Steve Brown’s Game: Round 2(1)

Returning to the game, here is the position after the start of round 2.

Close but no anti-smoking song

We got plenty of Diamonds plus a few Jacks and Kings – but unfortunately this falls agonizingly short of a cheevo as described in my previous post. Before discussing actual moves, I will give Steve’s own assessment of the position and your task is to assess how much of this is accurate:

One king was dealt at the start of Round 2 in column 1 (C1). With two kings now in the column it was very unlikely that I could turn any hidden card in C1 until late in the game. Now there were two columns which I was sure could not be easily turned: C1 and C4. Together they were harboring 7 hidden cards. My best hope for winning was to turn nearly all of the hidden cards in the other columns by the time of the final deal.

The situation in C1 was much worse than in C4 since the newly-dealt king blocked access to 17 cards, 13 or which were visible. Because of this, I now gave the extraction of the Ks high priority, although I knew that it could be quite sometime before I might accomplish the feat.

I got a very big break on this deal as the J098 of Diamonds all fell. What’s more they all fit so nicely onto the C3:Qd which fit onto the dealt C1:Ks. It takes very little studying of the starting game state of R2 to see that I would be able to turn both C3 and C6. If things went well, I stood a good chance of also turning C10. Unfortunately, the other 7 columns did not look as promising.

My analysis is below, following the usual spoiler-blocker.

The usual.

Steve correctly identified two guaranteed turnovers. The diamond run is nice, but we are a long way from completing the suit. Remember this is not poker. It takes 13 cards in a suit, not 5, before we hear a triumphant C major chord. Steve also correctly identifies a possible junk pile starting with the C1:Ks.

Steve says extracting the Ks from C1 is high priority. While I admire his long-term thinking, I think it is slightly inaccurate or misleading. Firstly, it contradicts his earlier statement “my best hope for winning was to turn nearly all of the hidden cards in the other columns by the time of the final deal”. Also, it would be more accurate to say “extract the Ks when the timing is right”. If we already had an empty column, pulling the Ks now wouldn’t achieve anything special – in fact, a beginner would rightly complain that we might not get back an empty column back soon. The timing just isn’t right. Those 13 cards in C1 will always remain visible even if the game were played by a team of three Crewmates and nine Impostors. We need more turnovers, which equates to more information, and hence better ability to judge when the timing is right. If we can get a rot13(fuvgybnq) of in-suit builds then it may be worth sacrificing an empty column – particularly if we also come close to completing a full suit.

Therefore, I would write “extracting the Ks from C1 is not high priority since we should just get on with the job of turning over cards and chasing an empty column – but we should be keeping an eye out for opportunities to pull the Ks if the circumstances are right.”

As an aside, before dealing 10 cards it was apparent that C7 was our best hope for an empty column and C2/C3 were our best bet for turnovers. It turned out C3/C6 came good instead. This illustrates the principle that the Captain Obvious option don’t always come to fruition and it pays to have multiple outs.

  • Move: ce,ca → 2d
  • Move: fa,fj,fh → 7c

After turning over the 2d, I would probably prefer to expose another card in column 3 to keep flexibility (the actual play involves two irreversible moves instead of one). But Steve’s play also has merit: he builds in-suit twice and turns over a non-atomic column before it becomes a problem. At least we can’t argue about the quality of both turnovers 😊

At last – we have an empty column.

I don’t know if Winston Churchill can wield a mean deck of cards or two, but he certainly knew something about great power and great responsibility. Once we get our first empty column, our options increase dramatically – and so do our chances of suboptimal play if we’re not careful.

That’s enough for today, and next we will come to the meaty part of the game 😊

Steve Brown’s Game: Round 1(3)

Here is the current position

Steve continues thus:

  • Move  ea, ga → 6d
  • Move fd, fd, gd → Jc
  • Move ge → 9c

Clearly, “ea” is essential otherwise the ability to build the 7h-6h in-suit is lost. Of course, had we known we were turning over a Six then better is “ge” keeping extra flexibility at no cost, but them’s the breaks if you pardon the terrible cliché. By shifting the Seven of Clubs onto column 4 Steve effectively treats column 4 as a junk pile, a reasonable decision since turnovers in that column ain’t happening any time soon. We need at least a second King (to make it legal) and then hope to avoid counterfeiting turnovers in column 3 (to make it worthwhile). That’s more ifs and buts than chasing runner-runner to complete the ignorant end of a straight when you’re already pot committed and facing expert opponents.

