“Yes,” continued Doctor 2. “there are things like viruses, worms, Trojan Horses and bugs that can cause you a lot of trouble if you are not careful.”
“We try to make it easier for you by watching out for them and preventing them from interfering with your travels,” concluded Doctor 3.
Tollie looked at Mouse questioningly. “This is true, Mr Mouse?”
“I’m afraid so, Tollie. I can only take you where you want to go, I cannot protect you from what may be there when you arrive. Doctor Yes, Doctor No, and Doctor Maybe are computer programming experts. They devise programs that can protect you from other programs, developed by more wicked people than are you, I or the good doctors.”
“It is perfect work for us,” said Doctor 1, “given the speech rules we have to follow.”
“This way we can work without having to be among a lot of people, and we are still doctors, so to speak, as our work is curing,” Doctor 2 elaborated.
Doctor 3 summed it up. “It would be hard for us to work among a lot of people, and, of course, all three of us would have to work together, or we literally could never finish saying anything.”
Doctor 1 changed the subject, as it was time to let their guests know the house rules. “You are very welcome here and we will do our best to show you a good time, but there is something you must first do to stay here.”
“Yes,” Doctor 2 took up the explanation, “anyone who comes here and is going to stay, must first identify us properly.”
“That is,” Doctor 3 said, “you must figure out which of us is which, Doctor Yes, Doctor No, and Doctor Maybe.”
“And you cannot directly ask us to identify ourselves,” Doctor One elaborated.
“You cannot ask ‘Are you Doctor so-and-so?’ directly, in other words,” Doctor 3 explained.
At that moment their attention was diverted by a one-foot high purple elephant galumphing through the room. He was a cute little fellow and Tollie and Mouse were quite entertained by the sight of his lumbering elephant form, flapping ears and his trunk swaying to and fro. He went to a bucket in the corner and took his fill of water, expelling it mightily toward them, so that they felt a fine spray across their faces. He then trumpeted his satisfaction, a fine strong baritone emanating from such a small creature, though he was surely intended to be much larger!
“He is ‘Li’l Jumbo,’ our house mascot,” Doctor 1 explained.
Li’l Jumbo galumphed back from whence he came, giving wide berth to Mouse. Tollie recalled that some elephants were indeed afraid of mice and laughed. Mouse smiled, as he was perfectly harmless and would not hurt a fly, or even a cat (though he did avoid them), much less a little elephant.
The three doctors smiled at their amusement upon seeing the little elephant, a rare sight. “Li’l Jumbo is quite a character and we treat him like a member of the family,” Doctor 2 said.
Going back to the challenge presented them by the doctors, Tollie said to Mouse, “I am thinking again of the Apples and Oranges challenge. As with the three doctors, we have three choices, so if we identify two of the doctors, as with the Apples and Oranges boxes, we will automatically be able to identify the third.”
“That’s right,” Mouse said. “We need only identify two doctors and then we will know which the third is. But what questions will we ask”
“Li’l Jumbo just gave me an idea about that,” Tollie answered. “Here, have a look at this.”
Tollie proceeded to draw a chart, in the form of a grid. Then another one. “These will outline some questions and their possible answers, which will identify our doctors.”
Mouse took a curious look at Tollie’s grids. His questions, listed in the grids, indeed concerned Li’l Jumbo, and could in no way be taken to be referring to the doctors themselves, directly or indirectly. So they were within the rules.
Grid 1
How Doctor Yes, Doctor No and Doctor Maybe would have to answer:
(Remembering that the doctors must answer as truthfully as possible, given their limitations.)
Grid 2
How the answers of Doctor 1, Doctor 2 and Doctor 3 would identify them:
Tollie explained.
“YYN is a possible sequence of answers for the first three questions. If so, we know Doctor 3 is N or M, as he answered “no.” So he must be N, since Doctor 1 and Doctor 2 could only be Y or M and, therefore, M is not available for Doctor 3. We can see this by looking at Grid 2. We know Doctor 3 is N, but we do not know whether Doctor 1 and Doctor 2 are Doctor Y or Doctor M, as this is still indeterminate, they could be either. Therefore, we require a fourth question.
