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The Problem of the Cryptic Clue

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It was Monday, 9:43 A.M., and traffic in the greater Los Angeles area was sticky after two maple syrup trucks leaked all over the 405. I was working the day watch out of Mathnet. My partner is George Frankly; the boss is Thad Green. My name is Monday. I'm a mathematician.

George was already in the office when I arrived, and I gave him my customary Monday morning nod. "Morning, George. How was your weekend?"

George gave me his usual Frankly morning smile. "Hi, Kate! Pretty good. Martha and I did some line dancing."

I took off my jacket and draped it over the back of my chair. "Sounds fun. Do you belong to a club?"

"No. Martha just paints a line on the living room floor and then we dance on it." George laughed, apparently recalling the great times they'd had. "You should try it, Kate."


"Good exercise." George paused to think. "Unless you've got a rug in your living room. Your landlord might not like you painting it."

The phone rang, and I answered it. "Mathnet. Monday."

"Hi. My name's Lisa. I have a problem, and I was hoping you could help me with it." Lisa sounded like she was about twelve years old.

"What kind of problem?" I asked. Solving math problems is our specialty, but if she had some other kind of problem--say her telephone service was disconnected--George and I wouldn't be the best people for the job. Although if her telephone service had been disconnected, we wouldn't have been having this conversation in the first place.

"We just opened a time capsule at my school. People at the school fifty years ago left messages and different things in the capsule."

I was taking notes. "Uh-huh."

"Well, the math teacher back then, Miss O’Euclid, left a rhyming note that doesn't seem to make much sense. I think it might be a riddle, but nobody else seems to think so."

Math riddles happen to be a specialty of George's and mine. "Do you have the riddle now?"

"They put the real copy back in the time capsule, but I wrote it down. I was hoping maybe you'd be able to come take a look at it?"

"Where and when?" I asked.

"I have recess at ten-thirty. We can meet on the playground if you can get to Central Average Middle School by then."

"We'll be there, Lisa," I said.

"Okay. See you soon!" She hung up.

George looked excited. "Got a new case for us, Pard?"

"Uh-huh," I said. "They've just opened a time capsule at Central Average Middle School, and one of the students thinks there's a math puzzle inside."

"Gee, that's better than the time capsule they opened at my school when I was a kid," George said.

"Why? What was in that time capsule?"

"Just a pile of watches," George said. "Central Average Middle School, you said?"

I nodded. "Let's roll."

We put on our jackets, took out our calculators, tapped out "shave and a haircut, two bits" on them, replaced them in our calculator holsters, and left the office, saying hi to Sam and Steve as we headed for the car.

As promised, Lisa was waiting for us on the playground when we arrived. The rest of the students were there too, but Lisa was the only one waiting for us.

"Thanks for coming," she said. "I wrote down everything from the note, but it still doesn't make any sense to me."

"May we see the poem?" I asked.

Lisa nodded, handing the poem to me. I read it aloud.

"I've something to add, so here's my point;
I'll make my meaning plain.
Just follow the path from school to school
and then to town again.
If that is your way, you'll get an A
and never will be square.
You'll be quite all right when you find the light
in the middle of Sea Square."

"Well?" Lisa asked.

"Kinda pretty," George said.

"Yeah, but what does it mean?" Lisa asked.

"That's what we're here to find out," I said. "Let's play 'What Do We Know?'"

Lisa frowned. "What do we know?"

"It's a way of going over information. Sometimes when you're trying to solve a problem, making a list of the things you know leads you to something new," George explained.

"Okay," Lisa said. "We know this note was left by the math teacher."

"She says she's making a point, so we know there must be some reason for her to leave this note," George said.

"Point." That reminded me of something. "Lisa, did you copy this down exactly as it was written?"

"No. I didn't get to look at it," Lisa said, looking crestfallen. "The principal read it aloud to us."

I pointed to one of the words. "George. Say this word aloud."

"Plain," George said. Then he realized what I was getting at. "Not plain. Plane!"

Lisa looked confused.

"P-l-a-n-e," I said. "Miss O’Euclid mentions a point in the first line and a plane in the second line. Which can only mean one thing."

I turned to look at George, and we both said the word in unison. "Geometry."

Lisa shook her head. "Why geometry?"

"Geometry's the mathematical study of spatial relationships," I said. "Within geometry, a plane is a flat surface that extends to infinity, and a point is a unique location in space."

Lisa's eyes widened. "She's telling us that there's a specific place she's thinking of! But where?"

"What comes next?" George said, looking at the poem over my shoulder. "'Follow the path from school to school and then to town again.'"

“Well, there are two schools in central Average—Central Average Middle School and Central Average Elementary School,” Lisa said.

