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.

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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

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