The Duplicate Spider Solitaire Club

“Minnie and her glasses did it again!” fumed Cy the Cygnet (*)

(*) Yes … I borrowed that idea from Frank Stewart’s excellent Bridge Columns

Minnie Mouse, the smallest member of the Duplicate Spider Solitaire club, wears second-hand bifocals that make her mix up same-colour suits, much to the chagrin of other players. Cy had been her chief victim.

“Now what?” I sighed. If I had a happy-face disc for every bad beat story someone told me then I swear I would never lose a game of Connect Four.

“The play had started well at my table. I had already turned over eight cards and I only needed one more good card to get an empty column.”

I nodded. Judging from the game state below, Cy hasn’t done anything majorly wrong yet.

duplicate_1

“Alas, the next card in column 8 was the other Ten of Diamonds,” continued Cy. “Column 1 didn’t yield anything useful either, a Four of Spades underneath the Ace of Clubs.”

In this hand there is a stipulation saying no cards to be dealt from the stock. I presume this is to help students improve by focusing on one concept at a time.

“Game over, +100.” I said. “How did Minnie go?”

“Minnie started the same way, but then she moved the Ten of HEARTS in column nine onto column 2.”

“Thinking it was the Ten of Diamonds,” I said.

“Minnie turned over a Nine of Clubs in column 9 and that was all she wrote, if you pardon the terrible cliché. It wasn’t even close.”

“I’m okay with terrible clichés,” I replied. “I use them time and time again.”

“Minnie’s play was wrong on two counts,” insisted Cy. “Not only did she misread the suits, but her goal was to expose as many cards as possible, not build sequences in suit.”

Actually Minnie’s play was correct. There are three guaranteed turnovers in columns 1,8,9 even if the worst possible cards turned up – provided the cards were played in proper order. Cy’s impulsive play meant that he was no longer guaranteed to turnover a card in column 9. If he shifts the Js-0h in column 9 first then the turnover in column 8 will not run away.

One might even make an argument of shifting the Ace in column 1 first. This “kills” column 5, but column 7 contains a suited 2-A. Therefore, we will only regret this move if we turned over two Threes (whereas we only need one King in order to regret shifting the Js-0h). The important point is Minnie’s play was better than Cy’s.

“Has anybody managed to expose all the cards for a single hand yet?” coos the Smart 65,83,83.

“Don’t ask,” replies the Dumb Bunny.

“Shush!” I say. “There are still animals playing.”

Duplicate Spider Solitaire is a fun variant, particularly for lousy players who never get close to winning a game at the highest difficulty level. Certain stipulations are also provided such as “score 10 points per turnover” or “do not deal any cards from the stock.” Therefore, if you get into a complete mess you can always hope your measly score is enough to beat the others players who must play the same lousy hands. You gain match points whenever you perform better than anyone else.

Unfortunately I am not aware of any existing Duplicate Spider Solitaire clubs anywhere in the real world. Perhaps some of my Bridge friends would know of one (or are willing to start one!). If so, then please leave a comment below 😊

The importance of move order (alternative version)

Yawn. Yawn. Yawn. Yawn. Yawn.

I could use a bit of sleep. It all started last night after the Bad Idea Bears suggested a long poker session with the usual suspects. After some thought I agreed, but only because they actually behaved well during the last week. One thing led to another and … anyways, you get the gist. Hopefully today won’t be too much of a disaster.

“Here is an interesting position,” I say. “What would be your play here?”

I pull out my i-Phone and show the position to my students. It’s a pity we don’t have whiteboards and chalk in the jungle.

pic1_08sep

The monkey takes out two decks of playing cards. After three minutes he is the first to offer an answer.

“I say it doesn’t matter what move we play. I’ve played 100 games thanks to my usual extremely-fast-metabolism and I estimate the winning chances are exactly zero”.

Groan.

“I believe we call this a self-fulfilling prophecy,” I reply. “Perhaps, if we thought that victory was actually possible and adjust our strategy accordingly then our chances would increase.”

