# Match Summary (Alternative Version)

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 😊

# Match To Five Points.

Following Ninja Monkey’s rather lame advice in our previous game and my spectacularly unsuccessful attempts at improving its win rate over the long weekend, I have decided to take things to the Next Level Of Trevor by proposing a Match-To-Five-Points. This would entail the use of a Doubling Cube, so games can terminate early if one side doubles and the other refuses. From earlier interaction with IM Bart and IM Bug, I can safely assume my readers have some familiarity with the mechanics of the doubling cube in Backgammon 😊

To be more specific, here are the rules for 5-Point-Match in Spider Solitaire:

• The readers (IM Bart, IM Bug plus others) are the “Heroes” and I am the evil Villain.
• The readers are responsible for making all decisions re card play. I will maintain the game state with my standard Windows Solitaire software. Both my readers and I participate in cube decisions.
• At any stage of the game, either side can double. The opponent must concede (and start a new game with cube in the middle) or accept (which gives him exclusive right to make the next double). Note that rejecting a bad starting hand is equivalent to conceding a point after the Villain doubles. Obviously restart/undo is not allowed.
• There are no gammons or backgammons. There is no reward for scoring more than 5 points so paying attention to the match score may be relevant.
• The Crawford rule applies. This basically means if one side is 1-away from winning the match then the cube is dead for one game only.
• Pretend you don’t know nothing about chouettes – we shall keep things simple!

# Further Clarifications

Just to cover my backside and avoid any unpleasant controversies, I will state the following clarifications up front.

• It is legal for either side to double before the first move is played. Beware that it is possible for a good start to sour. Conversely, it is equally possible to recover from a poor start with skilled play. It is usually much harder to obtain a reversal of fortunes (in either direction) after the stock is empty!
• To avoid the (rare-but-theoretically-possible) scenario where both sides wanna double simultaneously with the cube in the middle, you can assume that if I present the latest game state without explicitly mentioning the D-word then I forfeit my right to double.
• If the readers do not have a clear majority for a preferred action (card-play or cube decision) then I use a random number generator as tie-breaker. Obviously, I will not use my “better judgment” as tie-breaker!
• It is acceptable to concede after accepting a cube and regretting it (a typical Backgammon analogy would be the losing side resigning after barely avoiding a Gammon in a no-contact position)

# Compound Actions

To save time, a reader may wish to combine a cube decision and card-play decision. For instance Joe Bloggs may vote “Double: if Villain accepts then I play (ab,cd,ef)”. A combined decision is always split into two separate decisions, with cube-decision first. If the majority vote is against Joe Bloggs’ cube decision then his card-play decision is redirected to Tumbolia, the land of Dead Hiccups and extinguished lightbulbs. Obviously the card-play decision is N/A if the decision is “decline Villain’s double”.

Note that Villain actually gained information he was not entitled to because the Heroes gave a move sequence before Villain had a chance to accept/refuse the cube (but Villain must accept/refuse before seeing any turnovers resulting from the move sequence). For this reason, I recommend to use compound actions only when the card-play really is trivial.

# I Have No Idea If This Will Work But There’s Only 1 Way To Find Out

Here’s hoping there are some Backgammon addicts out there who are willing to join in the fun along with our two IM’s. If you know your Market Losers from your Post-Crawford Game then this should be right up your alley, even if you rot13(fhpx) at Spider Solitaire. In any case, that’s what team-mates like Bug and Bart are for 😊 So, without further ado Let The Games Begin Immediately!

# Choose Your Difficulty (alternative version)

“Are we there yet?” groans the Sand Griper.

“Do we have to do this?” asks the Dumb Bunny. Meanwhile the Eagle has no cause for complaint as she gracefully soars across the air.

“It’s good exercise,” I reply. Even a Spider Solitaire tragic like me has to get out once in a while.

I sit on a rock, giving myself a brief rest as the rest of the gang catches up. Ninja Monkey does a quick head-count and confirms I haven’t lost any of my students.

“If you judge this fish by its ability to climb a mountain it will live its whole life believing it is stupid,” quips the Smart 65,83,83.

“You’re not helping!” growls the lion. The long trek has clearly taken its toll and even the Bad Idea Bears are not in the mood for jokes. I allow a few minutes break for everyone. We have only another 400 metres to go.

“Are we there yet?”

I turn to the Sand Griper.

“Okay, to make this trip a bit more entertaining I will let you play a game called 20 questions.”

The Sand Griper perks up – not something I see every day.

