Before continuing with the game, I wish to highlight the situation in column 4: we have two Queens but no Jacks. This means if we do not shift the Ten of Spades soon, there is always a long-term danger of a Queen shortage later in the game (if all other things are equal). This explains why I was keen to “balance” column 4 in the previous round by moving one of the Jacks onto the Queen of Spades. To make matters worse, both Queens in column 4 are the same suit, which means we don’t need much bad luck to reach a situation where completing a suit of Spades becomes really difficult. We would prefer the Queens to be different suits. This may all seem trivial but I’ve been on the wrong end of too many close games – so I know that little details can make a big difference in the long run.

At the start of round 3, I recommend you should start thinking about overall game plans – even before making obvious moves (assuming the game isn’t close to trivially won or trivially lost). No need to rush – your obvious moves will never run away from you, and I assume you weren’t playing to win in the fastest possible time. Example plans may be

• Pick a suit (e.g. clubs) and focus on moving that suit to the foundations.
• Turn over as many cards as possible and hope to have a clearer idea of how to proceed in round 4.
• Get as many in-suit builds as possible and hope to have a clearer idea of how to proceed in round 4.
• Some mixture of the above three plans.

With apologies to Kevin Rudd, plans don’t have to be spelled out in detailed programmatic specificity: nobody expects you to write a Ph. D. on how to maximise your winning chances for a specific game state. But you should have some rough idea of what you’re hoping to achieve.

Note that round 3 was dangerously close to a “auto-deal”. If the Four of Clubs in column 8 were (say) another Five then dealing a new row is literally the only legal move. At least we have two turnovers (counting one turnover for an empty column) and a couple of in-suit builds.

Unfortunately, we’re not close to completing any suit. Yes, we might establish the K-Q-J-0-9-8-7 of Diamonds – if we were willing to shift the Ks in column 1 and burn an empty column. Alas, the lower half of that suit doesn’t look promising. So, we have to play on general principles: continue working on turnovers, empty columns and in-suit builds. But also look out for unexpected ways to “change the flow of the game” if an opportunity presents itself.

Steve points out this deal brought out the last Queen as well as a pair of Aces. With only three Twos versus six Aces, Steve has good reason to be worried.

• Move: gh, jg,fj,fd → Ks

I beg pardon of the reader who was expecting more moves in a single post, but I thought it was worthwhile to talk about general long-term planning rather than individual moves. Unlike DJDJDDJKDK or the Wikipedia version of Spider Monkeys, I believe this digression can actually help improve your game 😊

After a brief period of joy, it is now time to bid farewell to our only hole and we must choose between a number of depressing options. Still, our game state isn’t exactly in Limbo – at least our prospects are much better than the start of Round 1.

There are several alternatives to choose from:

• Turning over column 6 will probably get the vote of Captain Obvious.
• We can obtain a turnover plus an in-suit build in column 3. This motif comes up extremely often in practice – and will no doubt be familiar to experienced players.
• Maybe it is better to turnover column 3, but forego the in-suit build. This is because we probably wish to shift the As in column 2 – but having an off-suit 2c-As in column 7 would be very undesirable since we would then need two good cards to recover an empty column.
• We can obtain a turnover plus in-suit build in column 2, but that would commit us to shifting the Ace of Spades – costing a turnover if we reveal an Ace.

The observant reader may have noticed we have forfeited the option of building in-suit with the Qs-Js. An interesting question to ponder is: if we were allowed to perform the supermove “hd”, would it be beneficial? We get an extra in-suit build at the expense of putting more junk on column 4. Since column 4 doesn’t contain a King maybe it is worth hoping for turnovers in that column if things go well.

However, a hidden danger is that if column 4 turns into a junk pile then there is a long-term danger of a Queen shortage. If we played “hd” (assuming it was legal) then it would be less of a problem since we would be burying a Jack along with two Queens. It’s a minor defect in our overall position, but you never know when the word “minor” turns out to be a typo and the correct spelling was F-U-N-D-A-M-E-N-T-A-L all along. Therefore, I would play “hd”, if it were legal.

Steve elects to turnover column 6. In hindsight, Steve thinks column 3 was better because there are more Threes unseen than Kings. Also, threats of one-hole-no-card are starting to percolate so turning over all cards in column 6 isn’t necessarily all it’s cracked up to be.

• Move: fg → 9c
• Move: be → deal

And while we’re here, we may as well milk maximum value from any possible Limbo references. Spider Solitaire is a game full of traps, where the player is often tricked into believing everything is going smoothly – until it isn’t. With that out of the way, here’s looking forward to Round 3!

Continuing our game, after getting an empty column

Steve plays

• Move: gf, gh, di, di, d2=h1, fh → Qd

Technically, Steve should have turned over either column 3 or column 6 since the empty column isn’t running any time soon. But that’s only a minor quibble. It’s hard to imagine taking the hole in column 7 and regretting it later –  we would need some ridiculous “parlay of events” to prove Steve’s play was a mistake. Constructing such a parlay is left as an exercise for the reader. A more serious concern is the failure to extract all the “safe” in-suit builds.

By safe, I mean reversible moves that don’t commit to anything (e.g. a move such as bc). In the diagram below I have highlighted two off-suit Q-J pairs. Note that if we could swap the Jacks then we build in-suit in Spades for free. In fact, a good habit to learn is to look for such opportunities as soon as you obtain your first empty column. It turns out swapping “d3=h6” is in fact possible and I was surprised Steve missed this. I will leave finding the correct moves as an exercise for the reader.

One may ask why Steve did not take the turnover in column 3 since that builds in-suit in Diamonds. I’m guessing Steve wants to keep the option of turning over the last card in column 10. In any case the Seven of Clubs is on the correct side of the supply-demand inequality between Sevens and Eights and we are one face-down card closer to a second empty column. There is little to choose between columns 3 and 6.

• Move: da, df, jc, je → 4s

Steve indeed turns over the last card in column 10. This shows good insight: If we don’t obtain a second empty column before the next deal, it is much easier to salvage a bad situation if we know the last card in column 10 is the Four of Spades. As a general principle, exposing the last face-down card in any column is worth more than an “average turnover” if all other things are equal (one important exception is when you are risking the dreaded one-hole-no-card scenario).

Still, exposing an extra Ace is less than ideal so it might be better to turn over column 3 after all. Note that this maintains the option of turning over column 10 (at the cost of our only empty column).

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

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.

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 😊

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.

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.

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)

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).