Steve now has the following position:

I have a slight quibble with the last move. I would prefer to have the Jack of Clubs in column 3 instead of column 5. This is to avoid having an excess of two Queens in a single column. If column 3 becomes a junk pile (imagine e.g. a Five appears next round and we fail to shift that Five any time soon) then we could run into a shortage-of-Queens problem in the future. Still, given the spare King in column 8 we are most likely playing the move “ch” soon anyway, so this is probably moot.

In any case, our options are limited and there is only one turnover available in column 2.

  • Move ba, ij, bi → 6c
  • Move ch, deal

This completes round 1.

This is a pretty good recovery for the good guys. With only one column containing no face-down cards or a King, it’s not hard to guess plan A. Ideally we would want some plans for B, C and possibly D in case the Captain Obvious option does not materialise, but we take what we can get. At least we have multiple outs as far as turnovers are concerned.

Bad memes aside, Steve has turned over 11 hidden cards this round which is above (his) average. He estimates that he is now on par with his average game and I agree with this assessment.

Steve Brown’s Game: Round 1(2)

Continuing our game, here is the current position:

  • Move: gi → 5s
  • Move: hd → 8h

The next move sequence is a good illustration of Steve’s skill. The obvious option is “hf” since we are one good card away from claiming our first empty column. However, the downside of that move is polluting column 6. We would no longer able to perform the move “fa”. Assuming we connect the 8-7 of Hearts, this would also create a long-term problem of two Sevens in column 6 but no Sixes. This means if column 6 later becomes a junk pile, then we might have to start worrying about a shortage of Sevens. Therefore we might consider “fa,ha” instead. But this would be rather embarrassing if we turned a Nine.

Steve finds a good solution: noting that both Spade Nines are visible, he simply breaks the 9-8 of Spades in column 4 to form the “other” 9-8 of Spades in column 6. Now we get to have our anti-smoking song and sing it too since we will turn the last card in column 8 before committing ourselves to the move “fa”. Hence our best-so-far move is now “dh, df, hd”.

But Steve still isn’t done yet. He observes that he can shift some junk from column 4 to column 1. Since column 1 contains a King, we’re probably not turning over cards there any time soon. But there’s a chance we might get to shift the Q-Q-J-0 in column 4 if things go well. Admittedly this is a long shot, but no harm trying. In any case, Steve arrives at the following move sequence:

  • Move: dh, df, da, ha → Kh

In my opinion, Steve has done everything right here – except the final card in column 8 nullified all his efforts to obtain an empty column ☹ Steve observes that the King is not completely useless. At least there is an option of splitting the Queens in column 3.

Again, now is a good time to take stock. How would you assess our chances? More specifically, let’s pretend we are playing a mash-up of Spider Solitaire, Backgammon and Among Us and consider the following questions:

  • Would you rather be a crewmate or an impostor?
  • Should you double?
  • If you double then should your opponent(s) accept or refuse?

As usual, I have added a spoiler-blocker in case you wish to form your own conclusions before reading on.

This is the usual spoiler blocker for my blog.

Here is my analysis:

We have two turnovers, although one of them requires exposing an Ace and polluting column 10, our best chance of obtaining an empty column. We have fair chances of improving on our minimum guaranteed turnovers (I will leave computing the outs as an exercise for the reader).

It’s a pity we can’t quite perform the supermove “ah”. Despite Steve’s skilful play, there is a fundamental limit to how much awesomeness you can achieve when you don’t have an empty column! However, that is not a serious loss anyway. We are a long way from completing the Heart suit. We would almost certainly need an empty column before Hearts become a realistic possibility, but once we get an empty column there is a fair chance the K-Q-J-0 of Hearts will sort itself out anyway without us really trying or even noticing.

