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 😊
IM Bartacus and IM Bug had successfully beat the four-suit version of the Royal Game with the help of a few expert friends. Some of the advice was good, some not so good, but eventually they managed to remove all eight suits, albeit with some difficulty.
Meanwhile, Ninja Monkey had tested is new improved algorithm and reported a win rate of 6% inside a sanitised environment with Spider GM overseeing his every move. Now was the time to play with the big boys and see what it was really like.
Unfortunately, the first game did not get off to a good start. No sooner had the game started, Ninja Monkey was immediately ejected from the playing hall.
“What are those things?” says Ninja Monkey.
“Ngrmmph” replies Spider GM with the demeanour of a Scrabble player mega-tilting after picking up way too many consonants against a weak opponent.
“Eeeek!!! Monkey don’t understand Ngrmmph!!!”
“They are among my best and brightest students – but also the rudest” growled Spider GM.
“Tell me about it!”
“Rot13(Svpxyr nf shpx). One day Orange will play at GM strength. The next time Orange will play like a rank beginner and pin the blame on Dark Green whenever something goes wrong. Then everybody starts arguing for the better half of a minute. I don’t remember the last time somebody didn’t end up in detention!”
“This is the position when I got ejected from the playing group”, says Monkey.
“Okay, I see what happened”, says Spider GM. “You correctly calculated the minimum guaranteed evaluation score to be 61. That’s assuming 10 points for a turnover and 1 point for an in-suit build. In the worst case scenario we get six turnovers and one in-suit build guaranteed: even my Dad can do the math.”
Ninja Monkey nods in agreement.
“Now there are several ways to get 61. You can start with id, db, ec, eg or ei”. Any of those moves allows you to get 61, even if you turned over six Kings. Therefore any moveblock starting with the correct first move would score the maximum-minimum-guaranteed-score if you will.”
“Of course we should start with ib” replies Captain Obvious. “If our very first move is an in-suit build then we never lose any guaranteed turnovers.”
“Shifting the Five of Spades is entirely reasonable, since we have three Sixes” replies the Wise Snail, the world’s slowest player and Monkey’s best friend.
“When I ran my algorithm a second time,” says Monkey, “I indeed got the move ec”.
“Note that if we did get six Kings, all these opening moves are equally good,” says Spider GM. “But under normal circumstances, db is clearly bad because we lose a turnover if we expose a Nine. Monkey’s algorithm only considers the worst-case scenario when all cards turned over are bad”.
“Maybe you can think of a way to improve Monkey’s algorithm even further so it doesn’t start with moves like db”, says Captain Obvious.
“True,” replies Spider GM. “Unfortunately, even I have my limits, and I wanna enlist the help of some other friends – who either speak the Monkey’s language, or know something about the Royal Game, or preferably both.”
“The fact I win 6% of the time does mean looking ahead and calculating the consequences of bad cards is more important than getting the opening moves right” says Ninja Monkey.
“I agree you’ve come a long way since you first started Spider Solitaire with your famous random move algorithm”, replies Spider GM.
Spider GM leads his students towards a seedy-looking venue with a sign saying “Crazy House”.
“We call this place Git Hub” says Spider GM.
“Rot13(V jnf ubcvat sbe cbea uho)” moaned the Bad Idea Bears and rot13(Yhpl Gur Fyhg) in perfect unison.
“It’s a place for gits, pricks, dorks and old farts playing Bingo,” says Spider GM, “and the occasional unkempt computer nerd(s). A lot of us stay up late at night, against our collective better judgment. When you enter, you have no idea who you might bump into. Could be anybody from around the world …”
Confused looks from everyone else.
“I know it’s complicated but I’m hoping to find some really smart people among the nerds. Hopefully they will have some idea of how to play the Royal Game, or how to communicate with the Monkey to help him improve his win rate”
“Not sure if monkey like this!” squeals the Monkey
“Can’t be worse than those horrible coloured Blobs,” replies Spider GM.
“True,” says Monkey “Always true! Spider GM is always true!”
