Dr. Ecco: Mathematical Detective

Dr. Ecco: Mathematical Detective

by Dennis Shasha

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In this collection of original puzzles, games, and codes, the heroic Dr. Ecco, a mathematical detective and puzzle solver, takes on his archenemy, Baskerhound, and uncovers a plot that threatens the world. No sophisticated mathematical background is necessary to solve these challenges, which were inspired by the methods and thinking of researchers in computer science and mathematics. All you need is imagination and a passion for puzzles.
A graduate of Yale, author Dennis Shasha received his Ph.D. from Harvard and is Associate Professor of Computer Science at the Courant Institute of Mathematical Sciences at New York University. He includes complete solutions at the end of the book, and has rated the puzzles according to difficulty.

Product Details

ISBN-13: 9780486169316
Publisher: Dover Publications
Publication date: 02/05/2013
Series: Dover Recreational Math
Sold by: Barnes & Noble
Format: NOOK Book
Pages: 256
File size: 7 MB

Read an Excerpt

Dr. Ecco Mathematical Detective

By Dennis Shasha, Warren Linn

Dover Publications, Inc.

Copyright © 1992 Dennis E. Shasha
All rights reserved.
ISBN: 978-0-486-16931-6


The Letter

Whether the jug hits the stone or the stone hits the jug, it is bad for the jug.

Folk saying

More than two years had passed since I had seen or heard from Ecco. I stayed in touch with Evangeline, of course, and each weekend she came up to New York from Princeton. Together we'd pore over newspapers, magazines, and mail searching for a significant clue. So far, nothing.

Many people would have called us eggheads: Dr. Jacob Ecco, mathematical prodigy and the world's foremost omniheurist; Dr. Evangeline Goode, Princeton philosopher with a special interest in the logic of learning; and I, Professor Justin Scarlet of the Mathematics Institute here in New York, my field topology of structures.

By definition, an omniheurist is one who can solve all problems. That sounds like the claim of an advice columnist, but recall the Latin and Greek roots of the word: omnia],ITL all things; heuriskein, to find out. As developed by our friend, the science of omniheuristics applies a mathematical approach to the solving of problems. Omniheuristics is more a method, a way of seeing things, than a specific field of study. Specialists may supply critical data, but the generalist omniheurist, through insight and reason alone, finds the solution. The very word was Ecco's invention, and he was the first practitioner of the technique. He had employed it to help repair malfunctioning spacecraft, to catch spies, to design buildings. He had even used it in meeting challenges thrown down by the notorious and elusive Benjamin Baskerhound, a rogue genius and erstwhile Princeton professor whose warped love of puzzles had led him to crime. Futurists have acclaimed omniheurism as a science for the new millennium, pointing out that now that terrestrial frontiers have nearly vanished, the problems we face as human beings arise from the objects, systems, and relationships we ourselves create.

Ecco, Evangeline, and I had had exciting intellectual adventures together, which I had the privilege of chronicling for the public in The Puzzling Adventures of Dr. Ecco, and some good times windsurfing. Always, Jacob Ecco was our rigorous mentor and good friend. He was a quiet hero, a man who defied generals and blackmailing criminals alike if what they said was nonsense, untrue, or wrong. We missed his fine mind, his indifference to fame, his love of chess, his passion for cookies.

From the beginning of our search there had been offers of help, well meant, no doubt, but useless. The Director, with whom we had worked in some matters of national security, had told us he would put agents at our disposal if ever we had a promising lead. Personally, his sympathy meant little to Evangeline and me: the Director had a manner that was always arrogant and offensively mysterious, and neither of us liked him.

When Ecco first disappeared, the press was full of stories of the most speçulative kind. One actually proposed that Ecco had been sent up in a secret "Encounters Satellite" to communicate with aliens; others traced sightings of him in all corners of the globe, from Tibet to Tierra del Fuego. Usually, a few telephone inquiries were enough to convince us that someone had made a mistake or was playing a prank. We made several trips, as far as Delhi and Buenos Aires, but in every case the "evidence" seemed to evaporate when we arrived. After a time, the press lost interest, but the theories and tips of well-intentioned individuals kept us busy. The result—or lack of it—was always the same.

On this Sunday, the only relevant news was a piece in the New York Times Sunday Magazine by Cloe Anne Bennet about the fabulously successful Omniheurism Inc., a consulting firm that had risen to prominence shortly after Ecco's disappearance. "Since a client often puts the omniheurist's advice immediately to the test in matters involving great sums of money, hundreds of people, and huge projects, failure can be catastrophic," she wrote. "Success is all or nothing." She went on:

Only a handful of companies and individuals have survived this daunting challenge. Of these, Omniheurism Inc. has come to be the first, and certainly the priciest, choice of civilian and military government officials as well as of wealthy corporations and individuals.

