mathNEWS Issue 78.5: Friday, November 6, 1998

Spade Parade

Wednesday afternoon, the kind of day where you figure that you would have been better off staying in bed. Spade Parade's the name; I'm a private investigator. The money's lousy, but I take fewer risks than your average tightrope walker, so I suppose it balances out.

I had spent the morning rearranging my office, on the theory that my super was less likely to ask for the rent if the place was clean. I'd just put the last filing cabinet into an upright position, when there was a knock at the door.

A client. Well, worse things have happened.

I opened the door and there she was. I'm no poet, and so I won't even attempt to describe her. Think of all the synonyms for "beautiful" that you can, then add in "luscious" (my personal favourite) and you're getting warm. For that matter, so was I. I warned myself not to get too excited; the only reason that a looker like her would be at my doorstep is that she had a problem, and a big one.

"Spade Parade?" Her voice was low and husky, and sounded like she'd been crying. "My name's Ella Szticsdy, and I need your help."

I've always been a sucker for damsels in distress. Especially when they look like that. Thinking quickly, I undid an hour's work and cleared a chair of the stacks of papers that it had accumulated. "Help is my speciality. Take a seat, Miss Szticsdy," — those Hungarian classes in college were really paying off — "and tell me all about it."

She composed herself somewhat and began. "Victoria Shael and I... we never really got along. We've known each other since finishing school, and you would think that we'd be the best of friends, but there's always been this edge between us. Vicki's very conservative, and I don't think that she approves of my `flightiness', as she puts it. Still, we see each other almost constantly."

I interrupted, "Why do you stick around her, if you don't like each other so much?"

She gave me a look like she was a monkey and I was a blue banana. "I have to associate with someone of my own social standing, don't I?"

I shrugged slightly; some problems I don't have to deal with. "Of course, sorry to break your train of thought."

"At any rate," she continued, "we ran into each other at the Tannoy do the other night, and as usual started snapping at each other. I was talking about my whirlwind trip to Europe last week — she never travels anywhere, poor thing — and she started casting aspersions on my lifestyle. I mean really, just because I went to Paris for the unveiling of this fall's fashions..."

I tuned out, figuring that I'd recognize the point when I heard it. So she was a jet-setter, hey? If she was typical, I might just go and get my pilot's license.

"...and so she offered her ruby earrings against my priceless jewelcase that — oh, it's complicated — that I wouldn't be at the same spot at the same time two days running. She thinks that I'd get bored or something."

I pondered that for a second. "But that's easy. You sleep in your own bed, right?"

I realised as soon as I said it that I wasn't sure I wanted to know the answer to that one, but she brushed off the question. "It's not that simple, Mr. Parade. It has to be this Saturday and Sunday, between eight in the morning and eight in the evening, and I have to be awake."

I was still obviously missing something, and said as much. She looked like she was going to burst into tears. "But this weekend... my brother Ethan and I were going to hike to the top of Paul's Peak and camp there for the night on Saturday. It's a tradition... we used to go every year with our parents..."

I was beginning to see the light. "So in other words, because of this bet, you'll have to cancel your hiking trip? Because you have to be under this Vicki's scrutiny?"

"Oh, she doesn't need to see it, as long as I can provide proof that it happened. But I can't cancel on Ethan, yet I can't lose that jewelbox... what can I do, Spade?"

Time for some reassurance. "Don't worry, sweetheart, I think that there might be a way to have your case and Ethan too. Why don't you read the next sketch while I do some scribbling?"

As she glanced over to the next page curiously, I picked up my notepad and pen, raised my feet onto my desk, and thought.

I was pretty sure that I had it, but I had a couple of questions for this girl first.

"All right, Miss Sztacsdy, how long does this hike up the Peak take?"

She seemed a bit taken aback by the question, but gamely responded. "About twelve hours. In fact, Ethan and I had been planning on starting the hike at eight in the morning, which would put us at our campsite at around eight at night."

That was convenient. "And do you take a different path down?"

"Oh, never. That's part of the tradition; we have to come down the same way that we went up."

I couldn't help but grin. It all worked out just too perfectly. Ella saw my expression and brightened somewhat. "Oh, Spade, can you help?"

"I think I might be able to do that, Ella darling. Here, let me show you a little chart." I pulled an oversized graph out of the bottom of a big pile beside the desk.

[graph]

She seemed a bit confused, but I've never let that sort of thing stop me before. So I continued. "Now let's call the bottom of Paul's Peak zero, and the top where your camp is, one. So on Saturday, between eight in the morning and eight in the evening, you can consider your location – where you and Ethan are at any given moment – as a function of time. This function is at zero when you start, and at one when you stop for the night."

She looked like she might interrupt, so I kept going. "And on Sunday, if you break camp at eight in the morning, then the same thing happens, except you're at one when you start and zero when you stop."

She was quick, I'll give her that. "So those two lines are where we are at different times on Saturday and Sunday... and they cross! That means –"

"Right you are, sweetheart. Same place, same time."

"But how do I know that they always do that? I mean, we might break at different spots on the way up and the way down."

"Don't worry about it, Ella. If you subtract the two functions," I said while pulling out another chart, "then you get a function that starts out above zero, and ends below zero. The Intermediate Value Theorem guarantees that it must cross zero somewhere, as long as you don't teleport from place to place."

[graph]

"So I can go on my hiking trip, and still keep my jewelry box! That's wonderful, Spade! How can I ever thank you?"

I raised my eyebrow and smiled slightly. "I can think of a few ways, but let's wait until this sketch ends before trying them."

So much for cleaning up the office.



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