Although Steve has recovered somewhat after a poor start, I would rather be an Impostor than a Crewmate. But the difference is small and much will depend on the luck of the cards. I certainly wouldn’t be doubling the stakes any time soon (obviously I would expect an opponent to take a cube)

Steve Brown’s Game: Round 1(1)

Here is the start position, which also appears in my blog previous post.

Bart has kindly requested I redact the value of hidden cards to make it “more realistic” from the viewpoint of the player. Here is the game state after Steve reluctantly deals the first row of cards.

This is a dire state of affairs, much like our previous game involving a doubling cube and a similarly depressing round 1. I think an impostor Among-Us blob would be justified in sending a doubling cube over (assuming of course it is possible to double without revealing whose side you’re on!). We have only two guaranteed turnovers and most columns require more than one good card to get a turnover.

  • Move: gd,hd,ha → Jd

This is an interesting decision. The obvious option seems to be jg,jg which builds in-suit in Hearts and avoids exposing an Ace in Column 7. Moreover column 10 is one step closer to getting our first empty column. But Steve gives a valid reason for his play: column 8 is more difficult to turnover because we need a Jack and a King, whereas column 10 only requires a Jack because once we shift the Ten of Hearts, there is always the option of immediately shifting the Nine of Spades. Getting the more difficult task out of the way is a useful principle for expert play, and Steve shows good insight here.

However, if this were Among Us then I would vote “jg,jg” and let the whole world know that Steve is sus. Apart from the advantages listed, it also keeps some degree of flexibility. For instance, we keep the option of ca or da. At least Steve exposes a good card.

  • Move: da, hd → 7h
  • Move: ad, ja, jd → 8s
  • Move: jd → 2c

Although the next move is obvious, I wish to take stock and assess our chances. Our situation has improved quite a bit – we still have two turnovers and are getting closer to getting an empty column. We also have a small amount of flexibility (e.g. moves like fa,eh) and given our poor start we might need every advantage we can get. It’s a pity the 5-4-3 in Column 1 is buried under a rot13(xvat) but we can’t do much about that.

As an extra bonus, I get a chance to confirm that both IM Bug and IM Bart are both happy with the new format (gray question marks) before pushing forward.

NOTE: for inexperienced players, it is useful to observe how Steve is able to increase in-suit builds with “supermoves” despite the lack of an empty column.

I think it’s good practice to assess our game state regularly, even if the next move is obvious since it will improve your feel of how well or badly a game is going. If you’re willing to accept a Backgammon doubling cube centred at ‘2’ then your position isn’t that bad.

Well, that’s all folks and here’s looking forward to More Of The Same, coming soon to a place near u if you excuse the numerous terrible clichés!

Steve Brown’s Game: Round 0(1)

Here is the start position of Steve Brown’s game, which also appears in the previous post of this blog.

The observant reader has no doubt tried to assess the opening game state and concluded it’s worse than average. We have a run of length four (0987, mixed suits) and not much else. That’s three turnovers, which is less than average (just below 4). Given we also have two Aces showing, this definitely qualifies as a “bad three”.

With limited options available, the opening moves require little explanation:

Move: ea → Kc

Move: be → Ah

Move: af → 5s

Move: jf → 9s

The first interesting moment occurs after the fourth move. Steve explains he has a choice between “jf” (the actual play) and “da”. He chose jf because neither move was suited and the 7c is higher in rank than the 4d. Although Steve found the correct play, I don’t buy this explanation. The correct reason is that column 6 is already impure and there is no danger of losing a turnover if the next card is a Jack. Whereas “da” costs a turnover if the next card is a Six. Assuming no in-suit builds are possible, the higher-rank logic only applies when you have a full sequence like 9-8-7-6-5-4-3 rather than 9-8-K-K-K-4-3.

In general, when reading the entire book, I found that Steve sometimes struggles to articulate his thoughts properly and I’ve seen a number of strange typos such as “loses” instead of “losses”. Still, let us withhold judgment on Steve’s overall ability until the end of the game.