Imagine, if you will, a stately mansion in the elegant Georgetown section of Washington, D.C. A butler escorts the visitor to a large domed room, originally a ballroom. The room is unfurnished except for a single oak desk and a few chairs. Seated at the desk is Dr. Phillip Andrew Smartee, a graduate of Eton and Christ Church, Oxford.

A tall, handsome man, impressively tailored, Dr. Smartee begins all interviews by asking his clients to state their problem. Dr. Smartee speaks so little at this stage that the interaction resembles a psychoanalytic session. On the rare occasions when he has a question, he raises his index finger. When the client stops talking, Dr. Smartee states his question quietly, almost in a whisper. After the problem has been presented to his satisfaction, he raises his finger again and walks through a door in the back of the ballroom. The door leads to what he calls his Idea Chamber.

Sometimes he emerges to ask a few questions, listening to the answers as if deep in thought. Then he returns to the Idea Chamber. When he reemerges, he either provides a solution on the spot or asks the client to return in a day or so for the solution. On my first visit, Dr. Smartee described to an important Hollywood producer how to arrange film shots so as to minimize the expense of keeping famous—and highly paid—actors waiting. On my second visit, he suggested an elevator design to Takomoto, the construction company whose plans for a 200+ story office building had just been announced. Dr. Smartee's design allows elevator cars to change tracks the way trains do ...

Evangeline slowly put the paper down. "Professor Scarlet, will we ever see Jacob again?" Her black almond-shaped eyes filled with tears. Her face showed the heritage of both her parents: she had the straight narrow nose of her American missionary father and the fine-textured, delicate skin of her Chinese mother. Now, in her pain over the loss of Ecco, her Chinese ancestry seemed the stronger, her face the mask of a monk in deep meditation.

"Of course we will," I said. "Ecco is alive and well." I knew my voice lacked conviction.

I looked back at the article. Bennet did not give full details about the various problems presented, but Smartee's solutions bore an uncanny resemblance to Ecco's work, though Smartee's fees were far higher. Bennet had tried to ask this newly famous omniheurist about his background, but Smartee would answer no personal questions. The reporter had taken her investigation to the Pentagon, figuring that the military must have examined Smartee's background before entrusting him with top-secret material, but much of their information about him was classified. Still, Bennet had learned a few interesting things. Apparently, "Smartee" was a pseudonym, although she quoted a master at Eton as saying, "If this is the boy I think, we must have been utterly wrong about him. Though well-born and vain, he never struck me as intelligent."

As we discussed the article, Evangeline and I tried to encourage each other. But the last months had been difficult, and Evangeline's question showed we were both beginning to think what we had never admitted even to ourselves—that we would never see Ecco again.

"How stupid of me!" I exclaimed in an effort to be positive and banish such thoughts. "I forgot to check the mail yesterday." For two years I had instilled every trip to my mailbox with hope for a good lead. But two years of trips to the mailbox had resulted in nothing but disappointment. Why should today be different?

Evangeline went downstairs with me. The box was full of junk mail, but there was one air mail envelope, tattered at the sides. The ink had run, indicating it had been wet. However, its contents were perfectly legible, and we opened it to recognize the unmistakable hand of Jacob Ecco:

My Dear Evangeline and Scarlet,



Running Bulls

Be good and you will be lonesome.

Mark Twain

Evangeline and I took several hours to decode Ecco's letter. We were confused by the rotations, but we understood our friend's motivation and were grateful for his hints about the coding of future messages. I for one wondered whether there would be any future messages.

Ecco wanted to be rescued, there was no doubt of that. But rescues are dangerous and Baskerhound was a formidable opponent. We decided to suppress our dislike of the Director and ask for his help. He had given us a phone number and told us that an answering machine would take our message. If ever we had a lead, we should say, "Jake is awake." We made the call. Within twenty-four hours, the Director came to see us, accompanied by security men. He had not changed much in the past two years. He still carried his six feet two well, and most people would consider him handsome, with his strong jaw and chiseled features. But I always found myself looking at his small, humorless eyes.

"I hope you have a good lead," he said after a short nod of his head that passed for a greeting. "We could use Ecco's help in tracking down some international anarchists. Smartee is no substitute for the real thing. What have you heard?"

We told him that we had received a letter from Ecco and were going to Uruguay. To our surprise, he didn't ask to see the letter. He told us simply that we would have immediate help if we phoned the same number and left the message, "For Jake's sake."

"And if we can't get to a phone?" Evangeline asked.

"My dear Dr. Goode," said the Director, "we have already thought of that." He held up a small metallic object the size and shape of a credit card. "This is a solar-powered transmitter that sends out a distress signal. Our agents call it a pip-card. It emits a low-power signal that our planes, with their sensitive receivers, will pick up provided they are within two hundred miles of you. If you find yourself in trouble, put the pip-card out during the day or under the light of a lamp. Understand?" He handed the card to Evangeline and left. The next evening we took a flight to Montevideo.

