checkers on the 24 point can't afford to just sit there frozen like adeer caught in the headlights. Blue needs an advanced anchor, therefore heis going to have to split those back checkers. The question is how to splitthem. 24/21 puts a checker where Blue would most like to anchor, but Whitealso wants that point the most and has the ammo to make it almost impossiblefor Blue to get the four point anchor by route of 24/21. White may very wellpoint on Blue's head after any split, but I still think splitting is a musthere. If I'm going to get pointed on I'd at least like it to be on a pointthat White doesn't want as much. 24/22 ,24/23 also puts pressure on White'sblot without moving up to the point White wants most. I don't really likemoving 24/23 but I think it's better than anything else. I want to leavethat blot on Blue's 8 point. If White doesn't hit it and Blue can cover,Blue may get to do some priming of his own.
Hal Heinrich: 13/10
In a short play vs. play prop, I tried 13/10 against 24/23 8/6. What surprised me was how well Blue did witheither play. Blue's stronger home board, timing advantage, and ownership of the cube put a huge dent inthe advantage White's prime confers. 13/10 tries tocounter-prime White -- and is hugely successful when itworks! And when White hits the slotted eight point, Bluehas chances in a blot-hitting contest -- all without undue gammon risk.
Ron Karr: 13/11, 8/7.
Interesting problem. White has the better prime,which he's threatening to extend, and a more advanced anchor. I have abetter board, and I also have good timing because of the 3 spares onthe midpoint. So there are various possible strategies here.
I could try to interfere with White's prime building by splitting theback checkers. Normally splitting and slotting isn't a good idea, sothe only play that avoids that is 24/23 8/6. The problem there is thatit doesn't threaten White's blot, and also makes it difficult to everimprove my offense. Since I have the better board, I'd prefer to bemore aggressive with 24/22 and some ace. I'm already behind in therace, and White's numbers to hit on both sides of the board aresomewhat duplicated.
Another approach is to stay on the anchor and attempt to build my primeby leaving the 8 or 7 point slotted. This way, White can't attack as adiversion if he misses the blot, and if he does hit, I'll have goodreturn shot possibilities. 13/11, 8/7 looks best, because most ofWhite's 5s also make good points.
Since I own the cube, I'll prefer to stay on the anchor. I may be ableto win the priming battle; if I don't, I'll still have the ace pointgame, or I may be able to make an advanced anchor anyway.
George Klitsas: 24/21.
Taking into account one's assets - liabilities as well, which often are paired to relevant ones of his opponent but of opposite sign, is an art in itself in the game of backgammon. Here, for example, Blue has the upper hand in inner-board strength and the lower hand (if this can be said !) in a potential priming battle. Consequently, if other considerations are not strong enough to tip the balance in favor of a mutual priming game, Blue should make a play that exploits his asset (inner-board strength). The play that deploys clearly this theme is, undoubtedly, 24/21. This is not a play without a certain amount of (grave, sometimes) danger, especially if Blue is pointed on by the White spares alone - also, very strong for White are some double-hitting numbers like 6-2, 6-4 and 6-5 [but in the latter case Blue could turn things up with a four or 2-2 or 3-3 from the bar]. Still, I think Blue has to be involved in this blot-hitting contest, to undertake such a danger trying for the advanced anchor, backed up by his better inner board which, if he succeeds to hit a White checker, could make him the favorite. Plays that keep the anchor on the twenty-four point and try for a mutual priming game (the best of which seems to be 13/11 8/7) don't look to me particularly promising. Blue has to survive the immediate hit, then cover the slot and then win the priming battle from a worse still position. In-between plays don't seem promising, either. The best play from this group seems to be 24/22 24/23, involving some duplication in Blue's favor, but failing to reach the edge of White's forming prime - if White rolls a four, making a five-point prime, Blue will remain trapped. My play is 24/21.
Laila Leonhardt: 13/11, 8/7
Blue's timing is good, and does not need to move to the edge of the prime ifhe can manage to keep White primed. Getting pointed on or 2 checkers hitwill be devastating, but just having the slot bar point blot hit, will allowBlue to return hit from the bar and keep the pressure on White.
Rob Maier: 24/23, 8/6.
I've been learning the hard (expensive) way that you can't always make thebig play. This seems to be one of those times. I would like to play24/21, fighting to get to the edge of the prime while I still have abetter board and the prime is not yet unjumpable. The problem is I can behurt in too many different ways, and end up gammoned. This wimpy play maysacrifice some wins, but it also saves a great many gammons, and I thinkdiscretion is called for here. We still have opportunities to fight forthe advanced anchor or hit a fly shot, but fewer risks of being pointedon, not to mention no third blot to be sent back. Others of the groupmentioned that 8/5 was not included. The problem there is that White'sanchor isn't going anywhere for the next few rolls, so the distribution isnot nearly as useful as splitting the back checkers.
Steve Mellen: 24/22, 8/7.
Probably the most difficult problem. Blue clearly has theworse end of the priming battle because White's prime is further back andhis anchor is more advanced. I don't think Blue can afford to just sit onthe ace point and hope to counter-prime White's checkers. He will have toroll almost perfectly just to get a prime as strong as White's is now.Splitting to the 23 point does not seem to accomplish much; if this is thebiggest risk Blue is willing to take, he might as well hold the anchor andtry to avoid getting gammoned. On the other hand, splitting to the 21 pointmakes every number play well for White except aces. Splitting to the 22point in combination with 8/7 cuts down on White's good rolls, becausealmost all of White's 5's play well on the offensive side of the board.24/22 prevents White from leaving the 16 point slotted next roll andthreatens to anchor on the 22 point, which would be huge both in terms ofsaving gammons and retaining winning chances for the duration of the game.24/22, 8/7 is my play.
David Montgomery: 13/11, 8/7.
Lifting the slot looks too weak. Blue will probably be behinda 5-prime soon--he needs a 4-prime to fight back.