Unfortunately most of the students are sympathising with the Monkey. After all, nobody in the animal kingdom has managed to beat the game at the four-suit level.

“Anyone else have a better opinion? How about you Mr Snail?”

“I need some more thinking time,” says the Wise Snail.

Hmmm … this lesson ain’t off to a great start. Not surprisingly, the Wise Snail is the slowest player in the Animal Kingdom. At least I will give him credit for being a better player than the Monkey since the Snail hasn’t lost 50 quintillion games in a row.

“The position isn’t that complicated,” I reply. “There are only 11 cards in play and 5 legal moves.”

“Yes, but with 11 cards in play we have 93 cards unseen.”

“But what’s that got to do with the Fundamental Theorem of Calculus?”

“Well, we know that in Freecell the chances of winning is exactly 100% or 0% assuming perfect play,” replies the Snail. “This is because all cards are exposed. In Spider, if we ever reach a game state with only 2 hidden cards then the winning chances must be 0%, 50% or 100%. With 3 hidden cards, the winning chances will be some number divided by three …”

“Three factorial is six,” says the Smart 65,83,83. “Some number divided by six.”

“Whatever,” continues the Wise Snail. “Similarly one can compute the exact winning chances for any number of face-down cards”.

“I see where you’re coming from,” I reply. “Unfortunately with 93 face down cards, there are 1.156 * 10^144 possible permutations if we ignore cards with identical suit and rank. We only have half an hour remaining in this lesson.”

The Wise Snail pulls a frowny face.

“I wanna flip a coin, since there are no in-suit builds,” offers the elephant. “Unfortunately there are 5 legal moves and we don’t have a coin with five sides.”

Okay, +1 for humour but not exactly the answer I was after.

“Four of Hearts onto the Five,” says Bad Idea Bear #1.

“Only three more good cards and we get an empty column!” adds Bad Idea Bear #2.

“We can eliminate some moves,” offers the Jaguar. “Moving either Eight onto the Nine is equivalent, so pretend there is only one Eight. We shouldn’t move a Four onto the Five since that means we only have two guaranteed turnovers, not three. Therefore it’s a choice between 9-8 or 6-5.”

“That’s good,” I say. “Finally we’re getting somewhere.”

“So we don’t need a 5-sided coin after all,” says the Monkey.

At least the monkey is paying attention this time and knows a thing or two about humour. The Smart 65,83,83 gives the Monkey an oh-so-polite wink.

The eagle remains silent. He knows the answer, but wants to give the other students a chance to contribute.

The lion raises his front paw. It’s always a pleasure to witness the insights of the lion, one of my better students.

“If we move 9-8,” roars the lion, “then assuming we turn over a bad card we have to choose 6-5 next. But if we start with 6-5 then we can choose between 5-4 or 9-8 later. 6-5 it is.”

This is a good insight, but not the answer I intended.

“Every player knows that building in-suit is more desirable than off-suit,” I say. “When we build off-suit then (at least in the first few moves) most of the time we are effectively losing an out, assuming our goal is to expose as many cards as possible.”

“For instance, if we move a Ten onto a Jack then a Queen is no longer a good card. There are a number of exceptions: for instance, moving a Queen onto a King does not lose an out for obvious reasons and if we have e.g. a Two and a pair of Threes then again we avoid losing an out. Once all the easy moves are exhausted we have to choose carefully.”

I briefly glance at my notes, just checking I have the right game state.

“We have three guaranteed turnovers with 9-8 and 6-5-4. For simplicity let us ignore the fact we have duplicate Fours and Eights. Clearly we won’t move the Four onto the Five as that will bring us down to two guaranteed turnovers. Well done to the Jaguar for spotting this. Hence the choice is between 9-8 and 6-5.”

“Let us pretend that we have to make two moves before exposing any face-down cards. For instance, we might move 9-8, then 6-5 then turn over the cards underneath the Five and Eight. Or we might move 6-5, then 5-4 then turn over the cards underneath the Four and Five.”