Ninja Monkey leaps into the air and into the arms of Spider GM. He cradles the monkey and assures everything will be okay …
Spider GM and his students enter the Git Hub. Meanwhile, Rot13(Yhpl Gur Fyhg) is looking rather bored.
Julie sighed audibly. This was the 100th game she had lost in a row. On 57 occasions Simon called a card and it was Spiderwang. On the other 43 occasions, Julie called a losing card and the host declared it was Not Spiderwang. Yes, Julie could rot13(xvpx Fvzba’f nefr) at regular Spider Solitaire and a game with more dependency on luck was required to prevent things becoming boring. But a game show with unexplained rules was taking things too far. Julie had to reluctantly consider the possibility that Simon might be (gasp!) cheating with the cooperation of the host.
Julie discussed her predicament with her trusted group of Very Smart Friends. They eventually agreed on a plan. On the next few episodes her VSF would record the footage, take meticulous notes and work out the pattern. Once the pattern was figured out Julie could at least hope to fight on equal terms.
After burning much midnight oil, a pattern was found. Webb would declare Spiderwang whenever every card of a single suit was named at least once. Webb would declare Not Spiderwang if any single card was named for the third time.
Eventually some Math Ph. D.’s decided to join in the fun. They worked out that if nobody named the same card thrice and the probability of a new card being named was double the probability of a card already named once then Spiderwang should occur after 66.5 cards on average. The median was 67.0. The theoretical minimum was obviously 13 cards, and if e.g. no Kings were named within the first 96 cards we would achieve the maximum of 97 cards. On the last hand, Simon achieved Spiderwang after 60 cards, which corresponded to the 23rd percentile.
On the next day Julie is pleased when the host announces that Simon is going first. Her group of Very Smart Friends have done the math and with best play she would win her first game after 90 cards.
Julie takes out some pen and paper for round 1. Curiously neither Simon or Webb object to her taking copious notes during the game. Then again, nothing on this game show made much sense to her anyway.
Simon: Jack of Spades
Julie: Four of Diamonds
Simon: Queen of Clubs
Julie: Jack of Spades
Simon: Jack of Spades
Rot13(jung gur shpx) – why would Simon throw away round 1 just like that? Of course, it takes less than three nano-seconds for the host to confirm her worst fears.
Hester(*) was no stranger to staying up late and battling through miserable positions against her better judgment. But even she would have conceded the current game if tomorrow wasn’t a public holiday – or if an “opponent” were able to send over a Backgammon doubling cube with the “2” face-up at the most inconvenient moment.
With apologies to Nathaniel Hawthorne
Hester vividly recalled her first lesson from Arthur: Arthur drew a large scarlet letter “A” on a whiteboard and explained in no uncertain terms that Aces were not your friends. The colour scarlet signified danger – a fact even Pearl would know. Nothing could be moved onto an Ace. One or two Aces did not seem like much, but she knew from experience having too many of the rot13(saqbref) exposed meant limited options in the middlegame. Although Arthur had repeatedly drummed into her head that Aces were even worse than Kings, Hester had to learn this the hard way – losing 500 games in a row on a Spider Solitaire server that would later be shown to be biased.
Every time Hester exposed an Ace, a little voice at the back of her head would remind her that “A” stands for “AAAAAAAARRRRRGGGHHH!!!!”
Eventually Hester realised that Aces were sometimes useful to complete a suit. How annoying it would be when she followed Arthur’s teachings only to end up with a lovely run from King to Deuce in a single suit – with the missing Ace hopelessly buried under some random pile of rot13(fuvg). The secret was to plan ahead for the right Ace, before it was hopelessly buried under some random pile of rot13(fuvg). If the Ace was covered by only one other card, her options wouldn’t be so limited – and she would avoid the random pile of rot13(fuvg) problem. As Hester’s play improved, she would realise that “A” stands for Awesome. After many further months of self-study, Hester had even surpassed her tutor.
On the next day, Hester received the sad news: Arthur – the club champion in “A” Division, and Roger – her Autistic partner who introduced her to Spider Solitaire, had both passed.