Splitting while slotting looks too loose. Splitting to the23-point doesn't accomplish a lot, but splitting to the 22- or21-points lets White abuse Blue on either side of the board.
Both 13/10 and 13/11 8/7 are reasonable approaches for goingafter a 4-prime. I prefer the latter because it slots asolid blockade and leaves larger (that is, non-splitting) numbersto cover.
Snowie: 13/11, 8/7.
I don't have to sweat the priming battle when I have five checkers on themidpoint and he has only two. If I can contain his back checkers, my backmen will survive on their own. My play prepares to make the very importantbar point without risking getting battered around on the other side of theboard. It is no disaster if my blot is hit. In addition, I cleverlyduplicate some of the fives he could otherwise use to make his four point.
Kit Woolsey: 24/23, 8/6.
I have to do something with the back men before they get stuck. Being blitzedis not a major danger, since I have the stronger board. I don't want Whiteto be escaping with a hit, so 24/23, 8/6 looks like the logical choice.
Chris Yep: 13/11, 8/7.
The timing is close, so it may not be absolutely essential for Blue to move off his anchor. I like being vulnerable on only one side of the board. Between 13/10 and 13/11 8/7 I prefer 13/11 8/7. Both moves give White 13 single hit numbers and 2 double hit numbers. However, 13/11 8/7 duplicates some of White's 5s (White would prefer to make his own 4 point rather than his 3 point) as well as giving Blue a chance, if he is missed, to form a solid 4-prime instead of a broken one.
Summary: The panel voted to concentrate fully on the offense and letthe back men wait. Could be right since Blue has all those checkers on themidpoint, but with White having the more advanced anchor I'm not so convincedthat the timing will go Blue's way. An interesting problem, and a surprisingvote in my opinion.
Play Votes Score13/11, 8/7 7 10024/23, 8/6 3 7024/21 1 6024/22, 24/23 1 6024/23, 13/11 1 6024/22, 8/7 1 6013/10 1 6024/22, 6/5 0 50
Problem 5
| 136 148 | White money game Blue |
Chuck Bower: 21/13.
Hitting here doesn't look right. It gives up a valuable point and leavesoodles of shots. Even when it works Blue has a ways to go, and it's a favoriteto fail. I think hitting now puts too many eggs in a damaged basket.
Running 21/13 just treads water in White's home pool. Blue will still bebehind in the race. 8/2, 6/4 makes a good point but still leaves a messy board. In addition, this play doesn't put any pressure on White.
6/4 makes Blue's best remaining homeboard point. With that I like 21/15.Stare White straight in the eye and see if you can make him blink. Better yet,you may be able to spit some tebacky juice in it next time and have a strongboard to back up your aggression. If White sends this checker back it mightenter on the 24- or 23-point and further add to White's headaches as he triesto come home. Unfortunately my preference isn't even one of the finalists! Inthat case I'll go with my second choice, 21/13. Blue is only slightly behindin the race, has an OK holding game, and (big plus) owns the cube.
Steve Clark: 21/13.
Suppose we hit with 13/11 and any 6 that you like, and then suppose ourfoe rolls and 6-2, which is a pretty bad roll. How are we doing then?It seems to me that we are still rather a dog. This makes the hittingplays rather unattractive. If I am going to make a big play and itworks, I want to have a big equity gain to go with it. There is littlepoint to taking a big risk only to find that we haven't gained verymuch.
Suppose we play safe, like 21/13. We would not be very far behind inthe race. We will have substantial chances to get away and win the raceor to hit a loose blot once we have made a better board. If I made thisplay, I would judge that White does not have a double. If I hit, Iwould have difficulty taking a double. The safe play must be better.
Of the safe plays, I prefer 21/13 by a small margin. It is usuallyright to get a back checker safely away while we have the chance. Iwill play 21/13.
Malcolm Davis: 21/13.
Looks like the only reasonable play, especially owning the cube.
Ray Fogerlund: 21/13.
After I play these 8 pips, I am down 4. I have a solid anchor and the cube. I believe that makes me the favorite in the game. So, I won't be leaving double shots to hit while I am waiting for the winning set, thank you very much. I might consider just building the board with 8/2, 6/4 but this is the perfect opportunity to bring the man on the 21 point out without the danger of him getting hit and costing me ground in the race. The other plays are weak
Steve Hamilton: 21/13.
Any of the hitting plays leaves too big of a mess for my taste. My goal isto stay close in the race and either wait for double fives or sixes, or forWhite to leave a shot after I have had a chance to build my board.
Hal Heinrich: 21/13
Blue retains decent racing chances and some hittingchances by declining the opportunity to hit now. Hittingnow just gives White too many shots at too many blots --though being missed would be sweet! Blue is just notdesperate enough to hit this way. Of the non-hittingplays, 21/13 is best because it escapes a back man --which in turn enables Blue to hold the mid-point longer.
Ron Karr: 21/13.
To hit or not? Advantage: trap White's blot and hope tocontain it. Downside: get hit back, possibly multiple checkers, andmaybe get gammoned. If I don't hit, I'm down 4 pips in the race but Ihave some long-term hitting chances, so I'm a slight underdog. Can Ido better by hitting now? It seems unlikely. White is a favorite tohit, and could easily hit 2 or even 3 checkers. True, getting hitisn't the end of the world, since I can keep attacking, but I'll berunning out of ammunition. Even if he fans and I start covering myblots, one false move and it's all over, since I still have 3 checkersback.
If I'm not hitting, I may as well improve my flexibility with 21/13.I'm not likely to get a shot next time, and even if I do, 8/2 6/4leaves two home board blots.
George Klitsas: 13/11*, 11/5.
