# The World’s Worst Math Teacher (another short story)

“Another one of life’s disappointments.”

“Marking assignments, the bane of every teacher,” growls Ms. Spider, as she angrily scrawls the word “DREADFUL” on a sheet of paper. “This goose just divided by zero.”

I’ve always enjoyed math, but I am all too aware that it represents a bugaboo for many ordinary folk. Not everybody can have higher than average IQ and not everybody can play piano and solve Rubik’s Cube at the same time. I agree we have to Make Math Great Again.

“I s’pose I could improve my presentations skills or learn Statistics 101,” admits Ms. Spider.

“I confess I never studied stats at uni,” I respond. “I had to pick it up all by myself.”

“Learning stats 101 sounds too much like work. Surely there must be a better way.”

“You could make the exams and homework easier,” I suggest.

“We can’t make it too easy,” responds Ms. Spider. “I’m sure the good students wouldn’t mind an extra challenge or two,”

I steal a glance at the goose’s assignment. Yes the goose is below average, but one of the assignment questions are badly worded. Another question has kilometres as a typo for metres, and I have to suppress a chuckle. I can see why some of Ms. Spider’s students call her the WWMT.

“Actually,” says Ms. Spider, “I was toying with a more radical solution”

“Which is?”

“We could give different exams to different students”

“What a revolutionary idea!” I exclaim. “Nobody has ever thought of this before!”

“From each according to his abilities … “

“From each according to his needs,” we chant in unison.

I am impressed: this Spider is clearly well-educated, not just in mathematics. She knows her clichés and sayings.

“Does that mean,” I ask, “if an awesome student correctly answers 40 assignment questions in a row then he will get a very difficult exam?”

“Exactly.”

“Hang on, what if an awesome student deliberately flunks the assignments …”

“Well … we could give the exam less weight than assignments,” the Spider responds somewhat nervously. “Then there is no advantage to tanking the assignments.”

“That’s Dandy!”

“For this to work,” continues Ms. Spider, “we have to come up with some way of measuring the difficulty of certain questions.”

“I understand,”

I mull over this for a while. We all know that students can be graded according to some chosen system. For instance, a math student can be Outstanding, Exceeds Expectations, Acceptable, Poor, Dreadful or Troll. But how can we grade certain questions?

The Spider writes two math questions on a sheet of paper:

“Which of these problems is harder?” asks Ms. Spider.

“I think both are equally easy. After all, I participated in the International Mathematical Olympiad many years ago.”

Somehow, I think that was not the answer Ms. Spider expected.

Behind us, a monkey, eagle, mouse, elephant, lion and jackal are enjoying some Texas Holdem. As usual, the monkey has squandered away all his chips early, and the Eagle is schooling the rest of the field, having accumulated more than half the chips in play. The Spider eyes them warily: clearly they should not be privy to our discussion.

“You see,” says Ms. Spider. “Sometimes I find it hard to judge the difficulty of a single question. For instance, I expect problem X to be easier than Y, but for some reason the reverse holds when I mark the assignments.”

I mull over Ms Spider’s words. I am not really in a position to judge, given I have never marked any student assignments.

“I have an idea,” says Ms. Spider. “Let’s draw a table”

“For simplicity,” says Ms. Spider. “Let’s assume each question is either marked correct or not correct, hence there are no partial marks. I use blank instead of 0 for ease of reading. Sam is an awesome student since she answered most questions correctly. Owen is a Stupid student because he only scored 2 out of 9. Each individual is represented by a single row.”

“Okay.”

“But there is no reason we can’t do the same with columns if you pardon the double negative. For instance, only six people solved problem 8 but nine solved problem 9. Therefore problem 9 is harder than  problem 8 …”

“So even if you don’t understand the questions themselves you can still say things like Debbie is better than Anna”

“Exactly,” replies Ms. Spider.

“With 18 students and 9 problems, you don’t have a lot of data”

It’s a stupid observation, I know – but I am only trying to buy time as I try to digest her ideas.

“Well, the same logic applies if we had 1800 students and 900 problems.”

“I think I understand,” I say. “It’s like some kind of Mechanical Turk. Previous students have tried these questions (and of course you don’t have to pay them to do these exams!), so you can work out which questions are easy or hard.”

“Wasn’t the Mechanical Turk some kind of fake chess-playing machine by Wolfgang von Kempelen? What a disgraceful idea! I would never try to cheat chess players like that”.

Okay, didn’t see that one coming. We need to agree on a definition of Mechanical Turk.

“Do you think your students will eventually find out their exam papers are different?”

