Here is the start of round 5 after Steve deals the last ten cards from the stock.

In some sense, the last round is the “simplest” to calculate. Since the stock is empty, there are no more possible surprises associated with 10 new cards appearing simultaneously rather than sequentially. Most of the time, an empty stock implies the winning chances are either very very good or very very bad. Whereas, in previous rounds we have to make do with general considerations and “fuzzy evaluations” such as “we have good winning chances but not enough to justify sending over a Backgammon doubling cube with the ‘2’ face up”.

Of course, one can argue the last round is “hardest” to calculate since there are so many cards. To play the last round well you will need a LOT of calculation. Steve correctly points out the ability to calculate is a necessary (but not sufficient!) condition to achieving a high win rate at the Royal Game.

Another peculiarity of the last round is that most of the time at least one full suit will be visible (or it has already been moved to the foundations). For instance, every card in Hearts can be found in column 1, 5 or 8. Obviously we wanna work out if we can guarantee removing at least one suit. Most of the time, once you are able to clear any complete suit, everything else will collapse.

In the next post I will discuss how Steve proceeds from here. You may wish to analyse this game state for yourself and come to your own conclusions. If you do so then I recommend you give only a “general plan” rather than specific moves. Also, stop as soon as you turn over any face-down card. Your answer would look something like the following: (this is a template only and does not apply to the current hand).

(1) obtain two empty columns (2) Tidy up so we have a complete Heart suit missing only the King (3) Dump the K of Spades into an empty column (4) Win back another empty column (5) Complete Hearts (6) burn all empty columns to turnover a card in column 8.

Continuing our game. Steve’s only possible turnover is in column 2, albeit at the cost of splitting Aces.

In some cases it may be wise to spurn a turnover “for the greater good”, but not today. There is a fair chance of two turnovers, and the alternatives ain’t so attractive anyway.

Steve is able to turn both face-down cards in Column 2. They are the Eight of Clubs and Two of Diamonds. The good news is he has recovered the empty column.

The bad news is – yes you guessed it – this brings the familiar one-hole-no-card. Steve elects to shift the C8:4s into the empty column. Not a great choice, but I’m not seeing anything significantly better. Personally I would be loath to expose another Ace. I would prefer to move “jg”, maintaining the status quo and hoping to make progress after the next round.

This situation (one-hole-no-card) does happen more than we would like to admit. Usually what happens is an empty column gives us at least many different options to choose from and each option will be worth “some fraction of a turnover”. For sake of argument, suppose that we have a free and open source Spider Solitaire engine called StockSpider, and that StockSpider evaluates every turnover is worth an equity of +1.0. Building in-suit is worth +0.2 (on average), exposing an Ace is worth -0.2, having an atomic column is worth +0.1 and so on. Given so many options, sheer numbers dictates that we should be able to get some value from our empty column despite no turnover. Maybe some minor achievements here and there will add up to +0.5 or +0.6. Basically, one-hole-no-card isn’t necessarily the end of the world as we know it.

The observant reader will have noticed Steve failed to build in-suit with ed before wasting the empty column. Presumably Steve wants to get as much junk off column 4, since that is clearly where our most likely source of turnovers is coming from. However, it was actually possible to play “e3=d1” before burning the hole (an exercise for the reader). This allows us to have our anti-smoking song and sing it too: we build in-suit without adding more junk to column 4.

The other point is that at this late stage of the game we should (ideally) be thinking about removing a complete suit rather than our next turnover. Unfortunately, we’re in a bit of a fix and I don’t see anything spectacular here. One bright spot is that with so few cards unseen, there is a decent chance of drawing the cards we need. For instance, we might draw X5X5XX53X3, where X represents any card, take our empty columns and win the game. A little bit of wishful thinking never hurt anybody!

Continuing from last time, we reached a critical position and I asked the reader how best to proceed:

It turns out we can remove the diamond suit. Steve points out the following

• Move: ca, ga, fh, fd, af, a4=f6, a9=c0, da, a2=d1, dg, df, af, ga, hf, aj, a3=c8, c2=e2, jc,
• Move: ag, ag, ha, hg (clear diamonds)

The result is shown below (I have pretended the Diamonds were not moved to the foundations for clarity). We can then proceed with a turnover in column 1.

Bart found pretty much the same sequence (not the exact moves, but clears Diamonds and turns over column 1). I will not present Bart’s exact move sequence.

Unfortunately, Steve did not play this. He writes that “either I did not see the plan or I did not like the after-play”. It’s obviously impossible to reverse engineer his thought processes, not to mention that Steve had to record the moves for over 300 games, so it’s not really practical to record the reasoning behind every move of every game (just recording moves is already a significant effort). But judging from his actual choice, I’m guessing Steve didn’t see the plan.

Steve instead turns over column 9. This exposes three Aces and forfeits the Diamond suit. Not to mention burning both empty columns just for one turnover. The only advantage of this plan is it gets a difficult task out of the way (turning column 9) while it’s still possible. Personally, I would be extremely reluctant to expose three aces and knock back a full suit of Diamonds. I would need several good excuses to justify that, and this isn’t even close.

The exact move sequence is not important and I leave it as the proverbial exercise for the reader to verify this position is reachable from the previous game state.