Move da → 3d

Move da → Qs

This completes a disappointing round 0. Steve mentions that on average he will expect to turnover 12 cards in round 0, which is exactly double the six turnovers he has in this hand. The sample size is small (306 games) but I can’t accuse Steve of not keeping careful records.

From my experience, the real game starts in round 1, not round 0. With 50 cards remaining in the stock, it’s almost impossible for a half-decent player to make a fonumental muck-up and Steve is well aware there is plenty of opportunity for a reversal of fortunes (in either direction). This hand is no exception if you pardon the terrible cliché. In Backgammon/Among Us terms, round 0 is equivalent to memorising the correct plays for opening rolls and replies and it’s extremely rare for the luck-o-meter(TM) to surpass the “refuse-doubling-cube” threshold from the viewpoint of a Crewmate or Impostor.

One thing I should mention: the stock is read from left-to-right. That means the next 10 cards will contain two more Queens (as if we don’t already have enough problems in this stupid world).

Spider GM is back!

It’s been a while since I’ve posted in my Spider blog. I wrote that something has turned up at work a while ago. But it’s been fixed. More importantly, I didn’t have to “put in a special effort” to get rot13(fuvg) done (e.g. threaten to embarrass somebody if nothing improved).

The something I was referring to was the Gawler Line. Essentially the trains are running again and I can travel to work without worrying about less-than-stellar traffic or risk of car-pooling accidents etc.

Going back to the Spider Solitaire, the next exercise I wish to discuss is a walkthrough of a game by Steve N Brown – the author of Spider Solitaire Winning Strategies. This book has fourteen chapters with the final chapter being a play-through of a sample hand. Presumably Steve chose this hand as an example of overcoming a difficult start. He could also have chosen an example of safeguarding a strong start – also a useful skill for the expert – but that would be less appealing to the general audience. In any case, the reader will have a “spoiler” in the sense that he already knows that Steve won this particular hand.

In the next few posts, I intend to critique the way he played from beginning to end. This should give some insight as to how an expert player (other than Yours Truly) goes about playing four-suits sans rot13(haqb).

With decent luck I should be able to manage one post per two or three days. For further good news, I also have an opportunity to show off my mad coding skills: the image above shows how one can input a position to Ninja Monkey’s improved algorithm with a GUI that looks half decent.

Villain’s Double in Round 1

In this post I will discuss why Villain doubled after dealing the cards in round 1.

The position in round 1 is this:

There is only one way to get two turnovers. The best play is “fg,bf,bc,jc,jb” giving the following position (I used MS Paintbrush instead of explicitly undoing moves in the Spider Solitaire program).

Now let us compute the chances of improving our minimum guaranteed turnovers after turning the card in Column 10. For simplicity assume each of the 13 ranks are equally likely. Let us assume each rank is worth one happy star if it yields a turnover, 0.5 happy stars if you need the correct suit to get the extra turnover and 0 happy stars if no turnovers regardless of suit. Multiple turnovers are possible. For instance, 1.5 happy stars means you always get one extra turnover, with a chance of a second turnover if you were allowed to call the suit, etc.

Note that if the next card in Column 10 were an Ace or Four then we don’t get an extra turnover since we counterfeited the turnover in column 9. It turns out the only good ranks are Three, Nine, Jack and Queen. That’s only 4 happy stars out of a possible 13. Actually, we can increase that to 4 and a half, since a Jack of Spades gives us a double turnover in columns 8 and 10 to go with our turnover in column 9. Given that we only start with two turnovers, it’s about an even chance we get bad cards in both column 9 and column 10.

The other piece of bad news is there are no “atomic columns”. Note that we were forced to pollute column 6 to guarantee our two turnovers in the first place. If we assign the capital letters A and B to columns 9 and 10 respectively, then it’s hard to find decent plans corresponding to the rest of the alphabet. This also means that we need to turnover every card in column 9 or 10, not just the first to get a fighting chance.

In a nutshell, we have difficult short-term problems and long-term problems to deal with. This is why Villain doubled. If I were in Hero’s shoes, I would have passed the double.

Match Summary (Alternative Version)

Once upon a time, there lived a Beaver in the Animal Kingdom.