From the plays that don't hit White's checker on the eleven point, 24/13 looks more pleasing to the eye than 8/2 6/4. If White rolls a 5-2 or 5-5, then Blue would prefer to have the spare on his twenty-one point in order to hit conveniently should he roll a hitting number - the third inner board point should be an asset in that case, as well. This is a long shot, in my opinion (White leaves a shot [p1=1/12 - I don't count 2-2 as an awkward roll for White comparing these two plays, since, if White rolls 2-2, Blue is happy to have a spare on his midpoint rather than his twenty-one point], Blue hits [p2=1/3], White does not return-hit from the bar [difficult to calculate, let's say p3=2/3] so p1Xp2Xp3 is less than 2% and I slightly prefer the flexibility guaranteed by playing 21/13. From the plays that hit, I prefer 13/11* 11/5, leaving the least number of blots, better coverage of the outfield than 13/11* 13/7 and the least number of immediate good rolls for White. I played a number of games (36 in each case) comparing play a (21/13), representing the non-hitting plays and play b (13/11* 11/5) representing the hitting group. During these manual rollouts ( with a program of my own that includes all 36 rolls in every ply) I took care also to give Blue his fair share of hits, when it mattered. So these rollouts are equivalent to much more than 36 games each, but, frankly, I can't estimate how many. Anyway, the main profit from human rollouts is, I think, to isolate the features that play a part in each case and the kind of games (race, priming, back games etc) that tend to develop. What should be said is that play a (21/13) leads to quiet games, mainly races in which White's slight lead in that area score a number of points for him. Strangely enough, the fact that White's race lead is not overwhelming, permits Blue to win a number of games with the cube, if he happens to roll well. All in all, White won 17 plain games and 2 gammons, Blue 17 plain games. Play b, instead, leads to all kinds of games with a surprising number of well-timed back games for Blue. If White fails to hit anything on his very first roll (including the four dancing numbers) Blue can quickly close his inner-board points and try to win forwards, by attacking and/or priming White. If White hits only one checker on his very first roll and escapes, Blue retains good racing chances. If White hits without escaping, Blue can hit back and in a blot-hitting contest anyone could emerge as the winner. Blue won 21 plain games, White 13 plain and 2 gammons. A seemingly very risky play (thinking of the three blots strewn around, two of which on one's inner board) at a moment when one has decent chances in the race playing quietly, a play that could be characterized even as �unthinkable� by most players, is justified by these rollouts and, having also the feeling (from the games that I played) that it's probably correct, I vote for 13/11* 11/5, the bold play.
Laila Leonhardt: 21/13
A very close race, and no reason for Blue to jeopardize this, by leaving adouble shot. Blue owns the cube, and can wait for opportunity to escape orhit a fly shot later in the game.
Rob Maier: 21/13.
See problem 4. After hitting, White has 24 immediate returns, andprobably hits in about 3 of the other 12 games the following roll if hemisses the first time, for a total of 27 out of 36. That's a lot of hits,and they all hurt, with several double hits being especially painful. Theother 9 games aren't gin either. Take a breath, and be patient.
Steve Mellen: 21/13.
To me hitting looks desperate here and Blue's position is farfrom desperate. He is only 8 pips down after the roll, owns the cube, andhas a strong holding game. There are simply too many variations afterhitting where Blue gets another checker or two sent back and ends up buriedin the race. He will have plenty of chances to get a shot later or simplywin the race. When you own the cube, many times you do not have to attemptto win in one roll and can afford to gradually build up your position. Thisposition strikes me as a perfect example. While 8/2, 6/4 is the logicalnext step in building the board, Blue is running out of checkers to playwith and the third checker on the 21 point is not doing anything for him. Iwould take this opportunity to get that checker in play by moving 21/13.
David Montgomery: 21/13.
After playing Blue is down four pips in a very long race withsubstantial contact. Hitting looks unjustifiably desperate.
Getting the third checker out is important. It will allowBlue to wait longer for a shot when necessary, or to moreeasily run if 55 or 66 pops out.
Snowie: 21/15, 13/11*.
Hitting is thematic, since I am behind in the race. Jumping a checker outmay look risky, but a careful examination of White's numbers indicates thathe doesn't have all that many double-hitters. I'm willing to take thatchance to establish outfield control.
Kit Woolsey: 21/15 13/11*.
I am behind in the race and have a decent board, so it can't be right notto hit and give White a free ride home. Given that I am hitting, I mightas well spring a back checker. White's small numbers are all duplicated,so this isn't as dangerous as it might seem, and the outfield control is great.
Chris Yep: 21/13.
Could it be right to hit, breaking the midpoint and leaving at least 3 blots (including two inner board blots) when the race is still close? Maybe, but it looks too aggressive to me. I prefer the simple 21/13 which is better for Blue's flexibility than 8/2 6/4. Blue can clean up his board next turn and either play a holding game or a racing game (with appropriate doubles) depending on the dice.
Summary: Behind in the race, and everybody wants to race? I can't believethis. At least our resident bot agrees with me -- I was beginning to think Iwas the only sane one here.
Play Votes Score21/13 12 10021/15, 13/11* 2 7013/11*, 11/5 1 6013/7, 13/11* 0 508/2, 6/4 0 50
Problem 6
| 115 109 | White money game Blue |
Chuck Bower: 8/4, 7/2*.
Blue has a better board and White has a second blot to shoot at if theaggressive play results in White hitting from the rail. 8/4, 7/2*. Blue hasbeen given a chance to go for the throat. Without White's homeboard blot Iwould think harder. With White's vulnerability I like shooting firstand asking questions later.
Steve Clark: 8/3, 8/4.
I hate to leave a blot in this position. I am ahead in the race so Ifeel little incentive to to risk being hit unless there is somesignificant gain. If I make the safe play of 8/3, 8/4 I am wellpositioned to point on him next roll. Even if he rolls one 6 and getsaway, I will be ahead in the race and possibly a favorite. This lookspretty good to me.