At least Steve draws a good card (the Two of Spades) getting back “one hole + one turnover”. The next card in column 1 would have been much worse (to avoid spoilers, I will not reveal what that card was), and Steve’s post-mortem analysis says the “superior play” would have actually cost him the game in practice.

Steve is able to extract all face-down cards in column 9. There are (in order), 9h, 2h, 2c. The exact move sequence is trivial and not given here.

To summarise, Steve has managed to extract three face-down Twos in a single column, that was buried by random junk including three “unmatched Aces”. A lucky break if there ever was one, but I believe Steve has demonstrated enough skill (his book contains several examples, not just the profiled game) to earn some good luck!

Steve mentions he now has a full set of Spades – and I won’t expect a player of his calibre to muck up a simple exercise in card counting. Unfortunately, it’s not possible to extract Spades. One reason (possibly not the only reason) is because of the duplicate Queens and Fives in the same column which I alluded to earlier.

Hearts are missing only the Nine. Clubs and Diamonds are only missing the Five. Unfortunately it’s not even possible to “almost-complete” (let alone complete) a suit. So it looks like more short-term planning for now and hope for the best. True, our chances of turning over more than two cards are not great (or four cards if we’re willing to split the Aces in column 2 – generally very desirable in Blackjack but not so much in Spider Solitaire).

The only bright spot for Steve is this game has no “antagonist” and no Backgammon doubling cube.

• Move: cf, ch → 5d
• Move: ea, ge, fi, gi, f1=h1, if, cg → 6d

What a difference a card makes! From experience, I find that if I “hang in tough for long enough” (this really should be the lyrics of some song but my general knowledge of pop music is atrocious), then drawing just a single “joker” (i.e. the best possible card) will provide the good guys with a real fighting chance. And Steve has done exactly that.

This is the first time Steve had two empty columns. Technically, Steve also had two empty columns at an earlier stage – if he was willing to perform some reversible moves (but he chose not to). In any case the empty columns were worth little more than “one turnover one option” where option basically means you get to answer a multiple choice question but there are no correct answers because all of them equally rot13(fhpx).

This is also the first time we can realistically think about long term planning. Unlike last time, we can extract real value from two empty columns. We get several in-suit builds for free and every chance of pulling things around after a difficult start.

How would you continue here? This is a critical point of the hand so I recommend you take your time on this one.

We now have two empty columns for the first time. Obviously I’m counting an empty column if it can be obtained with reversible moves only since Steve is only playing to win, regardless of the number of moves required.

Usually, two or more empty columns mean we get to tidy in-suit builds without committing ourselves to any irreversible moves. For instance, if we had two long sequences such as K-Q-J-0-9-8 and K-Q-J-0-9-8-7-6 in different columns then chances are you will be able to swap two cards of the same rank (such as both Jacks) and increase the number of in-suit builds for free. Unfortunately, we don’t get to do that here. Ergo, the advantages of two empty columns boil down to guaranteeing two turnovers (plus the knowledge these columns never contain face-down cards for the remainder of the game).

The obvious (and correct) plan is to turnover column 2. This avoids exposing a new Ace in column 3 and also allows us to shift the C1:5h if desired. Unfortunately, we don’t get any new in-suit builds since we have “duplicated” the Ks-Qd-Jd. At least our option of turning over column 3 implies we have some leeway before the dreaded “one-hole-no-card” scenario.

• Move: gf, bg, bf → As
• Move: af, jf, cj, ch → Js
• Move: ce → 3h, deal

Note that Steve chose to deal immediately after turning over the Three of Hearts. Normally, we get to make some “final tidies” once there are no more turnovers available, but not today (actually, it is probably a good idea to play hf, to have the majority of the Diamond suit contained in a single column. From experience, I find this does come in useful at the long run).

We only got three turnovers from our two empty columns. Steve avoided dumping the Ace in column 2 into an empty column because there is already an Ace in column 7. This “diversification” increases our chances of recovering an empty column in the next deal. On the other hand, Steve exposes the Ace of diamonds. This is tolerable since (i) we have two exposed Threes, (ii) we desperately need turnovers in column 3 to avoid one-hole-no-card (iii) after the next deal, the Ace of Diamonds will be covered anyway.

The last point is worth remembering: whenever you expose an Ace, you are “forgiven” to some extent as soon as a new row is dealt (LINK). As a corollary, you can afford to expose more Aces if chances are you will soon be forced to deal a new row of cards.

Steve criticises his choice of moving the 2-A of diamonds to column 6 instead of column 8. Unfortunately, Steve has only recorded his moves but not the logic behind them, and it would be difficult for Steve to reverse engineer this logic (especially considering he did the same for over 300 games). Steve can only say that it is quite possible that he did not notice column 8 is missing a Five.

Finally, Steve concludes he is unhappy with his prospects. He has 18 face-down cards, but of the 149 victories (in 306 games) he had only 13 face-down cards at the end of round 3. I agree the game state is poor. If I were an Impostor, I would be salivating at the sight of a Backgammon Doubling cube – but probably in private, otherwise I would be rot13(pbzcyrgryl fperjrq) as soon as someone calls an emergency meeting.

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 😊