“Another way of looking at it is that one of the doctors must answer twice if we are to get a comparison that would identify that doctor. One answer from each doctor will not give enough information to identify them or any one of them. Once a doctor gives two answers, however, his identity is known, as can be seen by examining Grid 1 above.
“Since they must tell the truth, insofar as the rules allow them to, the optimum strategy is to vary the questions between those that assume a true premise (is it an elephant?) and those that assume a false premise (is it a dog?). This will force them to answer differently according to the rules they each must follow, allowing us to deduce which Doctor is which.
By looking at Grid 1, we can see that a “yes-maybe” pair of answers to questions 1 and 4 means it is Doctor Yes. A “maybe-no” pair means it is Doctor No. A “yes-no” pair means it is Doctor Maybe.
There are three possibilities. Once we know one doctor’s identity, we can examine the grid to determine who the other two are.
If Doctor Yes, the doctor who answered “maybe” to question 2 will be Doctor No.
If Doctor No, the doctor who answered “maybe” to question 3 is Doctor Yes.
If Doctor Maybe, the doctor who answered “yes” to question 2 is Doctor Yes.
Once we know the identities of two of them, just as in the case of the boxes of Apples and/or Oranges, we will know the identity of the third doctor.
The three doctors nodded approvingly. “Quite logical,” Doctor 3 said, as Doctor 2 had been the last speaker. The good doctors had good memories.
“I especially like the way you made use of the grids to solve the problem.” Doctor 1 said.
“Carl Friedrich Gauss would have approved,” said Doctor 2.
“You know about Carl?” Tollie asked in wonderment.
“Oh, yes,” affirmed Doctor 3, “we had to approve him for the World-Wide Web you explored, to make sure he was the real Carl Friedrich Gauss.”
“Of course,” Tollie said. “I see.”
Doctor 1 extended his hand. “I am Doctor Yes.”
“And I am Doctor No,” said Doctor 2.
It was Doctor 3’s turn. “You’re method was excellent, so we might as well let you know without making you go through all the questions, and I am Doctor Maybe, of course.”
“Didn’t you just sneak in two sentences in the guise of one?” Tollie asked mischievously.
“Maybe,” said Doctor Yes, equally mischievously. Of course, if he had passed his turn, which would acknowledge that Doctor Maybe had taken two turns, Doctor No would also have said “Maybe,” Tollie realized. Hoist by his own petard, he grinned.
The doctors waved them into their house and they followed in anticipation. They knew by now that it would be an interesting experience.
It was a big shambling house of many rooms, each with its own character. They entered the modern living room with the expansive curved view, but, beyond, they could see bits and pieces of other rooms of a quite different character. There was a library visible, for example, containing full walls of bookshelves filled with new and old books of all sorts, with sliding ladders arranged to allow access to the higher volumes.
From the expansive window, they could see the ocean, much of the island, and, very clearly, the path they had ascended to get here, including even the pond where they had stopped to rest. They could see, too, that
this path was the only way to get to the doctors’ house. So they would always know in advance that someone was coming.
Tollie’s eyes returned to the library. All those books! They fascinated him, for some were visibly old books, and he loved old books. They had knowledge from ages past that had either developed into our modern knowledge or had been mostly forgotten. But they were still of interest, since they gave an idea of how we as people thought of the world through time.
Seeing his interest, Doctor No said, “Please, you are very welcome to come and see our library.”
Doctor Maybe: “Yes, of course, we must show it to you.”
Doctor Yes: “Come, come!”
CHAPTER 15
MR QUIRM
Many a Quaint and Curious Volume of Forgotten Lore
Entering the library, the three doctors, Mouse, and Tollie were greeted by a long smiling creature, small enough to fit in your hand, but long enough to wrap itself around your thumb. He was perched on a shelf directly in front of their noses.
Doctor No: “Mr Quirm, allow me to introduce our new friends, Tollie and Mr Mouse.”