So said Hester, and glanced her sad eyes downward at the scarlet letter. And, after many, many years, a new grave was delved, near an old and sunken one, in that burial-ground beside which King’s Chapel has since been built. It was near that old and sunken grave, yet with a space between, as if the dust of the two sleepers had no right to mingle. Yet one tomb-stone served for both. All around, there were monuments carved with armorial bearings; and on this simple slab of slate – as the curious investigator may still discern, and perplex himself with the purport – there appeared the semblance of an engraved escutcheon. It bore a device, a herald’s wording of which may serve for a motto and brief description of our now concluded legend; so sombre is it, and relieved only by one ever-glowing point of light gloomier than the shadow:
“ON A FIELD, SABLE, THE LETTER A, GULES”
The End.
(*) Surnames have been withheld to maintain confidentiality
Okay, so after three weeks it seems that most of my readers/followers have some experience in writing short stories but none of them have any experience in playing Spider Solitaire. Nobody has left a single comment so far. So maybe my next project should be to play a game of Spider to the best of my ability, then pretend my move sequence is the plot of a really lame story, told from the viewpoint of one of the cards in the tableau.
THE GREEN SCREEN
So here we are in the Green Screen (for lack of better
name). Our task is to somehow escape from this simulated reality and teleport
back into the real world as physical objects. Perhaps we have to follow a white
rabbit, enter a night club, meet a hot 67,72,73,67,75 named Trinity and take
things from there – which would be insanely cool. Or I could be completely
wrong.
Ten of us have been chosen as the “Starting Hand”, whatever that means. I don’t know which ten they are, mainly because I am still asleep. I don’t even know my name because my memory has been erased. All I know is I am rectangle-shaped and my back is blue. I am in the Fourth Column, covered by one other rectangle-shaped thing. There are 54 of us, and we have to somehow escape.
Ever heard of The Maze Runner by James Dashner? Or the movie of the same name directed by Wes Ball? The protagonist, 16 year old Thomas, wakes up in some intricate maze along with several other boys. He has to work out his role in their society (as does everyone else), work out the rules of the maze as they go along, solve a number of puzzles … only to find out once they escape there are more challenges ahead (it’s a trilogy after all so what do you expect?). Okay, that’s probably not the greatest analogy but it’s the best I can come up with right now. At least I am not a Slopper and there are no toilets to clean every day.
Apparently my name is Queen Of Hearts. Goodie – at least I’m
a human. Most of my friends are numbers. 83,85,67,75,83 to be them. I’ve moved
onto the Diamond King. It’s not the same suit, but at least I have someone to
call a friend. I no longer feel alone. It also means I am further away from the
dreaded Queen of Spades. She’s always grumbling about too much demand for
something and not enough supply. Or the other way round. Whatever. I was never
any good with Economics at School. I would later learn it’s not just her –
everyone here often grumbles about cards of the same rank for some strange
reason. <sarcasm> That’s what friends are for right? </sarcasm>
These guys are weird.
Okay I get it. Cards arrange themselves in descending order.
At least that explains why the Kings and Jacks seem to treat me with respect.
All the other spot-cards don’t seem to care much about me. I watch as the other
cards gracefully dance around the tableau. Apparently they know the rules
better than I do.
Just for fun, I try to scamper across to the adjacent column, onto the Ace of Diamonds. No – the laws of physics don’t like that, and I am immediately whizzed back to my original position. Rules are rules.
What the 70,85,67,75 was that?!?!?!? Ten new cards just
popped out of nowhere! So that’s worse than I thought: if we don’t escape from
this contraption then ten new cards will periodically appear every 5 minutes or
so and eventually the whole place will get flooded – it’s probably not
something I’d wanna think about … and I’ve just been covered by the King of
Spades. I’ve heard plenty of bad stuff about them. They tend to appear at the
worst possible moments. They sit on your face and stay there forever – or at
least it feels like forever. I don’t mind the kings so much, but I much rather
have their 65,82,83,69 next to me instead of on my face. At least there are no
monsters in this world, unlike The Maze
Runner.