Uh oh. The Sloth is snoring. I think nothing of it: after all he’s not the sharpest tool in the jungle out there if you excuse the terrible cliché and/or mixed metaphor. In fact I don’t recall the last time he didn’t fall asleep.

“Observe that in the first case we have lost two outs since Tens and Sevens are not as good as before (even though they are still good). But in the second case we only lose one out (the Seven). Therefore the correct move is 6-5. Well done Lion!”

“Roughly speaking, making two moves before exposing face-down cards corresponds to a worst-case scenario when a useless card comes up (e.g. an Ace). If a decent card came up then we might reconsider. For instance, after moving 6-5 we might expose a Two and then we must choose between 5-4, 2-A or 9-8.”

The Eagle is desperately trying to suppress a chuckle. Something is out of character: my best student doesn’t exactly have a reputation for lame puns, knock-knock jokes or pranks.

“As a general rule,” I continue, “building a long off-suit sequence of cards means you generally have more safe moves before you start losing outs. For instance if you had 3-4-5-6-7 within the first ten cards then playing 7-6 loses an out, but then you can build 6-5-4-3 within the next three moves without losing any extra outs. Of course the fickle Spider gods might eventually give you an Eight and an empty column, and you find you are still unable to move the 7-6-5-4-3 onto the Eight –”

79,72,32,70,85,67,75.

I’ve just realised that EVERYBODY HAS FALLEN ASLEEP EXCEPT THE EAGLE. Maybe quitting my day job and teaching various animals how to play well at Spider Solitaire ain’t what’s it cracked up to be. Or perhaps my teaching skills need a bit of work. Or perhaps I should learn to say “NO” to the Bad Idea Bears whenever I have to teach the following day.

Now it is my turn to pull a frowny face.

THE END

The importance of move order

Every player knows that building in-suit is more desirable than off-suit. When we build off-suit then (at least in the first few moves) most of the time we are effectively “losing an out”, assuming our goal is to expose as many cards as possible. For instance, if we move a Ten onto a Jack then a Queen is no longer a “good card”. There are a number of exceptions: for instance moving a Queen onto a King does not lose an out for obvious reasons J and if we have e.g. a Two and a pair of Threes then again we avoid losing an out. Once all the “easy moves” are exhausted we have to choose carefully.

Consider the following position. What would be your play here?

pic1_08sep

We have three guaranteed turnovers with 9-8 and 6-5-4. For simplicity let us ignore the fact we have duplicate Fours and Eights. Clearly we won’t move the Four onto the Five as that will bring us down to two guaranteed turnovers. Hence the choice is between 9-8 and 6-5.

Let us pretend that we have to make two moves before exposing any face-down cards. For instance, we might move 9-8, then 6-5 then turn over the cards underneath the Five and Eight. Or we might move 6-5, then 5-4 then turn over the cards underneath the Four and Five.

Observe that in the first case we have lost two outs since Tens and Sevens are not as good as before (even though they are still good). But in the second case we only lose one out (the Seven). Therefore the correct move is 6-5. Well done if you chose this move.

Roughly speaking, making two moves before exposing face-down cards corresponds to a worst-case scenario when a useless card comes up (e.g. an Ace). If a decent card came up then we might reconsider. For instance, after moving 6-5 we might expose a Two and then we must choose between 5-4, 2-A or 9-8.

As a general rule, building a long off-suit sequence of cards means you generally have more “safe moves” before you start losing outs. For instance if you had 3-4-5-6-7 within the first ten cards then playing 7-6 loses an out, but then you can build 6-5-4-3 within the next three moves without losing any extra outs. Of course the fickle Spider gods might eventually give you an Eight and an empty column, when you are still unable to move the 7-6-5-4-3 onto the Eight – but that’s beyond the scope of this post.

Until next time, happy Spider Solitaire playing!

Building Complete Suits

One of the hallmarks of a winning Spider player is the ability to consistently clear at least one suit, even on difficult hands. Often players get caught up in the minutiae of trying to turn over as many cards as possible or to “tidy” things up by arranging suited builds. This is all well and good near the beginning but when you have several cards in play it’s time to think about building suits. This often requires “whole board thinking” and long term planning.