“That shouldn’t be an issue,” says Ms. Spider, as she squirms in her seat. “If a poor student finds out, he has no reason to complain. If a good student finds out then deep down in his heart he already knows he is better than the poor student, so the exam result doesn’t matter.”

Somehow I think her logic is very, very, unsatisfactory. But I do know that many of the greatest insights precisely come from those who are willing to suggest ideas that sound utterly outrageous. For instance Rivest, Shamir and Adleman are your average computer scientists, but with a bit of luck they might one day become famous, known to every student of cryptography. So I should cut her some slack.

In fact, I am more than looking forward to the results of her revolutionary teaching methods. After all, I’m not the teacher and I don’t set the exams. I was especially careful not to suggest any drastic ideas of my own. If the radioactive 83,72,73,84 hits the fan and grows to fill the size of the entire house then I am more than happy to watch, knowing my 65,82,83,69 is fully covered.

Bring. It. On.

# Another short story (yay!!!!!)

Hooray! I have finally quit my day job and found something I really enjoy: teaching students how to play 4-suit Spider Solitaire well. According to legend, nobody in the Animal Kingdom has managed to beat the game at the highest difficulty level. Even the Ninja Monkey with an amazingly fast metabolism couldn’t achieve it despite 50 quintillion tries (and counting).

In my first class I have 6 students: a mouse, lion, Jackal, Elephant, Eagle, and last but not least, the monkey.

I have already gone over the basics: empty columns are good, suited connectors are good, but aces and kings are usually not your friends. I notice the monkey is taking copious notes. He is an ideal teacher’s pet, if you excuse the lousy pun.

“Take a look at this position,” I say. “Do you think we should win with correct play?”

“I think so,” replies the mouse. “I would bet 3 dollars.”

“That means you think it’s not possible,” quips the lion.

“What do you mean?”

“Your bet is too small,” replies the lion. “If you thought this is a win, you would be betting \$30, not \$3.”

“I would raise it to \$60 at least,” offers the Jackal.

“Yeah sure,” says the elephant.

“We do have 4 suits removed and two empty columns. Sixty dollars says we win this.”

“I wouldn’t bluff if I were you,” replies the Elephant. “Look at that pile of 83,72,73,84 on Column 4.”

“Come on guys,” says the eagle. “I think we need to analyse this seriously. This game is about math, not people or animals.

I turn my attention to the monkey, who as so far been silent.

“What’s your opinion?” I say, putting him on the spot.

The monkey looks embarrassed.

“This would be easier if it were one-suit.”

Everyone laughs. The monkey looks like he wants to kill himself.

“But that’s way too easy!” exclaims the mouse. “Even I would go all-in.”

“The monkey raises a good point,” I say.

Stunned silence. All the other animals look at each other, unable to believe what they heard.

“Okay,” I continue. “Let’s change the rules: the game is one-suit but we cannot remove suits to the foundations until all cards are exposed.”

“But that’s cheating!” shout all the animals in unison (except the monkey). “You’ve already moved four suits to the foundations”.

Don’t ask me how the animals manage to speak in perfect unison without proper rehearsal. I guess everyone has their own unique talents.

“For purposes of this exercise, let’s assume I changed the rules mid-game and you have to Deal With It.

The animals discuss this for a few minutes.

“K to 3 in column Five,” says the mouse. “9 into column Seven. 10 to A onto the Jack in -”

“Not-so-fast,” replies the eagle-eyed eagle. “There are two aces in column 4.”

“But we can create another empty column,” says the Jackal. “4-3 in column Eight onto 8-7-6-5 in column Nine.”

“Once we reach the 4 of Clubs the rest should be easy. Column 2 becomes empty and 4 of Clubs into Column 2 etc.”

“And it doesn’t matter what order the hidden cards are in.”

I am pleased that all my students are participating in the discussion.

“So we can win at 1-suit,” I say. “Note that if we couldn’t win with 1-suit we can deduce it probably won’t be possible at 4-suits (unless we can quickly complete a suit).”

All the animals nod in agreement.

“Now back to the original problem,” I say. “How to continue at the Four-Suit level?”

The animals quickly discover the right plan. Once the King of spades lands in an empty column we can recover an empty column by moving the 8-7-6-5 onto the Nine of spades. We then have to shift the Nine of spades and Ace of clubs into two empty columns. Then we have to hope we have enough empty columns to finally shift the 10 of Hearts and reach the Four of clubs. Of course, all this is easier said than done, if you excuse the terrible cliché.