“The rules are simple,” I say. “You can ask as many questions as you like – except ARE WE THERE YET can only be used twenty times”.

The Sand Griper returns to being his usual grumpy self. Apparently he’s also not in the mood for jokes.

Finally I see a wooden sign and everyone soon reaches the top of the mountain, including the stockfish.

We immediately enter a tunnel. We follow the path and soon find ourselves at a large Games Room. All the animals marvel at the immense variety of board and card games, ranging from the prosaic Snakes and Ladders to the ever-popular Die Siedler von Catan or the ethereal strategic complexity of Risk. Not surprisingly the usual suspects are keen for a game of Texas Holdem after a long trek up the mountain.

“This is different,” says the Stockfish.

Stockfish is looking at a chessboard, except there is something unusual about the Black pieces.

“White has a large advantage” says the letter Alpha.

“Not so fast,” says the Dumb Bunny. “Black only needs to capture the King to win, but White has to capture everything.”

“I say White is completely winning,” replies the letter Zero.

The Eagle notices something unusual about the adjacent Backgammon board: one of the Green dice has the numbers 1,2,3,4,5,5 instead of the usual 1,2,3,4,5,6.

“Oooh look!,” I squeal. “My favourite game!”

Even better – the cards are already dealt, sparing me the arduous task of setting up the start position.

The Wise Snail seems pleased with the initial position. There are four guaranteed turnovers and two guaranteed in-suit builds.

“Jack of Clubs onto the Queen,” says the Elephant. “It’s in-suit and we also have a spare Queen.”

“Well done,” I reply. “You’re learning fast – no wait, I think this game could be rigged.”

“Why is the game rigged?” asks the Eagle. “Yes, there are two exposed Aces but …”,

“My favourite card!” squeals the letter Alpha. Clearly, he is new to the game. But from what I’ve heard these Letters and Numbers are capable of learning a new game with only four hours of self-training.

“But you have taught us many times the initial position is a poor indicator of whether a game will be easy or difficult,” continues the Eagle. “Besides you have four guaranteed turnovers and two in-suit builds.”

“There are other indicators,” I reply. “Remember the backgammon board with the faulty Green Dice, and what about the chessboard with unequal armies? If that’s not rigged then I’m challenging RIGGED whenever somebody plays it in Scrabble!”

“Look at this!” squeals Minnie Mouse. “Texas Holdem is also rigged. Take the Queen of Spades from the deck. Hold the back of the card to your nose. It should be blurry. Focus as though you are looking through the image into the distance. Very slowly move the card away from your face until the letter Q appears …”

Meanwhile the Bad Idea Bears are engaged in a fierce battle of Snakes and Ladders. They eventually realise that every square between 83 and 88 (inclusive) contains a snake and no ladder reaches a number higher than 88.

“So does that mean every single game here is rigged?” asks the Eagle.

“I will assert with 95% confidence every game is rigged, including Spider Solitaire,” I reply. “Welcome to Peak Stupid. But at least we know the game is rigged before moving a single c-”

“But that’s outrageous!” says the Eagle. “I refuse to play”.

“I know you are one of my top students but I want you to understand carefully: I have no problem with the game being “rigged” if the player knows in advance the cards are not properly shuffled. Think of it as an extra challenge – we already know it is possible to beat four-suit Spider Solitaire without boop if the cards are properly shuffled.”

It takes some convincing, but my students eventually agree to play the game out.

< several moves later >

• Round 1: three Kings appear simultaneously
• Round 2: A very awkward Q84KA84Q20 with lots of evens.
• Round 3: three Threes
• Round 4: four Fours
• Round 5: at least I didn’t draw five Fives. But three Sevens and three Tens are awkward.

“You’re right,” says the Eagle. “You correctly predicted the game would be rigged. I’m not sure whether trips and quads in every round is a true indicator of difficulty and we haven’t even considered the permutation of unseen cards in the tableau but it is apparent someone did put in the effort to rig the cards”.

“Despite our best efforts we couldn’t win without the help of boop,” I say. “We obtained two empty columns at one stage and came close to completing the Heart suit. Pity that both Jack-of-Hearts were hiding behind two Kings in Column Four though.”

Hang on, I think to myself. Stockfish’s fishbowl has somehow moved right by a good half-a-meter when nobody was paying attention. I soon figure out this mischief was due to Ninja Monkey (thanks to his extremely fast metabolism and lightning reflexes he was able to avoid suspicion for quite a while). But at least I’ve worked out how the stockfish was able to ascend the mountain without violating the laws of physics.