Either of the other plays offers up a real good hit. Even if the hit isnot necessarily decisive (after all he does offer return shots at hisblot on his 2 point), I hate to give up the safer play. If I hit on the2 point I will get more gammons but his outfield checkers are not so faraway from his infield that I will get many gammons in any case. It ishard to judge the relative merits of the two big plays but I am notheaded in that direction anyway. 8/3, 8/4 is my choice.
Malcolm Davis: 18/13, 18/14.
Not my usual style, but it looks like I pretty much win the game with no 2 by my opponent, and I have some return shots if I get hit. If I owned the cube I would seriously consider 8/3, 8/4 and double next time if my opponent rolled a non-6.
Ray Fogerlund: 8/3, 8/4.
Leave a blot, find a way to lose. Up 15 pips after this roll, my only worry is 66 by White. I am worried about any number that hits with the other plays, and I am still in the game if White rolls his joker here.
Steve Hamilton: 18/13, 18/14.
After moving 18/13, 18/14 Blue has a big advantage if White does not hit witha two. If Blue moves 8/3,8/4 and then White escapes with a six Blue is leftin a very bad spot, with those two checkers stuck over on White's barpoint.Blue will probably end up crunching his homeboard or leaving a shot afterWhite has had a roll or two to build up his board. I'd risk the hit now with18/13, 18/14 rather than risk White escaping with a six.
Hal Heinrich: 8/3, 8/4
Blue has this position well in hand, and doesn't need toexpose any blots. If White escapes with a six, Blue won't like it, but will still be ahead in the race withthe chance to hit a fly shot. Getting hit with either ofthe other plays is a much bigger loss of equity.
Ron Karr: 8/3, 8/4.
Tough choice,though a nice dilemma to have: I'm ahead in the race, as well as havinga strong offense.
8/4 7/2* goes for the jugular now. If it succeeds, I've maximized mygammon chances by freezing White's outfield blots. If White hits back,I'll still have some counterattacking chances, but things could getmessy. I'm pretty much committed to going for the closeout.
8/3 8/4 leaves no shots. If White escapes, I'm still ahead in the race(except after boxes), although I could have trouble getting off theanchor. And if he doesn't escape, I'll have lots of numbers to pointon him or at least hit loose and keep the 5-prime (also, it's nice thatthe 1s and 3s which hit inside are the numbers blocked on the otherside of the board).
18/13 18/14 solves the problem of escaping the back checkers by doingit now. If White misses, I'm in great shape, though not guaranteed tocover next time. If he hits, I still have some life.
I'm uncertain about this one, but I'll go for the play that maximizesracing equity, makes a solid 5-prime, and leaves attacking chancesopen.
George Klitsas: 8/3, 8/4.
I am sure that the third option, namely 8/4 7/2* is clearly worse than any of the other two. Blue wins a number of gammons in excess, but these gammons don't compensate for the dramatically increased number of losses for Blue. Having many blots strewn around is a liability in a vast number of positions as it is definitely here. The difficult part of this problem is, actually, deciding between the remaining plays. Play a (18/13 18/14) offers a direct route to victory. If White does not roll a hitting two, Blue is a huge favorite to make a six-point prime (huge but not certain, though, since 6-6, 6-5, 5-3, 4-4, 4-3 all fail to make the prime). Play b (8/3 8/4) instead, is designed for attack, if White fails to roll immediately an escaping six. Apart from the pointing numbers on the two point (numerous with four builders), in that case, Blue will hit even loose, even when White has made his own two point (look at the duplication of sixes, though and the fact that White can't make it with the roll of 4-1), since,White will need a two AND a six after that to escape. Comparing plays a and b is basically to compare White's good number in each case. The hitting two's as in variation a, cause more damage to Blue (White often becomes the favorite) than the escaping sixes, as in variation b (White remains the underdog) - and the fact that escaping with a six offers White a number of optimal recubes, if the game becomes virtually a race, is of little importance. There is still another (subtle) consideration in favor of play b. After play b, Blue is ready to attack on his very next roll having a five-point prime with adequate builders on it. After play a, Blue actually loses a tempo, for he must first make his four point and then attack (on his very second roll). This fact has two major consequences. First, Blue wins fewer gammons after play a, for White has two rolls to bring in some of his checkers reducing the gammon danger. Second, White has two rolls to make his inner board more menacing, a fact that should deter Blue from hitting loose in many cases, in his effort to extend forwards his made six-point prime. I vote for 8/3 8/4, noticing as a last thought the power of a five-point prime.
Laila Leonhardt: 8/3, 8/4
Blue is up in the race. No reason to leave White a shot that can turn thegame around.
Even if White should roll a 6 and escape, Blue will still get an indirectshot and still be ahead in the race.
Rob Maier: 8/3, 8/4.
White will have play if they roll a six, but nothing compared to the playthey get when they hit. We're winning the race and White has a strongposition, not a good time to get cute.
Steve Mellen: 8/3, 8/4.
I can't see any reason not to beef up the board with 8/4. Ishitting too big? If this were a position where Blue was likely to end uphitting loose within a roll or two anyway it would definitely be right tohit now because White has a blot in his board and Blue wins more gammons bykeeping White from bringing his outfield checkers in. The key to theposition, however, is that Blue is up in the race by 15 pips anyway. Hewins a lot of games by simply bringing his checkers around and winning therace. After 8/3, 8/4 Blue has a lot of pointing numbers next roll, andretains the option of just leaving White alone and racing. White's 11escaping numbers are not nearly as dangerous as his 14 hits after the otherplay. Blue's position after 8/3, 8/4 is way too strong in every area of thegame to warrant putting everything on the line with a loose hit.
David Montgomery: 18/13, 18/14.
Over the board in a fast chouette I would probably play 8/3 8/4.I'm ahead in the race with very favorable contact after thisplay. The other plays need large gains on the good variationsto make up for the terrible swings on the bad variations. Theposition isn't that gammonish even after hit/dance, so I wouldplay safe and hope to attack next time.