The Beaver had just beat the highest difficulty level of Spider Solitaire – four suits sans undo. He felt he had played well after a difficult start, but it was hard to judge his overall ability at the game. After all, one wins and zero losses does not a large sample size make. And the fact none of his friends displayed any aptitude for the Royal Game certainly didn’t help. So, the Beaver decided to have a chat with his best friend, the Raccoon, who was known for his extensive knowledge of all things mathematics.

“It’s hard to judge your playing strength after one game,” said the Raccoon. “You need to play a large number of games to prove your victory wasn’t just beginner’s luck.”

“Suppose I played 129 games in a row,” replied the Beaver, plucking a three-digit number at random. “Then we can tally up my wins and losses and then we have a much better understanding of where I’m at.”

“Agreed,” replied the Raccoon. “Right now, the only thing we can agree on is you can play a hell of a lot better than I can.”

The Beaver chuckles, and he soon notices Captain Obvious is eager to join in the conversation.

“The only problem is it will take a long time to churn through 129 games,” says Captain Obvious. “Spider GM probably doesn’t wanna hear this but we all have better things to do in our lives than playing the Royal Game all day.”

“True,” says Raccoon. “Very True.”

Hang on, thinks the Raccoon. 129 happens to be a power of two plus one. This has me thinking – what if we can involve powers of two somehow? Let us say some games can be worth more than others. Suppose that each individual game was worth N victory points, where N was a power of two. A series of 129 games is equivalent to “First to 65 wins”. This should speed things up considerably. But Captain Obvious will gleefully point out Spider Solitaire is a game for one player, not two. Hang on (***thinks for a while***) I think I might have something.

“Okay I have an idea,” says Raccoon.

“What is it?” asks the Beaver and Captain Obvious simultaneously.

“Let us pretend Beaver is the protagonist,” says Raccoon. “Only Beaver can move any cards. I am the Antagonist and I am willing Beaver to lose.”

Using a stick, the Raccoon sketches a hypothetical cube with all powers of 2 between 1 and 32.

“Initially, each game is worth 1 Victory Point. If Beaver thinks he has a good position, then he can double the stakes. I must concede 1 VP or agree to play on for 2 VP. Similarly, if I think Beaver has a poor position then I can double the stakes and Beaver has the same choice of refusing or accepting.”

“Sounds interesting,” says Beaver. “But if my game state were really bad, can’t you just double the stakes after every move? That wouldn’t be very interesting”

“That is correct,” replies the Raccoon. “Therefore, I propose another rule: if either side doubles the stakes and the opponent accepts then the opponent has the exclusive right to make the next double.”

“So that means, if I get a poor position, you double, I accept, then I turn the game around, then I can redouble and play for four VP?”

“Quite correct,” replies the Raccoon.

“Wait a minute,” says Captain Obvious. “If first to 65 wins then is it possible to get more than 65 if the doubling cube is more than 1?”

“Yes,” replies the Raccoon. “It doesn’t matter if you’re above 65 or exactly equal to 65. And before you ask, it’s perfectly legit for someone to double near the end of the match regardless of the game state because the math says he has nothing to lose.”

“Just to touch base,” says the rot13(fzneg nff) as he gleefully pokes the rot13(nff) of Captain Obvious, “does that mean only Beaver can moves cards, but both Beaver and Raccoon participate in cube-decisions.”

“That’s correct,” says Raccoon. “Even though I don’t move any cards, I can still participate in evaluating the winning chances of a given game-state. Win-win for everybody since I get a chance to improve my game as well.

This idea proved quite successful, and soon Raccoon was discussing the implications of the doubling cube with his friends, many of whom were also avid mathematicians. They had independently discovered some interesting theory and concepts such as market losers, the Crawford Rule, Jacoby Paradox, Woolsey’s Law for Doubling and so on. Not surprisingly, much of this theory is well-known to expert Backgammon players today.