In easy or medium (1 or 2 suit) level, if a player turns over enough cards and gets and empty column or two then complete suits will take care of themselves. But this is not true at expert level. A good player should be thinking about building suits at virtually every stage of the game.

What happens if you get 1 or 2 empty columns, a few suited connectors scattered here and there but are never able to remove a complete suit onto the foundations? The following diagram should give you a pretty strong hint 😊

To clear a suit, two things must happen:

  • All 13 ranks of that suit must be visible
  • It must be possible to organise them into a single column.

 The first condition is easy to check, since it’s just an exercise in card-counting. The tricky bit is answering the second condition, assuming the first condition actually holds.

Here is a simple example, which you may recognise from my (admittedly lame) short story from a previous post.

We have already cleared the club suit and there are three empty columns. This game should be easily winnable, but we may as well use this example to illustrate the concept of building full suits. Every rank in the Heart suit is visible. We have K-Q-J-0 in column 2. The remaining cards are found in columns 3,4 and 9. With three empty columns it is not hard to verify the Hearts can be collected into a single column.

As a fun exercise, try to do it with less than three empty columns. The following table should give an estimate of your playing strength

If you can clear Hearts with Then
Three empty columns Well done
Two empty columns You are already above beginner level
One empty column You are probably an International Master
Zero empty columns or less Your name is probably Chuck Norris

If you wanna get really good at 4-Suit Spider, you should try to visualise what happens after clearing a suit. After all the aim of the game is to complete eight suits, not just one. But that’s a lesson for later. As usual, it’s best for a beginner player to focus on learning one thing at a time.

In practice, it is often wise to think about complete suits before all 13 ranks of a particular suit become visible. As an example, consider the following two diagrams and answer the questions below:

  • What is the difference between two diagrams?
  • Are they equivalent? That is, given one diagram can you reach the other?
  • Assume your next move is shifting the 10 of Hearts to an empty column. Which diagram would you prefer and why?

These diagrams are the same, except columns 8 and 9 have some cards switched. If we assume that each suited connector is worth 1 brownie point, both diagrams would score the same number of BP.

The difference of course is that in the second diagram we already have a run from K-Q-J-0-9-8 in spades. If, somehow, we get a run from Seven to Ace, then the difference between the two diagrams becomes manifest. It is true that we are a long way from getting 7-6-5-4-3-2-A in Spades, but there is no harm setting up the run from K-Q-J-0-9-8.

Those with an attention to detail might have noticed it took me 15 moves just to swap the Q-J-0-9 in columns 8 and 9 for some nebulous gain. But I recommend that the serious player should get into this habit of striving for perfection even at the cost of playing numerous moves and losing score. Once the player can get a decent win rate (e.g. 30% without 85,78,68,79) he can start to think about optimising score.

Many close games are lost because a player is stuck with a “twelve-suit” instead of a complete suit, and it is quite possible the loss can be blamed on poor planning at an early stage of the game.

I hope you found these lessons useful. If your Spider win rate has dramatically improved in the last three weeks, please leave a comment below 🙂

Endgame calculation

Hooray, You’ve finally made it to the endgame. After N consecutive losses for sufficiently large N, you have cleared two suits, there are few hidden cards and victory is in sight. Provided you draw the right card(s) that is.

Okay I lied. This is the continuation of a game where I felt cocky after 5 wins and started by dealing 20 cards from the stock instead of the usual 10

Endgame example

A little thought should show we are in some trouble. We have only one hidden column. A little thought shows we can turn over a card in column 1 but this is kadoban. If we draw a bad card then game over.

With only 10 face down cards, one can enumerate the possibilities and calculate the probability the next card will 83,85,67,75. For the purists among you, let me state at the outset that counting cards is not cheating. Counting cards is second nature for serious Bridge players and tile-tracking is even allowed in Scrabble. I’ve even heard of “professional” Spider Solitaire programs that does card-counting for you (but I’ve never played them).