Near the end of the match in the finals of a big tournament, I'dcompare the plays by assigning equity-wins for each of severalcontinuations. Here's one such breakdown:
| Play | Variation | Rolls | EquityWins | Total |
| 8/4 7/2* | dance | 16 | 14 | |
| inside hits | 8 | 2 3/4 | ||
| outside hits | 6 | 3/4 | ||
| mis-entries | 6 | 4 3/4 | ||
| 8/4 8/3 | escape | 11 | 7 | |
| contained | 25 | 19 1/2 | ||
| 18/14 18/13 | hit andleap | 2 | 3/4 | 27 |
| hit and cover | 5 | 1 1/4 | ||
| just hit | 4 | 2 | ||
| no hit | 25 | 23 |
I have a hard time believing 7/2* 8/4 is so weak. I probablybotched those numbers, but even so this is enoughto convince me that 7/2* 8/4 isn't right by a lot.
One vig in favor of 18/13 18/14 is that White will cash mostof his good variations, mitigating the gammon danger of leavingthree blots.
Snowie: 18/13, 18/14.
28 numbers to complete the prime and claim if I'm not hit. How much more couldI want for one play?
Kit Woolsey: 18/13, 18/14.
I have to get the back men moving sometime, and this looks like the right time.A lot of numbers to complete the prime if White misses the shot, and ifhe hits I still have a good chance to contain him.
Chris Yep: 8/3, 8/4.
If Blue plays 8/4 7/2* he will be attacking while he has a stronger board and White has an inner board blot. Still, Blue will leave two blots of his own. If White enters with 2-5 or 2-6 and Blue dances, White will be playing for a gammon. If White enters with 2-2 and Blue dances, White will redouble Blue out. If White enters with 2-1, 2-3, or 2-4 and Blue dances it looks like White will have a very efficient double - he will have 4s to cover and 5s and 6s to hit another checker.
18/13 18/14 works out well if White doesn't roll a 2. However if White rolls 2-x followed by Blue dancing, White can claim with a recube. I like 8/4 8/3 which makes a solid 5-prime and gives Blue good attacking chances if White doesn't roll a 6 next turn. If White escapes with a 6, Blue will still have a solid racing lead with some coverage of the outfield. White will have to either repeatedly give Blue indirect shots at his straggler or move it to safety, after which Blue will usually be able to turn the game into a race (any combination of 4s, 5s, and 6s, or double twos will allow him to disengage).
Summary: An interesting and difficult choice. I still like the idea ofpreparing to complete the prime next roll, but the more conservative approachmay well be right.
Play Votes Score8/3, 8/4 9 10018/13, 18/14 5 808/4, 7/2* 1 60
Problem 7
| 119 52 | White money game Blue |
Chuck Bower: 8/7*, 4/2*.
Best problem of this set, IMO. My initial reaction was: "why leave theshots? 4/2*/1 is safe now and may end up never leaving future shots."I didn't see the double hit here, which may be the best reason for votingfor it! When it works Blue is on the way to a gammon. But when it fails Whiteoften will have a juicy recube rather quickly. 8/7*, 4/2* leaves the samenumber of shots (11) as 8/7*/5, but three blots here is a pretty big liability.Does this play deliver enough extra gammons?
I did a Kleinman-like count of hits over the next 36^3 = 46,656 games forthe 3/2*/1* play. 20% of the time White hits a shot but 25% of the time Bluestill has the 8-point to clear. So adding 1/3 prorated hits (based on the 3::1ratio of clearing 75% of the time and still holding the 8-point 25% of the time) estimates the additional hits to clear that point at 7% and says White'stotal hits occur 27% of the time as White clears the 8-point. Compare that tothe rolls 26, 36, 66, 25, 35, 65 or 11/36 > 30% hits when Blue clears the 8-point now (8/7*/5). It doesn't look like the extra diversity when Blue survives after 8/7*/5 is enough. Note that secondary consideration of Blue's increased gammons after 8/7*/5 are offset by the value of getting White off Blue's 2-point forever (sometimes) with 4/2*/1. Of the two single hit playsit looks like 4/2*/1 gets the nod.
Now, back to the double hit. Sometimes things go so well for White thathe can actually play on for the gammon. Although this partially negates White's cube ownership equity, it still is an added option worth some gains. That argues against the double hit. But surpringly, even when White hits AND enters BOTH checkers on his first roll Blue can still scratch out some wins (even gammons) by entering with a 5. If White hits but one of hischeckers remains treed, even if Blue fans I don't think White can cube.OTOH, White's failure to hit is no guarantee that Blue wins a gammon. A sequential 36 trial rollout gave the following results for Blue: gammon wins = 16, simple wins = 10, simple losses = 8, gammon losses = 2. Unit cube equity = +30 for Blue. Equivalent numbers of rollouts for the other two plays gave similar equities. As usual, hand rollouts don't give statistical significance.
This one is close but my gut reaction after studying this position is thatthe double hit is the winner. And I'm betting that will NOT be the panel'schoice. Any takers? (Too late. You've already seen the tally. :)
Steve Clark: 4/2*, 2/1.
If I play 8/7*, 7/5 and I get hit, I will still have a significant amountof life. I will probably have a number of chances to get in on his 5point while still leading the race. The problem with this play is theeven if he misses me, my position is not particularly safe and he islikely to have more blots to hit in the future. I will look forsomething better.
4/2*, 2/1 has the theoretical objection that we are burying anotherchecker. But that is not so important here. We are not trying to builda board necessarily. We are just trying to get a couple of guys insafely. There is actually a lot of potential gain from the play. Hemight flunk. That would be terrific. He might come in but not on the 2point. That is good as well. I would have a much safer bearoff and abetter opportunity of getting in safely as well. With this play I havea good chance of never leaving a blot. This seems more attractive than8/7*, 7/5.