“Hello, Mr Quirm,” Tollie greeted the creature, politely as was his way.
“Mr Quirm,” Mouse acknowledged.
Doctor Maybe: “Mr Quirm is our librarian.”
“That I am,” Mr Quirm agreed. “My name is Booker Quirm, but they call me Book for short, Book Quirm. As you can see, my name is similar to my occupation. I am a bookworm. I could simply devour volumes and still want more books. But do not worry, I only eat those parts of books that have no print, mainly the margins.” He giggled merrily at the thought of all those delicious musty pages. “What could be a better career for me than being a librarian?”
Doctor Yes: “Mr Quirm keeps all our books in order so we can find what we want among them.”
Doctor No: “In exchange, we let him eat away to his heart’s content.”
Doctor Maybe: “As long as he keeps his promise and eats only those parts of the page that are without print.” He gave Mr Quirm an admonishing glance.
“I am about to have lunch, as a matter of fact,” Mr Quirm announced. “So, you will see that I keep my promise. I only eat through the blank margins of the pages. They are more delicious anyway! None of that ink. Ink is yukky.”
He performed a comic side-winding slide toward a section of the bookshelf on which he had been perched, approaching an imposing looking ten-volume set.
“That is lunch, I presume,” Tollie queried.
“Oh, yes, and a feast it will be!”
Mouse looked at the little fellow, wondering why he could not just eat cheese, like any sensible creature.
“Now, Tollie,” Mr Quirm continued. “I have been watching, along with the good doctors, your progress through our World-Wide Web and the many challenges you have faced and solved along the way. I now have a challenge of my own for you.”
“Oh?” Tollie felt a sense of wonderment that even this small but friendly enough worm would present him with a puzzle, if puzzle was what it was to be.
“I am going to eat my way through these ten volumes for lunch. It is a ten-course meal, you might say.”
Tollie noted a certain pomposity in the little worm, but tolerantly accepted it. Mr Quirm, was, after all, proud of his well-developed gastronomical tastes.
“Your challenge is to tell me how far I shall travel as I eat my way through the ten volumes, from page one of volume one to the last page of volume ten!” Mr Quirm announced all too portentously for his size. ”Each volume is two inches thick, with the covers being one eighth of an inch thick each. You might say that the pages of each volume make up one and three fourths inches, while the covers take up one fourth of an inch. And if you said that, you would be right. Each volume would still be two inches thick in toto.” He swelled his chest in pride, having showed off his skills in the dead language of Latin.
Latin was a subject Tollie had not found endearing. It was more something he had been enduring. Nevertheless, he put up with Mr Quirm’s quirky verbal mannerisms. His interest was piqued by the puzzle presented.
“Wouldn’t you be eating through 20 inches minus two covers, one eighth of an inch each or a quarter of an inch total, so you are traveling nineteen and three fourths of an inch?”
Mouse nudged him. “Think it through a little more, Tollie. Where is page one of the first volume and where is the last page of the last, the tenth volume? Think of the books as they are on the shelf and then as they are when you pull them out to read them!”
“Of course!” Tollie said, realizing it was not as simple as he had thought. “The first page of volume one is on the right side of the book and the last page of volume ten is on the left side… This is looking at them as they are on the bookshelf versus how they are when you pull them out and read them.”
Mouse smiled helpfully. Mr Quirm literally squirmed, realizing the jig was up and Tollie would solve his riddle. The good doctors smiled genially.
“Mr Mouse is very good to you, Tollie,” observed Doctor Yes.
Doctor No: “You do realize he can only give you hints?”
Doctor Maybe: “But they help you along your way, even though in the end you are left to figure it all out yourself.”
“Oh, I appreciate Mr Mouse very much indeed,” Tollie allowed. “He is my guide and friend here.”
Mr Quirm wriggled himself into a more comfortable position. He was smug enough, though, as he knew that in the end he would have his book feast.