For the record, the Beaver managed to win 66-42, although that may have been a function of Raccoon’s limited understanding of the Royal Game (and hence sub-optimal decisions with the cube). At least it was a lot better than the 8-65 drubbing that Raccoon received when they reversed the roles of Protagonist/Antagonist. Initially the Raccoon thought the best equaliser for a mediocre player is to play each game at high stakes and hope to get lucky, even if the game state rot13(fhpxrq) since a long match would allow the antagonist to “grind” his way to victory. But the Beaver thought it was better to be aggressive with even marginal advantages – for instance if an intermediate player starts with six guaranteed turnovers or a “good five” then he should immediately double. Then at least he is fighting from a position of strength. If the protagonist thought his chances without a doubling cube were 50-50 then he is probably better off grinding and should hope to win on skill, not luck.

And the less said about Ninja Monkey’s first Match-to-65 and his infamous random move algorithm the better 😊

The End

Match Summary

Well, that didn’t last long. Hero got off to a bad start after Round 1 of Game 1 and decided to go all-in on a lousy hand, reasoning that if Hero’s chances of winning an individual game were less than 50%, then the chances of winning a long match would be rot13(fuvg) regardless.

I can see where Bart/Bug are coming from, but if the “protagonist” believes his chances of winning an individual game are less than 50% I think a better strategy is to be aggressive with even marginal advantages. For instance, Hero could insta-double if the initial game state allowed six guaranteed turnovers or a “good five”. That way, Hero would at least be fighting from a position of strength. If I had to play a 25-point match against Kit Woolsey or Paul Magriel, I would certainly consider a similar strategy – looking for any excuse to double from a position of strength. I would be less sure about accepting/refusing when my opponent doubles, but at least I would avoid any kamikaze redoubles unless the value of the cube is already enough to give opponent a win.

Bart has had a go at analysing the mathematics of an unbalanced match where Hero has, say, a 25% chance of winning a match to 5 and tries to equalise the match to some extent through judicious use of the cube. I will have more to say about this in a future post.

I don’t have much to say about the actual card-play. Hero had a reasonable position after round 0, managing to turn over every face-down card in column 7, albeit without getting the empty column. It’s hard to make bad decisions when we have very few face-up cards, no spaces, and all face-up cards are arranged in descending sequence. But after dealing a less-than-stellar 10 cards in round 1 the situation was already desperate. Rightly or wrongly, Villain immediately doubled the stakes and before we knew it, the number on the D-cube exceeded the number of points required for victory.

When you have a lousy position, the good news is it’s hard to make a mistake because your options are so limited. But we all know what the bad news is. Indeed, I couldn’t find any serious card-play mistakes by Bart or Bug for the entire game. Maybe Hero could have taken an extra turnover in round 3 at the expense of exposing two more Aces, but that’s always easier to say with hindsight. I think it was one of those games we were destined to lose. I’ve had similar games when playing with Microsoft Spider Solitaire and – dare I say it – I have no reason to believe this version is biased.

In the post-mortem, I will discuss in some detail why Villain doubled after we dealt the initial 10 cards in round 1. The TL;DR version basically says “I know from vast experience this is probably not gonna end well”.

Final position, Hero concedes the game and the match.

Match to Five Points, Game 1 Rounds 4 and 5

xxx 0c 4c Ah 2h As 2s/3h 5h/xxxxx Ad 5c 4s 3d 2s Ad 3s Ks 4d/xxxxx 4h 3h Kd Qd Jd 0h 9d 8h 7h 6s Kh Qh 2d/Kd 6s/xxxx 7d 6d 5h 4s As 8d 7d 6d 6h/7c 6c Kh Qh 0s/x Qd Jh 0h 9s 0s Ks Qs Js 0c 9c 8c 7c 6c 5c 4c 3c 2c Ac 8s/4d 3d 2d 8s 7h 8d/x Kc Qc 0d

Checksum: 9 + 2 +14 + 18 + 2 + 13 + 6 + 20 + 6 + 4 + (1*10) = 104

Actual play (6 April, score = 432): bf, bc, jb (deal)

Actual play (7 April, score = 428): ???

Spider GM Comments: Since the last few rounds have been quite short, I’ve decided to merge Rounds 4 and 5 together (and skip the Round 4 Summary) – but this does not look healthy at all ☹

Game Over

Hero concedes the game and the match. Villain wins 5-0.