Captain Obvious is not always right

However, before turning over column 1 that we should check for any other less-obvious options. Options include turning over columns 6 or 9 or completing a suit.

  • Column 6: clearly impossible, we would in fact need 3 empty columns
  • Column 9: Impossible because of the suit breaks (we can’t shift the 9-8-7-6-5-4-3-2-A despite having a free Ten).
  • Complete Clubs: It is usually impossible to complete a suit twice until all other suits are completed once and this is no exception.
  • Complete Diamonds: impossible: the Nine is not visible.
  • Complete Hearts: same as clubs
  • Complete Spades: Oh-so-close! If we had a spare Deuce we can expose the Spade Ace in column 9, then swap with the Club Ace in column 2 and Bob Brown is Julia Gillard’s uncle. Or something like that.

Now that we have verified that there are no options we can expose a new card in column 1.

You should find the missing cards are: A447990JKK, ignoring suits.

You would immediately notice these are probably not the first ten cards you would wish to start a game with, since there is only two turnovers. As a general principle, if FOO is a set of 10 cards you don’t wanna see at the start, then EVERYTHING – FOO is probably not a set of 104-10=94 cards you wanna see at the end. Similar principles apply if we had some number of face-down cards other than 10. We have completed two suits, have one empty column and we are still in trouble.

Let us assume we play <18><19>. Assuming columns are labelled 1234567890 from left to right, you should have no trouble working out what that means.  It turns out any Ace, Ten, Jack or King spells 76,79,83,69. Any Four Seven or Nine means we are still alive.

This is bad news. Five out of 10 cards guarantee a loss, and we haven’t verified that any of the other five guarantee a win. We estimate we are a serious favourite to lose this game.

Fighting For Scraps

Actually, there is another option we haven’t considered. In the right-most column we notice the possibility of replacing 6-5-8-7 with 8-7-6-5. There might be some advantage in doing so … or there might not. Notice that we must commit to 6-5-8-7 or 8-7-6-5 before exposing a card in the leftmost column.

You might also notice the possibility of building the Spade Five onto the Spade Six in column 7. Thus we can move the Five of Clubs off somehow and then move <17><19>. This is possible, but requires us to compromise our position in some way.

The difference between these options may seem small, but at the end of the day it might allow us to snatch victory from the jaws of defeat, if you excuse the numerous clichés. In the end, I decided the obvious option is the best option.

EXERCISE: Assuming best play, what are the chances of victory in the original diagram position? This is not an easy question and I do not claim to have an exact answer. Let me know your thoughts by leaving a comment!

And it’s a …

10 of Spades … 70,85,67,75!!!!!!

Lesson learnt – no more cockiness and no starting the game with 20 cards (except for teaching purposes)

Empty Columns (a.k.a. holes)

All Spider players know that empty columns (aka holes) are one of the most valuable commodities in the game. Any card can legally move onto a hole (not just Kings). Having a hole means so many extra options for manoeuvring cards. Of course, more options also imply more chance of making a sub-optimal play 😊 In fact, I believe the hallmark of a winning player is the ability to take maximum advantage of holes.

There are three main use cases for holes:

  • Turning over a new card
  • Moving a sequence that is not in-suit
  • Tidying cards so they are in-suit.
How would you play this position?

Examples of the three use cases are:

  • we can turn-over a new card in columns 4 or 5. Further thought shows that column 6 is also possible since the 6-5-4 in diamonds can fill the empty column and the Q can go on one of two kings. Similarly column 3 is another option.
  • We can also turnover column 1. A little thought that if we have at least one empty column any length-2 sequence can be shifted to a non-empty column regardless of suits. This is the simplest example of a supermove.
  • We can also swap the deuces in columns 1 and 8. This increases the number of in-suit builds (3-2 of clubs).

You should immediately notice that options 2 and 3 mean we improve our position for free since we keep the empty column. Therefore option 1 is the worst. It might be tempting to turn over Column 3 so we can get a straight flush in Diamonds but that is a serious error. Not only is the Six-high straight flush the second-weakest of all possible flushes in poker, but having a King in an empty column means we would be a long way from securing another empty column (we need a minimum of three good cards).