Oh, and I’ve just noticed my twin sister has appeared in
column 1. It then occurs to me: there are TWO decks of playing cards. I was
expecting something like a single deck plus two jokers, and once the jokers
turn up then Good Things Will Happen. Jokers can move onto any card, and any
card can play onto a joker. But apparently they don’t exist in this Green
Screen. At least empty columns seem to be useful: any card can use them, not
just a king. That’s almost as good as a joker.
I scan the tableau. There are two-dozen cards yet to be
turned. They don’t know the laws of physics that govern the Green Screen. They
don’t even know their own name. The exposed cards are dancing around, apparently
making no effort to free them. Ten more cards will magically appear on the
tableau every 5 minutes or so. At this rate, they will never see the light of
day, if you pardon the cliché. I was one of the luckier ones, having started
near the top of a column. I was able to observe most of the proceedings so far.
At least I have some idea of what’s going on.
“Stop moving around aimlessly!” I yell.
“We’re not moving around aimlessly!” said the Five, Four,
Three, Two and Ace of Diamonds, all in unison.
“We need to come up with an overall grand plan.” I said “We
need to consider the state of the whole board, not just a single col-”
“We’ll sort ourselves into suits first” said the Nine,
Eight, Seven, Six, Five and Four of Spades also in unison. “Then we can take it
from there”.
I watch as they gracefully leap from an empty column onto
the Ten of Hearts. It now occurs to me why the cards were “wasting time”
organising themselves into suits. In the early rounds, cards were able to move
only one at a time; maybe two or three if we were lucky. In mistaking the trees
for the forest, I succeeded in missing the details. I feel stupid.
Still, I think that there is some truth about “thinking
about the bigger picture”. I start to wonder, what happens if we get a complete
run from Ace to King, all in the same suit. I’m a bit rusty at Texas Holdem but
I believe that is called a straight flush. The Club suit is looking good.
YOWZA!!!! All the Clubs have escaped! A triumphant C major chord pierces the dreary silence and fills us all with hope. Three suits to go, this should be easy as … no wait a minute. I’m still covered by the Jack of Clubs. My twin sister is still sitting in column 1. I remind myself there are two decks in this game. And we are still covered by a King. The King of Spades can move into an empty column, but for some reason he seems reluctant to do so. I guess we just have to wait then.
Okay, I get it now. Two decks of cards make 104. We started
with 54. Every now and again, 10 new cards magically appear out of nowhere.
After 50 cards are added, we have the right number to complete two full decks. Without
two full decks, there is no way we can complete suits from Ace to King. So
those cards appearing out of thin air are a blessings in disguise, if you
pardon the cliché. I’ve finally figured this out. Who needs jokers when you’ve
got the smarts like me?
Of course if more than 104 cards appear in play then we are REALLY 83,67,62,69,87,69,68.
“Off with the hearts!” I yell.
Everyone looks at me in disbelief. This isn’t the right time
for a lousy pun.
“We have a complete suit from Ace to King”, I continue.
“King-Queen-Jack-Ten in the second column, Nine-Eight in the right …”
“I’m not sure if that will work” said the Three of Clubs.
“Even if it did work, it will cost three empty columns just to reach the 7 of
Hearts in column 9”.
“There’s no choice. We’re gonna use up at least two columns
to expose a card”, said the ace of spades.
“Even if it doesn’t work”, I say, “we still get to partially
tidy up that mess in column 9 which is worth something. Spider is not all about
turning over as many cards as possible”.
Yes, I just contradicted myself about earlier feeling sorry
for the two-dozen face down cards that don’t know the laws of physics that
govern the Green Screen. I get that.
The Nine of Spades leaps from column 2 into an empty column,
taking the 8-7-6-5 with him. He is clearly eager for the Nine of Hearts in
column 9 to take his spot on top of the K-Q-J-T of the same suit.
“Not so fast” says the Nine of Hearts. “The Spider Grand
Master does not look kindly on 85,78,68,79,73,78,71 moves”.
“Who is this Spider GM you speak of?” I ask.
I shudder at the thought that we are being controlled by
some “higher being” and we are pawns in a bigger chess game (or cards in a
solitaire game if you wanna take things literally). Then again, if there is a
higher being who is a GM at Spider Solitaire and he is playing to the best of
his ability then that can only increase our chances of winning. So perhaps we
shouldn’t be complaining.
