*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 âŠ anyways, you get the gist. Hopefully today wonât be too much of a disaster.

âHere is an interesting position,â I say. âWhat would be your play here?â

I pull out my i-Phone and show the position to my students. Itâs a pity we donât have whiteboards and chalk in the jungle.

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

âI say it doesnât matter what move we play. Iâve played 100 games thanks to my usual extremely-fast-metabolism and I estimate the winning chances are exactly zeroâ.

*Groan.*

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

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.

âAnyone else have a better opinion? How about you Mr Snail?â

âI need some more thinking time,â says the Wise Snail.

Hmmm âŠ this lesson ainât 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 hasnât lost 50 quintillion games in a row.

âThe position isnât that complicated,â I reply. âThere are only 11 cards in play and 5 legal moves.â

âYes, but with 11 cards in play we have 93 cards unseen.â

âBut whatâs that got to do with the Fundamental Theorem of Calculus?â

âWell, we know that in Freecell the chances of winning is exactly 100% or 0% assuming perfect play,â replies the Snail. âThis 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 âŠâ

âThree factorial is six,â says the Smart 65,83,83. âSome number divided by six.â

âWhatever,â continues the Wise Snail. âSimilarly one can compute the exact winning chances for any number of face-down cardsâ.

âI see where youâre coming from,â I reply. âUnfortunately 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.â

The Wise Snail pulls a frowny face.

âI wanna flip a coin, since there are no in-suit builds,â offers the elephant. âUnfortunately there are 5 legal moves and we donât have a coin with five sides.â

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

âFour of Hearts onto the Five,â says Bad Idea Bear #1.

âOnly three more good cards and we get an empty column!â adds Bad Idea Bear #2.

âWe can eliminate some moves,â offers the Jaguar. âMoving either Eight onto the Nine is equivalent, so pretend there is only one Eight. We shouldnât move a Four onto the Five since that means we only have two guaranteed turnovers, not three. Therefore itâs a choice between 9-8 or 6-5.â

âThatâs good,â I say. âFinally weâre getting somewhere.â

âSo we donât need a 5-sided coin after all,â 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. Itâs always a pleasure to witness the insights of the lion, one of my better students.

âIf we move 9-8,â roars the lion, âthen 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.â

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

âEvery player knows that building in-suit is more desirable than off-suit,â I say. âWhen 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.â

âFor 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.â

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

âWe 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 wonât 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.â

âLet 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.â

*Uh oh.* The Sloth is snoring. I think nothing of it: after all heâs not the sharpest tool in the jungle out there if you excuse the terrible clichĂ© and/or mixed metaphor. In fact I donât recall the last time he didnât fall asleep.

âObserve 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!â

âRoughly 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.â

The Eagle is desperately trying to suppress a chuckle. Something is out of character: my best student doesnât exactly have a reputation for lame puns, knock-knock jokes or pranks.

âAs a general rule,â I continue, âbuilding 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 ââ

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

Iâve 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 ainât whatâs 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