Playing 8/*7, 4/2* is like playing football without a helmet. Some ofthe time you are going to end up with a very bad headache. The meritsof this play are clear. If he misses me, I am likely to be able to putmy position together safely, and I will have a terrific chance ofgammoning him. Of course the 3 demerits of this play are the blots Iwill be leaving. If he hits me, the gammons will be going the otherway. Initially I was quite attracted to this play and I would make itstill if the technical merits of 4/2*, 2/1 were not as great as I think theyare. In the end, however, I will make the safe play.
Malcolm Davis: 4/2*, 2/1.
Can't imagine another play! Is anything else safer this roll? I will save my energyfor a later eecision.
Ray Fogerlund: 4/2*, 2/1.
Never in my younger days would I have done this, but why volunteer shots when you may not have to leave any at all? Pay later.
Steve Hamilton: 4/2*, 2/1.
There is no way I would risk leaving a blot here for White to put on the baragainst a five point board. Yes, there is a gap on the five spot but it isstill a five point board. I can't count high enough the times I have spentfive or six rolls on the bar against just such a board.4/2* ,2/1 is my choice. White may just dance herself, or if she rolls highmight have to move a checker out past Blue to avoid crunching that fivepoint board, making it easier for Blue to bear in safely next roll. Whoknows, next roll Blue might get lucky and solve all his bear in problems byrolling some nice doubles.
Hal Heinrich: 8/7*, 7/5
This is the right time to take a small risk to come homesafely. 8/7*, 4/2* is just nuts -- more shots and moreblots! Picking and passing is plausible, but generatesmore shots overall.
Ron Karr: 8/7*, 7/5.
This is themost straightforward play, bringing 1 checker home to a flexible placeand preparing to bring the last one home & start bearing off. It leaves11 shots (if I counted correctly) but if missed I'm in good shape; ifhit, I should still have several rolls to enter (and only 1 blotexposed) before I really start worrying.
8/7* 4/2* doesn't leave any more shots, but it leaves 3 blots, whichmeans if I'm hit I'm worrying immediately. The upside is that I'llprobably win more gammons, but I don't think it's worth it.
4/2*/1 is safe for this roll but it doesn't solve my long-termproblems. Even if White is forced to enter on the 6 point, I couldstill have trouble clearing the 8 point safely.
George Klitsas: 8/7*, 4/2*.
Even a small rollout (36 games) is, in unusual positions like this one, perhaps more fruitful than abstract thoughts or a mental process that divides the rolls into groups and tries to assign rough values in the relevant outcomes. I played 36 games starting with 8/7* 7/5 (play a) and then another set of 36 games starting with 8/7* 4/2* (play b). In the first case, Blue's main problem is the difficulty of playing two's, during the bear-off, which sometimes results in leaving even two blots (for example having three checkers on the four point and the rest on the ace point and rolling something like a 6-2). If Blue is hit early, he's in big trouble, as White will slot his own five point with a covering man in range and (if the slot survives for one roll) claim with a recube that tends to be optimal. Blue won 9 plain games, 10 gammons, 1 backgammon, White 16 plain games, for an equity of 0.88 points per game for Blue on a two-cube. After play b, White is entitled to go for an un(re)doubled gammon in many cases,but, in exchange, seems to win enough gammons in excess himself, that more than compensate for this fact. Blue won 6 plain games and 16 gammons, White 10 plain and 4 gammons for an equity of 1.12 points per game for Blue on a two-cube. For the sake of completeness, I played a few games adopting play c (4/2*2/1), noticing that Blue can't win many gammons after that overcautious play and is a favorite to leave lots of shots in the future, especially with rolls containing a five. I can't be sure, but I must vote for the bold play, 8/7* 4/2*, the typical winner, although I always argue against leaving many blots strewn around. This interesting position might well be an exception.
Laila Leonhardt: 4/2*, 2/1
White has a 5 point board and owns the cube. Blue needs to try clearingwithout leaving a shot, so not advisable to leave 11 indirect shots. Byhitting the blot on the 2 point and play safe, White may not reenter on the2 point, and Blue will have a better chance to clear safely and worse casebe forced to leave similar amount of numbers as he just avoided givingvoluntarily
Rob Maier: 4/2*, 2/1.
Zero shots, and a decent chance to keep it that way. How bad could it be?
Steve Mellen: 8/7*, 7/5.
Between the two plays that hit the outside checker, 8/7*/5looks clearly superior. Both plays leave the same number of shots; however,leaving only one blot means that Blue will still be favored after 8/7*/5even if he gets hit, whereas after 8/7*, 4/2 a hit will put him in gammondanger. Blue will win a lot of gammons after 8/7*/5 even without takingthis risk. Playing completely safe with 4/2*/1 is tempting. It is tough toestimate the amount of future jeopardy this play leaves, but it should beclear that it will be very hard for Blue to bring this awkward position homewithout any risk. Even if White dances Blue will have 12 numbers that forcehim to volunteer a direct shot and several others which leave just as manyindirect shots as playing 8/7*/5 directly. In addition Blue will win fewergammons with this play because it fails to start bearing off as quickly as8/7*/5 and most of White's rolls will enter and safety the blot on the 7point. I would play 8/7*, 7/5.
David Montgomery: 4/2*, 2/1.
Blue doesn't want to get hit. He has a plausible chance ofavoiding a direct shot the rest of the way, and a good chanceor avoiding double shots and/or multiple blots. If the follow-upwere worse, I might volunteer, but not here.
Snowie: 8/7*, 7/5.
The eight point is so difficult to clear that I'm better off paying now witha few indirect shots than risking a real calamity later. I can probablysurvive one checker being hit now, and if I'm not hit life will be mucheasier in the future.
Kit Woolsey: 4/2*, 2/1.
Ugly, but I am willing to delay the evil day for another roll. By rippingWhite off the two point I may make my future plays easier. Since thevolunteering plays are far from gin when they work, it looks best topay later.
Chris Yep: 8/7*, 7/5.