The Jacks, Queens and Kings engage in a long discussion.
This is a critical point in the hand. Make the wrong decision and we are
trapped forever. All the little cards shy away from discussion: they are unable
to visualise a long complex sequence of moves. They know full well it is better
to remain silent and be thought a fool than – well you know the rest of the
cliché!
“Okay, I’ve got this”, I say. “Queen of spades goes to King.
Rearrange cards so that we have Nine through Ace in column 9. Clear the hearts,
dump the six of diamonds to an empty column. We still have one empty column
left. Swap Ace of Spades with Ace of Diamonds, move the 3-2-A onto the 4 of
spades, four of hearts onto 5 of diamonds … Oh 67,82,65,80, we don’t have an
eight.”
“Yes we do”, says the Eight of Clubs in column 2.
Of course we do. We cleared the hearts. Duh.
We execute the plan. Fortunately we have visualised everything correctly and things go as expected. I no longer feel stupid. I farewell my twin sister as her suit gracefully whizzes to the bottom left of the Green Screen. Only six more triumphant C-major chords to go and we are done. Looks like we got this!
As expected there are no further difficulties and victory is
a mere formality. Just for fun I do a little endgame calculation. There are ten
cards missing. As long as nothing stupid happens like all even cards on the
last deal …
Clubs = A4 Diamonds = 67TJ, Hearts = 2, Spades = 3JK. Yep, this is a lock.
“So what happens after we win?” asks the Eight of Clubs.
“Well, we go back to the physical world as plastic cards”, I
reply.
“Would we become inanimate objects?”, asks the Nine of Spades. “Would we lose the ability to talk to each other and move around according to certain rules”
“Perhaps,” the Three of clubs says, “we would be fondled by
grumpy old computer-illiterate farts in a retirement village who only know how
to play Klondike.”
Always the cheery one,
I think to myself. Maybe going back to the real world ain’t what it’s cracked
up to be after all.
“Or perhaps,” adds the Two of Diamonds, “one of us gets a
68,73,67,75 pic after a bad beat in Texas Holdem.”
The Seven of Hearts gives the Two of Diamonds an
oh-so-polite wink. No card higher than a Nine is amused.
I rest my case.
At this very moment, the last ten cards magically appear
onto the tableau. We easily clear the remaining suits and win the game.
Some lame music plays and two pills immediately appear on the table. I’m supposed to choose one and swallow it. The Orange pill means we all stay in the Green Screen. The Blue one means we go back to the real world as inanimate objects. Both options 83,85,67,75.
I hold one pill in each hand and recite to myself: “Eenie meanie minie moo, smoking very bad for you, drinking is bad for you too, eenie meanie … ah 70,85,67,75 it”.
In one last act of defiance, I swallow both pills
simultaneously. Hah!, bet they didn’t think of that did they? A burning
sensation sizzles my tongue, and I feel ill. I feel the system crashing about
me, as I teleport to God-knows-where. The colour drains from the Green Screen
and I throw up. I believe it’s called a technicolour yawn except it looks more
like 50 Shades Of Gray. Okay, I probably shouldn’t have done that.
THE END … or perhaps not?
OKAY THAT PROBABLY DIDN’T WORK
So there you have it. There’s probably a reason or three why I haven’t won any meaningful short story competition yet. But at least I had a go. Do you guys think I have potential as a budding short story writer? Or should I stick to just playing Spider Solitaire to the best of my ability, and leave the writing to the Short Story GM’s? On second thoughts, if I can improve your win rate at Four-Suit Spider by a substantial amount then I don’t care how lousy my short story is bwahhahahahahahahaah 🙂
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.
Okay, so I ascii2word([70,85,67,75,69,68]) up. Apparently WordPress automatically converts xx-xx-xx-xx to hyperlinks (e.g. thinking it represents an 8-digit phone number). So instead of writing xx-xx-xx-xx I shall use the notation ascii2word([xx,xx,xx,xx]) instead.
