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Monte Carlo and Humble Pie

I have been teaching mathematics in an Australian High School since 1982, and I am a contributing author to mathematics text books.



Len was throwing darts in the staffroom. “What topic are you on?” he enquired as he carelessly hurled a dart over my head in the general direction of the dartboard.

“Measurement,” I replied, “but I want to do something different rather than use the same old formula for the area of a circle.”

I watched five darts land on a dartboard secured to a large square cork splashback.

“Well, at least three landed on the dartboard,” Len commented. “That’s 60% accuracy.”

“Yes! Yes!” I yelped and jumped upright. “Thanks, Len. You’ve given me my next lesson plan.”

I walked out excitedly, leaving Len surmising that only an apparition could produce the kind of reaction he witnessed.

“Today is circle day,” I stated cheerily to my class. “What do we know about circles?”

Steve opened the discussion. “That they’re round?”

“And they don’t have a beginning or an end,” Yao continued in the same uninformative vein.

Rosa also thought a tautological response was appropriate. “Their centre is at the middle.”

I squeezed my head and made a long face to express lugubriousness. This was open-ended questioning at its most comical level.

“Sir has a headache again,” Helda remarked.

“Come on,” I urged passionately. “Think. What is the inside of a circle?”

“Its area,” Alex proposed.

“Good. Now how do we work it out?” I asked quickly in an effort to maintain the momentum.

“Area is pi times the square of the radius,” Teng said insightfully.

“Right,” I smiled in appreciation of his contribution, “but what if we don’t know the radius and there is no ruler to measure with?”

“Google it?” Dave offered helpfully.

“We’ll start with a square circumscribing a circle,” I began.

“You’re doing it,” Janice commented.

“What am I doing?” I asked.

“Using words that scare us,” she noted.

“Circumscribe means one shape goes around another shape and touches it without intersecting,” I clarified.

“Oh,” she said, unenlightened still.

“We shall use the Monte Carlo method,” I began.

“Which class is he in?” Jimmy asked out of curiosity.

“Who?” I asked, confused.

“Carlo. You said we’ll use his method and that he’s from Monty.”

“No,” I said.

“Well, then, is he related to the Count of Monte Cristo?” Jimmy persisted.

“No, Jimmy. Monte Carlo is the name given to a heuristic simulation algorithm stating that the number of random hits is proportional to the area.”

“You’re scaring us again,” Janice reiterated.

“Think of it like this,” I began as I snuck into the store room and returned with a dartboard and magnetic darts which I rested against the whiteboard near the door. I took a marker and drew a square around the dartboard. Now there was bourgeoning interest.

“Suppose I throw 100 darts and all of them land on the dartboard or between the square and the dartboard,” I proposed. “What would happen?”

Steve was resource-conscious. “You’d put holes in the whiteboard.”

I laughed. “Applying the Monte Carlo method gives an estimate for pi. It is four times the number of darts hitting the board divided by the total number of darts thrown. So, if 80 of the 100 darts hit the dartboard, our estimate for pi is 4 x 80/100=3.2”

The flavour of this discourse stirred Jimmy to reminisce.

“Sir, didn’t we do this when you were tossing chopsticks?” he recalled not very factually.

“No, Jimmy,” I replied. “You’re thinking of Buffon’s Needles. That’s an entirely different method.”

“Yeah, that’s right,” Jimmy remembered, and his expression became pensive. “But sir, didn’t they have anything better to do hundreds of years ago than to find values for pi?”

“Because they didn’t have TV,” explained Sarah.

I squeezed my head again. “In fact, Jimmy, Monte Carlo was a method named by scientists in the 1940s.”

“But why did they call it Monte Carlo?” Janice wanted to know.

A History and Geography lesson was warranted. “The method is based on random number generation. Monte Carlo is a city in the Monaco famous for its games using a roulette wheel, which is really a random number generator. Scientists were conducting experiments involving random numbers relating to nuclear particle collisions.”

“Can we go to Crown Casino and investigate the roulette wheels?” Steve suggested optimistically. Others backed his recommendation.

“Yes. We’ll meet there when all of you turn 21,” I stated. “Until then, let’s practise what we’ve learnt.”

Jimmy attracted my attention by raising his hand. “Sir, can I have one go at throwing the dart?”

“Sure,” I said, and handed him one of the magnetised darts.

Jimmy projected the missile but it did not reach its intended target. Instead, it bounced off the smooth surface of the whiteboard and followed a trajectory of self destruction. The magnetic tip zeroed in and attached itself like a mollusc on the metallic spectacle frame complementing the visage of the Principal as he walked in unexpectedly.

Incredulity took the form of silence, even superseding the ludicrousness of a dart attached to spectacles dangling from a most senior administrator’s proboscis. I knew I had a lot of explaining to do.

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