The monkey brings out a set of cards and arranges them into the diagram position. He tries and tries but to no avail. Despite his repeated failures the other animals are amazed at the monkey’s dexterity and eidetic memory as he quickly reorders the cards into the starting position without error.

“Let me have a try,” says the eagle.

The eagle quickly discovers a truly remarkable solution to this problem, which this blog is unfortunately too small to contain. All the animals applaud loudly as he smacks the last suit onto the foundations with a satisfying thud.

“How the 70,85,67,75 did you do that?” asks the Monkey.

Uh oh. I’m beginning to have doubts about the quality of the monkey’s copious notes. His strategy is still as lousy as before. He is not really a model student after all.

“Let me have a look at your notes,” I say.

I confiscate the monkey’s book and riffle through his “oodles of doodles”, none of which have any artistic merit. I am about to 83,80,65,78,75 the monkey in front of everybody and rip his doodles to shreds. But at the last minute I suddenly remember that I owe the monkey a tremendous debt. Without him I wouldn’t have been able to publish a paper in the International Journal of Arachnids, Primates and Other Predatory Species. Then I stumble upon a very strange picture:

So the monkey does have occasional flashes of brilliance when doodling after all. Okay, the happy star isn’t exactly centred properly, but it’s not a bad job for someone with only pencil and paper. I doubt that any human could produce art like that. I return the book to the monkey and bow before him.

Seeing that the monkey got off without even a warning, I guess it’s only fair that the last word belongs to Shakespeare: All’s Well That Ends Well.

Class dismissed.

I like my new job. Okay, the pay isn’t great and the diet leave much to be desired, but at least I’m my own boss and I get to set my own hours. All the animals are easy going and real friendly, and we can cuss and swear with impunity. And best of all, nobody smokes around here (because smoking is bad for you). 70,85,67,75 89,69,65,72!

# 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.

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.

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 🙂

# Continued …

The reader may be familiar with Klondike and Freecell. In Klondike, exposed cards are always ordered. For instance, if an exposed Seven of Hearts is covered by another exposed card, the latter must be one of the two black Sixes. This means the number of legal game states is much smaller than a game with “random rules”. The “non-trivial” part of Klondike of course comes from the face-down cards. Freecell is a famous game because it should almost always be won with perfect play. With all cards exposed, any initial game state can (in theory) be analysed to a certain win or certain loss before making a single move. The “non-trivial” part of Freecell comes from the fact exposed cards are not necessarily in order. For instance, the Seven of Hearts can be covered by any other card, not just the black Sixes.

As you might have guessed, Spider Solitaire combines the non-trivial aspects of both Klondike and Freecell. With half the cards face-down there is no question of certain victory or defeat at the start. The fact that exposed cards are not necessarily in sequential order yields several orders of magnitude of extra possible game states, and hence much greater scope for interesting strategy.

Oh, did I mention that Spider is played with two decks instead of one?

Local vs Global

The last point I wish to discuss is the concept of local vs global. If you have played board games like Die Siedler von Catan or Agricola, you probably know that the early rounds of a game have “small-scale plans”, but the middle game is where “deep strategizing” and “maximum tension” occurs. The endgame is where the tension ceases, presumably because everybody knows one player has a decisive advantage and it is basically impossible to lose, except by tanking. The ideal curve is shown below (disclaimer: I only got ‘C’ in Year 10 art).

Note that the peak is not centred, but is closer to the end than the beginning. Although I am not the world’s greatest expert on Board Games in general, I think the most successful games tend to obey this curve. Obviously if you play Catan 100 times, not every game will have the ideal curve: for instance, one game might be easily won by Player 3 thanks to some lucky rolls at the start. But most of the games will be “fair” and everyone feels they started with a decent chance to win.

In my experience, I think Spider Solitaire has a reasonable time-tension curve. In the beginning the player is concerned with short-term plans such as exposing as many face-down cards as possible. In the middle game, the tension increases because there are a large number of cards in play, and the player is usually aiming to clear a complete suit or two. Spider only allows complete suits (instead of single cards in Klondike or Freecell) to be moved to the foundations and it is rarely possible to achieve this with short-term planning alone. Assuming the player is successful, the tension decreases because the player is practically certain of victory. Technically one can still search for the very best moves, but by this stage the player is probably playing on auto-pilot.

So there you have it. Hopefully my first blog post gives you some idea of why I consider Spider to be the Cadillac of Solitaire. Okay, this entire post probably didn’t make much sense because I haven’t even explained the rules … I should probably start with that on my next post 😊

Toodle pip and piddle too, ciao 😊