“No Central Average High School?” I asked.

Lisa shook her head. “Everyone in Average goes to North Average High School.”

If there were only two schools...but I needed to see it on paper. “Lisa, are there maps in your school library?”

Lisa nodded. “I’ll show you where.”

In as much time as it took me to write this sentence, Lisa took us to the geography section in her school library. We took out an atlas showing the greater Average area, and I took some tracing paper and a pencil out of my pocket.

“The middle school is there,” Lisa said, pointing to a location on the map. I made a corresponding dot on the tracing paper. “And the elementary school is there.” I made another dot, noting that the elementary school was directly south of the middle school.

“But you’re forgetting the last part,” George said. “‘Then to town again.’ Where’s town?”

“East of central Average is known as Average Town,” Lisa said. “They’re actually kind of annoyed that they don’t get to be just plain Average, but when they were founding this place, everybody wanted to be Average.”

“And the center of town?” I asked.

Lisa pointed to Average Town on the map, and I made another dot. Then I took my straight edge out of my pocket—a mathematician always needs to be prepared to chart, graph, or diagram—and connected the dots. “What kind of angle does that look like to you, George?”

George peered over my shoulder. “I’d say 90 degrees, wouldn’t you?”

“Uh-huh,” I said. “Lisa, what was the rest of the riddle?”

“This is the part that doesn’t make any sense,” Lisa said. “‘If that is your way, you'll get an A, and never will be square. You'll be quite all right when you find the light in the middle of Sea Square.’ But there’s no place called Sea Square anywhere in Average. We don’t even have a beach!”

I read the verse again. “In the middle of Sea Square.” But she hadn’t seen the poem written down, so...oh. “Not Sea Square. C-squared!” I turned to look at George. “The Pythagorean theorem!”

“My gosh, Kate, I think you’re right,” George said.

“What’s that?” Lisa asked.

“It’s a rule about right triangles—triangles with a 90-degree angle as one of their angles. If you multiply the length of each leg of a right triangle by itself—that is, square it—and then add the two products together, your total will equal the square of the hypotenuse—the third leg.”

“But we don’t have a triangle,” Lisa said. “All we have are two lines.”

“That’s just it,” I said. “This is leg A.” I retraced the line from the middle school to the elementary school. “This is the right angle—the 90 degree angle—and this is leg B.” I retraced the line from the elementary school to the center of Average Town. “So if we draw a line from the center of town to the middle school...”

“The hypotenuse,” Lisa gasped.

“Exactly,” I said. “And if we add A squared—getting an A, from the poem—to B squared—not being square, as the poem says—we’ll get C squared.”

“The last part mentions the light in the middle of Sea Square—or rather, leg C,” George said. “There must be something exactly halfway along that leg.”

“What do you think is there?” Lisa asked.

“Probably a streetlight,” George said. “Or maybe one of those really nifty neon signs.”

“No, I you think Miss O’Euclid left something there fifty years ago? Something for us to find?” Lisa asked.

“There’s only one way to find out,” I said.

The school bell rang and Lisa sighed. “Can we find out at lunchtime? I really want to come with you.”

We waited until lunchtime, and then Lisa rode with us to the place we’d determined was the exact middle of the hypotenuse of our triangle. Sure enough, there was a streetlamp there, with a wide pillar at its base. If anything were going to be hidden there, it would have to be in that pillar. We circled the pillar, examining it closely.

It was Lisa who found the panel. “Hey, I think I got it!” She pressed hard on the panel, and it sprang open, revealing a secret drawer.

“A secret compartment,” George said. “Neat!”

Lisa reached into the compartment and drew out a device both George and I knew very well. “What is it?”

“It’s an abacus,” I said. “People used these to make numerical calculations before pocket calculators existed. The earliest historical references to the abacus are from thousands of years ago.”

“Wow,” Lisa said, wide-eyed. “And she left this here for somebody to find fifty years ago.”

George chuckled. “Abacus, huh? Guess we really did get an ‘A’ after all.” He chuckled again.

“Can you show me how to use it?” Lisa asked.

We gave Lisa a few beginners’ lessons in how to use an abacus and returned her to school in time for her to eat lunch. Then we drove back to Mathnet HQ.

"Well, Kate, I have to admit I'm a little disappointed," George said as we walked into our office.

I turned to look at him. "How so?"

"It normally takes us all week to solve a mystery," George said. "But we solved this one and it's still Monday. What are we going to do Tuesday through Friday?"

"Probably paperwork," I said. "But that doesn't make a very interesting story, no matter how you tell it."

George started to laugh. After a moment, I did too, and we gave each other our customary end-of-case high five.