EDIT: This is only relevant for mobile phone devices
By this stage the impatient reader probably wants to see some “action”. Here is a possible starting hand in Spider:
Let us try to find the best move in this position.
I recommend that a beginner player should start by asking the following questions: (i) how many cards are we guaranteed to turn over even if the worst possible cards turned up? Another useful question is (ii) What are the chances that the first new card turned over will be “good”? We will take good to mean “increasing the number of guaranteed turnovers”. Of course there is more to Spider Solitaire than counting guaranteed turnovers but if you’re a beginner then simplicity is the mother of self-improvement … or something like that.
Strictly speaking, it isn’t necessary to ask these questions
to arrive at a good first move in the start of the game. If the first two
columns are 3-4 of Hearts, then you could move the 3 onto the 4 regardless of
the other eight columns: if it’s not the best move then the difference is
small. But these questions will be good practice, and it will come in handy as
the game progresses.
I hope you answered “Four cards” for the first question. Ignoring suits for now, we have J-0-9 for two turnovers, 7-6 for a third turnover and finally 4-3 for the fourth. Obviously we can’t count multiple turnovers for the three Sixes since we can’t stack them onto the same Seven without violating the laws of physics! Similarly, we only count one turnover for two Nines. Assuming we don’t ascii2word([70,85,67,75]) up the move order, we will turn over at least four cards before being forced to deal another row.
For the second question, there are 13 possibilities for the
next exposed card (if we ignore suits). An Ace is clearly useless since we have
no deuces, but a deuce I’d like to see since we have a three … okay that’s
probably not the best way to start a rap song, but you get the gist.
Continuing in this fashion we get the following good cards: 25780Q. The chances of getting a good card is therefore 6/13. Note that 5 and 8 are especially good since we get two new cards instead of one. But the question defined good as “allowing at least one extra turnover” and it didn’t ask for “how good”. Assuming you have completed Year 3 or better in school, you should know by now that it is always wise to make sure you are answering the correct question! The observant reader may have noticed an error (okay, maybe one-and-a-half errors) in the above calculus. Before proceeding further I invite the reader to figure it out. To protect against accidentally reading spoilers I have inserted an image consisting of happy stars and blank spaces. Each happy star represents a point I obtained for a short story competition I entered some time last year, with a maximum score of 100. Unfortunately I didn’t win anything, not even a Honorable Mention. Perhaps the judges secretly docked 5 happy stars for the protagonist’s terrible Dad joke but we’ll never know. ascii2word([70,85,67,75])!!!!!
The first error is I have assumed each of the 13 cards from
Ace to King occur with equal probability. This is not correct since we already
know e.g. there are three Sixes and no Fives visible. Hence Fives are much more
likely than Sixes. The probability of 6/13 is therefore only an approximation
of the true probability of getting a good card. As a general rule, failing to
take into account cards already exposed will almost always underestimate the
true probability at the start of the game. With only 10 cards exposed, this
error will probably not contribute much to All The Problems In The World As We
Know It.
The other half-error is we must choose our move before seeing the next card and this may
affect our chances for the worse. For instance, suppose we move the Ten in
column 8 onto the Jack in column 5. Any Queen is no longer a good card unless
the 10 and Jack are the same suit. Fortunately we are in luck here since they
are both diamonds. This is why I only counted 1 and a half errors instead of 2.
Clearly, moving the Ten onto the Jack is good because we “don’t lose any
outs”.
Note that if we moved the Three onto the Four we don’t lose any good cards despite it being off-suit, since if we draw a Five it can still be played onto a Six. But obviously we want a Five to be “very good” (two new cards) instead of “just good” (only one card). This might sound overly technical, but this kind of deduction must become second nature if you aspire to kick ascii2word([65,82,83,69]) at Spider.
Okay, this example fails the Duh Test since one can arrive
at the best move by observing it’s the only move that builds in-suit. But my
point was to illustrate the concepts of counting guaranteed turnovers and
calculating outs.
FUN FACT: Assuming perfect shuffling, a player should have on average 3.96 guaranteed turnovers at the start of every game.
Edifying Thoughts of a Spider Solitaire Addict 