Tollie proceeded. He had the measurement of the distance Mr Quirm would travel in eating his way through the set of books. “From page one of volume one all the way to the last page of volume ten, Mr Quirm will travel, as he eats his way along, a distance of sixteen and one fourth inches.” He stated it baldly, as a fact. Mr Quirm looked a bit put out.
This is because page one of volume one is on the right side. He will eat his way through one eighth of an inch to get to volume two. He will eat his way through volumes two through nine, a total of sixteen inches. He will then eat his way through one eighth of an inch of volume ten to get to the last page of that volume. This is a total of sixteen inches for volumes two through nine. He eats that plus one eighth of an inch, the front cover of volume one, and one eighth of an inch more, the back cover of volume ten.”
Mr Quirm set out on his meal. His riddle had been solved and he was hungry. And, yes, it would be sixteen and one fourth inches of savory, musty pages and covers that he would be eating. He set at his lunch with gusto, and they watched as his tail disappeared into the front page of volume one. They could hear him now as he started munching his paper trail.
Now Doctor Yes gestured toward the game tables. “We have all kinds of games here,” he said.
Doctor No: “This one here is our chess table, the board is in the middle, and you can see we even have timers, so we can play official chess.”
Doctor Maybe (gleefully): “But we have a new problem for you!”
Tollie, Mouse, and the good doctors moved over to the board.
CHAPTER 16
CHESSBOARD AND DOMINOES
“Is it a chess problem?” Tollie asked. “I like them but I have trouble solving them.”
Doctor Yes: “It is not a chess problem, but let us call it a chessboard problem.”
Doctor No: “And a domino problem!”
Doctor Maybe: “Yes, a chessboard and dominoes problem.”
Doctor Maybe gestured around the room now.
Tollie looked around and realized that the game room adjoined not only the library but also the living room. The great curved window swept through here also, and he could see the wonderful view the doctors had of the island, the ocean, and the path they had ascended to get here. It was an architectural accomplishment and somehow Tollie knew the doctors had designed the entire property themselves, the gardens and paths as well, as their home and personal workspace. Since they patrolled the World-Wide Web they needed a wide vista, so they could
see the many things that were there to be seen. His attention focused on the chessboard on the table.
It was inlaid into the table, which was a normal game table, on which any game could be played. He noted the green felt top, which could cover the table if you wanted to play cards, for example. Or dominoes. He noted also that the doctors had many games stored in the under-structure of the table. They were clever fellows indeed. Chess, checkers, go, dominoes, backgammon, Scrabble, decks of cards, all were there. At present, the chessboard was blank. The chess pieces had not been set up.
Doctor Yes pulled out a decorative wood-inlaid box and slid its cover open, revealing a set of dominoes.
“Note that each domino covers exactly two squares of the chessboard,” he observed to Tollie and Mouse. To illustrate, he covered two squares of the board with one domino.
Doctor No: “You see, a chessboard has 64 squares, 8 X 8, and we can cover it completely with the dominoes, using 32 of them, since each one covers exactly two squares.”
“But,” Doctor Maybe took up the presentation of their puzzle, “suppose we remove two squares from the chessboard, specifically the two diagonally opposite squares!” He said this with considerable glee, much as Mr Booker Quirm had presented his puzzle.
Tollie marveled that they spent their time with such entertainments as a matter of course. He was glad to live in the real world, he thought, even though he greatly enjoyed visiting the virtual World-Wide Web. “All right, he said, suppose we do remove those two squares?”
“Well then,” Doctor Yes finally posed the challenge, “could we still cover the chessboard completely with the dominoes, using 31 now, since two squares have been removed?”
Tollie squinted at the board. He did not immediately understand the problem. “Why not?” he asked. “Each domino covers two squares and we have removed two squares, so we still have an even number of squares. 31 dominoes should do it.”
“Well, let us see you do it,” Doctor No challenged.
Tollie began covering the board with the dominoes, two squares at a time. But, try as he might, he found that no matter how he did it, the last domino was left without two squares to place it on. At first he thought it was a matter of his not arranging them properly. But after repeated attempts, he began to suspect there was some reason it could not be done.