Punta Ballena is a ridge rising from the mouth of the Rio de la Plata. Approached from the sea, it looks like the profile of a whale (ballena is the Spanish word for "whale"). The spine of the ridge is the area's paved road. Dirt paths lead from the main road to private farms and ranches, some expensive and well kept, but most ramshackle.

We tracked down the horse-breeder with only minor difficulty, finding him in the stables of one of the largest estates in Punta Ballena. When we asked him in halting Spanish whether he raced his lame horses, he laughed and invited us in for "maté," pointing at his mug of thick tea. "Es bueno," he assured us, "muy fuerte."

The horse-breeder told us about the "crazy gringo." "He takes long rides in the arboretum, sometimes coming back with plant clippings. Whenever he is around, the arboretum dog gets sick and vomits. A week ago, the dog nearly died. Sometimes the gringo brings a younger red-haired man with him. The younger man is almost a prisoner. He is always surrounded by big men with small heads, me comprende?"

Evangeline took out a color snapshot of Ecco, his red hair very evident in the sunlight. "Si, es el rojo," said the horse-breeder.

"They went away," he said. "Last Saturday. El hombre with the red hair left you this." He handed Evangeline a letter. We thanked him, relieved that we didn't have to drink more mate, which tasted to me like hot wet grass. As we were saying our farewells, an exquisitely dressed gentleman, the picture of a Latin aristocrat, came riding up on a stallion. Our host raced to meet him as he dismounted.

"Pedro Alcatraz is my name, honored señor y señora," the newcomer said with a bow to Evangeline. "I have just heard that you are friends of the poor red-haired muchacho whom that crazy Baskerhound is keeping prisoner. I have heard rumors that this red-haired man is the great Dr. Jacob Ecco and that you are his colleagues, Professor Justin Scarlet and Dr. Evangeline Goode. Is this true?"

We both felt that a denial would arouse more suspicion than it would allay, so we nodded hesitantly.

"Please do not worry. I hate that Baskerhound. He is always digging up the plants of my family's arboretum. You see, I am the great-grandson of the great corsair."

He smiled at our surprise. "What is the use of protesting my great-grandfather's innocence? In our family, small boys play 'hide the stolen treasure.' My great-grandfather made his money by turning off the light in the local lighthouse during storms and waiting for shipwrecks. He used the money to collect rare plants and trees. Somewhere in the arboretum is a chest of rare coins, too, but we have never found it. For a time, I suspected Baskerhound was looking for that chest, but he never digs deeply enough.

"Anyway, we have used our wealth for many humanitarian causes in the last few generations. Perhaps you will help me accomplish the goal of my own humble lifetime: to eliminate bullfighting in Uruguay."

Evangeline and I exchanged glances.

"Oh, I've tried everything," Senor Alcatraz said. "Petitions to lawmakers, newspaper advertisements, everything. But the people aren't with me. They need the entertainment. I have given up moralizing. My goal now is to interest the people in bull races instead of bullfighting. Will you help me?"

"We can try," I said uncertainly.

"I am sure you can help," said Señor Alcatraz. "Let me tell you a little history. Two years ago, I built four parallel running tracks on my large estancia near Montevideo. My idea was to release four bulls simultaneously at the start of each track. Each would run to the end of its track. I built a high stadium around the tracks, so the spectators could watch the bulls. Unfortunately, the people found it boring to watch four bulls on four independent tracks. So, I built cross-tracks that allow the bulls to go in any direction they like. The trouble is that the bulls have a terrible habit of fighting with one another. The people love this, Dios Mio, but it is very expensive and defeats the point of my efforts." He shrugged. "One of the fans suggested that I create races in which the bulls start in one track and end in another. My engineers say that if we build bridges or lay barriers across some of the intersections, we could keep different bulls from ever coming into direct contact with one another. The people would like it and the beastly bull fights would stop."

While Senor Alcatraz was speaking, Evangeline had been busily sketching on a piece of paper.

"Let me make sure I understand," she said, showing us what she had drawn. "Here are the bridges and barriers you allow [see top figure on p. 14]. Here is the original design, where the tracks are labeled A-A', B-B', C-C', and D-D' [see bottom figure on p. 14]. Here is your present design with cross-tracks" [see figure on p. 15].


Excerpted from Dr. Ecco Mathematical Detective by Dennis Shasha, Warren Linn. Copyright © 1992 Dennis E. Shasha. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents

In this collection of original puzzles, games, and codes, the heroic Dr. Ecco, a mathematical detective and puzzle solver, takes on his archenemy, Baskerhound, and uncovers a plot that threatens the world. No sophisticated mathematical background is necessary to solve these challenges, which were inspired by the methods and thinking of researchers in computer science and mathematics. All you need is imagination and a passion for puzzles.
A graduate of Yale, author Dennis Shasha received his Ph.D. from Harvard and is Associate Professor of Computer Science at the Courant Institute of Mathematical Sciences at New York University. He includes complete solutions at the end of the book, and has rated the puzzles according to difficulty.

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