It looks close between paying now and paying later. I have a slight preference for paying now before White gets better outfield coverage by moving the checker on Blue's 7 point forward . White will only have 10 shots (2-5, 2-6, 3-5, 3-6, and 5-6). In addition, Blue puts a useful spare on the 5 point. 8/7* 4/2* is another way of paying now, but leaves 3(!) Blots. 8/7* 7/5 looks best to me.
Summary: The panel voted to pay later in a very difficult decision.Rollout results of this problem should prove to be interesting.
Play Votes Score4/2*, 2/1 8 1008/7*, 7/5 5 808/7*, 4/2* 2 70
Problem 8
| 165 193 | White money game Blue |
Chuck Bower: B/21, 22/14*, 8/4*.
Blue needs to take maximum advantage of this great roll. A lot of players(myself included) have a tendency to play doublets too quickly, especially withthe euphoria accompanying GOOD doublets. Finding all the candidates is much harder for doublets than non-doublets. Fortunately in Kit's quiz we have theaid of his listing the candidates for us.
Here we are forced to play b/21. That leaves three more 4's. We'vegot to hit at least one blot, and if we hit on the 14-point we might as wellhit the other one (on the 4-point). So no matter how we play the last two 4's,I like 8/4* as the second 4.
If we now cover the 4-point (that is, we play b/21, 8/4*(2)) then we don'treally have a very good final 4. I guess it's 24/20. But by playing b/21, 8/4*, 22/14* we get full use out of every one of our four 4's. The downside isthat we may not get the homeboard point, but 2/3 of the time White doesn'treturn hit there and we have a chance to eat our cake, too! If one opponentchecker on the bar is good, isn't two there double good?
Steve Clark: B/21, 22/14*, 8/4*.
Suppose I play B/21, 22/14*, hitting. This would leave me with one more4 to go. None of the potential 4th 4's are very constructive. The onlyone really worth considering is the second hit with 8/4*. Of courseputting 2 men on the bar is always good but normally I prefer a moreconstructive play such as 8/4(2)*.
In this position the constructive plays involving 8/4(2)* are not asattractive as usual. The problem is that we are not making anadditional point; we are just shifting from the 8 to 4 point. This isnot all bad, of course, particularly if our opponent rolls a combinationof 6's and 4's, but that doesn't happen all that often. Suppose I makethe 4 point, how would I play the final 4? None of the possibilitiesattract. 8/4* gives up diversification and advances the checker toofar. The other plays just put out blots to be hit without reallyaccomplishing anything. Non of this seems very appealing.
Well what about hitting on the 14 and the 4 point. Most of his rollsmiss me and I will be very happy after any of them. This play doesleave a large number of Blue blots around but White will have two men onthe bar and doesn't have any inside points yet. Kent Goulding'sconcept, "The man with the most blots wins," doesn't work very wellwhen your opponent has a 5 point board. But it does have substantialmerit when he has made no points whatsover.
Malcolm Davis: B/21, 24/20, 22/14*.
Sure looks pretty. Don't want to break the 8-point this early in the game bymaking the 4-point. Everything else also appears to be inferior.
Ray Fogerlund: B/21, 22/14*, 8/4*.
Ha ha, 7 choices and my play is not included! Recently, in studying my matches I have noticed that Snowie likes structural plays in the outfield. I have incorporated this thinking into my game. So, my first three 4s are now, automatically, B/21 and 13/9 (2). Then I would just step up, I guess, but hitting would not be considerably worse. Alas, neither of these plays are among the choices... So, I just hang on to the 8 point, and flail away in an effort to keep White off balance as much as possible. Then she will be on the defensive for another turn, at least, and if she rolls a 6 I may be able to develop something... (like an outfield point/prime???) !!
Steve Hamilton: B/21, 24/20, 22/14*.
Keeps maximum pressure on the outfield? Keeps open the possibility ofgaining the 4 or 5 point anchor? Heck I don't really know about this one. Isuppose this move just looks right to me.
Hal Heinrich: B/21, 22/14*, 8/4*
Entering and hitting on the four point are clear. Afterthat Blue can either hit on the fourteen or make the four point and find a final four. While making the fourpoint is a big gain in the early going, here it does soat the expense of the eight point. Hitting a second mangains in the race, freezes White's checker on the barpoint, retains the eight point, and starts the four point. That's more than enough to convince me.
Ron Karr: B/21, 22/14*, 8/4*.
It would be great to have an extra 4 so I could hit on 14 AND make the 4point. In the real world, hitting both blots seems best. The 4 pointis good, but I'd have to give up the 8 point. Hitting the blot on 14is very helpful, gaining in the race and hurting White's offense.Hitting loose on the 4 point is the equivalent of making the point 2/3of the time anyway (if White misses), so it seems like the best ofworlds. If White misses, I'll be very close to a cube, and if he hitsit's still a tossup.
George Klitsas: B/21, 22/14*, 8/4*.
From the long list of conceivable moves, only two seem to me like serious candidates, namely B/21 24/20 8/4*(2), and B/21 22/14* 8/4*. The first one makes the second-best inner point, at the cost of breaking the eight point and risking to lose it by a White hit. Other plays that make the four point seem inferior to me, for they either break Blue's defensive anchor on the three-point (you never know what accidents lurk in innocent-looking positions like this [and then one thinks he is unlucky!] ), create additional blots that tend to be liabilities than assets (13/9), or, finally, are structure self-destroyers (8/4*(3)). The second play (B/21 22/14* 8/4*) has the big advantage of hitting two White checkers, keeping him off balance and forcing him to roll a four in order to survive the immediate assault. White's checker on Blue's bar point becomes vulnerable, particularly if White partially dances. I think that both plays are strong and playable, but I will go with the double-hitting play (B/21 22/14* 8/4*) as a matter of style.
Laila Leonhardt: B/21, 22/14*, 8/4*
this is early in the game, neither side has established a point yet. Puttingtwo checkers on the bar can turn the game in Blues favor if White rolls a 6next time. and else keep the pressure by the flexibility of the manybuilders.