Option 3 is also completely safe because it is reversible, so an experienced player will make this move immediately. I am assuming we are only playing to win without regard to score (e.g. -1 penalty per move), otherwise more thinking would be required. Option 2 is the only way to turnover a card without using the hole.

Although option 1 is the worst, most of the time it is available as a fallback option. In other words a hole (usually) implies you always have at least 1 turnover.

EXERCISE: Assume you get 1 brownie point for every suited-connector (e.g. 6-5-4 in diamonds is worth 2 brownie points). From the diagram position above, what is the maximum brownie points you can get without losing the empty column or exposing any new cards?

Again I will use the happy-star method for avoiding the reader unintentionally reading spoilers. For those who don’t recall from a previous post: each happy star represents 1 point in a short story comp, and I have no idea if the judges docked 5 points for the protagonist’s terrible Dad joke.

Did I lose 5 points for a terrible Dad Joke?

Answer: we currently have 8 brownie points and get 2 more (3-2 of clubs and 9-8 of spades).

Well done if you answered correctly (or found an error in my counting). If you aspire to kick 65,82,83,69 at Spider Solitaire, finding opportunities to tidy up suits “for free” must become second nature.

That’s all for now, toodle pip and piddle too 🙂

Continuing from the previous post

We established that there are 5 guaranteed turnovers and we could, if we wanted to, obtain the following position if we ignored the identity of the newly-turned cards.

Obviously in practice we would never ignore the new turnovers, but this shows a “worst-case scenario” and is a useful benchmark to keep in mind (on second thoughts if every face-down card is a politician then perhaps we should be aiming for the fewest guaranteed turnovers).

Bad jokes aside, we can immediately tell that if any of the new cards is an ace we get an extra turnover. Similarly, a 3, 5 or Q is a good card. Further thought shows that 7 is good. Are there any other good cards?

Suppose we go back to the start position. Recall there were three Sixes but only two Sevens, so we could choose only two out of the three columns containing Sixes. One of the columns involves exposing a Queen and moving it onto a King. Aha! That means a Jack is also a good card.

I believe there are no other good cards (check this!) In summary: going back to the start position

  • if we turn any one of A,3,5,7,J,Q then we are guaranteed more than 5 turnovers.
  • If we turn any one of 2,4,6,8,9,0,K then we only have exactly 5 turnovers

Well done if you came to the same conclusion. Even better if you found an error in my analysis 😊

Recall that we had to choose a move before looking at the next card. The most obvious option seems to be column 7 which builds two suited connectors (5/6 in diamonds and 6/7 in clubs), and this does not cost any outs, assuming I haven’t missed anything.

Okay, I agree this analysis was probably a bit over the top for experienced players, but I think it’s useful for a newbie to get used to this kind of thinking. Of course there is more to Spider than counting guaranteed turnovers (especially when you are not close to being forced to deal another 10 cards). In some cases it is wise to “sacrifice” turnovers if we can gain some other advantage, such as getting an empty column or completing a suit. But that lesson comes later😊

Note that Cy the Cynic or Unlucky Louie (borrowing characters from Frank Stewart’s excellent Bridge website) would probably choose to turnover column 7 without doing any calculation – two easy suited connectors and a turnover can hardly be a serious error at this early stage. But I think the calculation exercise is useful because

  • we have some idea of whether the game is going well or badly.
  • You never know if some trick play can be used to gain an edge over the obvious play.

Here is a possible outcome after replacing politicians with actual cards. One can readily compute we get 2 more turnovers on top of our guaranteed 5.

EXERCISE: Start a new game by dealing 10 extra cards from the stock. Calculate the guaranteed turnovers and the probability that the first newly turned card will yield at least one extra turnover. To keep things simple, (i) assume you get to look at the new card before making your first move, (ii) any rank occurs with probably 1/13 (iii) when you see the new card you are allowed to call the suit (but not the rank).

That’s it for now. Next lesson: Empty columns (aka holes)