Rob Maier: B/21, 22/14*, 8/4*.
A great time to hit two. No opposing board, and two additional blots toscoop up. Making the four point is nice, but isn't worth losing the eightpoint with such an attractive alternative.
Steve Mellen: B/21, 24/20, 8/4(2)*
I do not like hitting two with B/21, 22/14*, 8/4*.If White hits back Blue will hate having all those blots, and even if hefails Blue will have a major cleanup job to do. I would prefer to make thefour point immediately and have an asset for the rest of the game. Withboth sides having so many blots it cannot be bad to start beefing up theboard. 8/4(3)* strikes me as too stingy, however. I would rather play24/20 with the last four and aim at White's blot on the 14 point. If Blue'schecker on the 8 point gets hit Blue will have an excellent chance to make asecond anchor in White's board and give himself a solid position to playfrom.
David Montgomery: Bar/21, 8/4(3)*.
If Blue hits on the 14, the last 4 is probably 8/4*. But Blue istoo light up front to blitz. Making the 4-point and safetying theblot preserves all of Blue's offensive material to attack with nextturn.
Snowie: B/21, 22/14*, 8/4*.
A bird in the hand may be worth more than two in the bush, but two blots inthe hand with more to come is worth more than one point. Even if he hits backI'm in decent shape, and if he doesn't I will have the whole board at myfingertips.
Kit Woolsey: B/21, 24/20, 8/4(2).
It sure looks right to point on White's head on the four point. For the finalfour, coming up to the 20 point puts him under maximum pressure on allparts of the board while he is on the bar. I expect to do well in ablot-hitting contest.
Chris Yep: B/21, 22/14*, 8/4*.
With White having 4 blots and no board, I like hitting twice. Hitting twice gives Blue a racing lead, and still lets him shoot at two blots next turn. If White doesn't roll a 4 next turn, Blue can make the 4 point himself with a strong initiative. In fact if White dances with 6-6, Blue probably has an efficient double, although this has only minor importance in this position.
Summary: The panel strongly voted to hit twice and ask questions later.I don't know. Could be right, but that made four point is awfully tempting.
Play Votes ScoreB/21, 22/14*, 8/4* 10 100B/21, 24/20, 22/14* 2 70B/21, 24/20, 8/4(2)* 2 70B/21, 8/4(3)* 1 60B/21, 22/14*, 22/18 0 50B/21, 22/18, 8/4(2)* 0 50B/21, 13/9, 8/4(2)* 0 50
1 2 3 4 5 6 7 8Chuck Bower 24/23, 10/5 24/20*, 20/16, 13/9(2) 18/13, 18/15 13/11, 8/7 21/13 8/4, 7/2* 8/7*, 4/2* B/21, 22/14*, 8/4* Steve Clark 24/23, 6/1* 24/20*, 20/16, 13/9(2) 13/8, 6/3 24/23, 8/6 21/13 8/3, 8/4 4/2*, 2/1 B/21, 22/14*, 8/4*Malcolm Davis 24/23, 6/1* 24/20*, 22/10 13/8, 6/3 24/23, 13/11 21/13 18/13, 18/14 4/2*, 2/1 B/21, 24/20, 22/14*Ray Fogerlund 24/23, 6/1* 24/20*, 22/10 13/8, 6/3 13/11, 8/7 21/13 8/3, 8/4 4/2*, 2/1 B/21, 22/14*, 8/4*Steve Hamilton 24/23, 10/5 24/20*, 20/16, 13/9(2) 13/8, 6/3 24/22, 24/23 21/13 18/13, 18/14 4/2*, 2/1 B/21, 24/20, 22/14* Hal Heinrich 24/18 24/20*, 20/16, 13/9(2) 13/8, 6/3 13/10 21/13 8/3, 8/4 8/7*, 7/5 B/21, 22/14*, 8/4*Ron Karr 24/23, 10/5 24/20*, 22/10 13/8, 6/3 13/11, 8/7 21/13 8/3, 8/4 8/7*, 7/5 B/21, 22/14*, 8/4*George Klitsas 24/18 24/20,* 20/16, 13/9(2) 13/8, 6/3 24/21 13/11*, 11/5 8/3, 8/4 8/7*, 4/2* B/21, 22/14*, 8/4* Laila Leonhardt 24/23, 8/3* 24/20*, 20/16, 7/3(2) 18/13, 18/15 13/11, 8/7 21/13 8/3, 8/4 4/2*, 2/1 B/21, 22/14*, 8/4* Rob Maier 24/23, 6/1* 24/20*, 20/16, 13/9(2) 13/8, 6/3 24/23, 8/6 21/13 8/3, 8/4 4/2*, 2/1 B/21, 22/14*, 8/4*Steve Mellen 24/23, 6/1* 24/20*, 20/16, 13/9(2) 18/13, 18/15 24/22, 8/7 21/13 8/3, 8/4 8/7*, 7/5 B/21, 24/20, 8/4(2)*David Montgomery 24/23, 10/5 24/20*, 20/16, 13/9(2) 13/8, 6/3 13/11, 8/7 21/13 18/13, 18/14 4/2*, 2/1 B/21, 8/4(3)*Snowie 11/10, 6/1* 24/20*, 22/10 13/8, 6/3 13/11, 8/7 21/15, 13/11* 18/13, 18/14 8/7*, 7/5 B/21, 22/14*, 8/4*Kit Woolsey 24/23, 10/5 24/20*, 20/16, 13/9(2) 13/8, 6/3 24/23, 8/6 21/15, 13/11* 18/13, 18/14 4/2*, 2/1 B/21, 24/20, 8/4(2)*Chris Yep 24/23, 10/5 24/20*, 22/10 13/8, 6/3 13/11, 8/7 21/13 8/3, 8/4 8/7*, 7/5 B/21, 22/14*, 8/4*
