“It doesn’t matter,” Tollie said. “He now has two doors each with a fifty-fifty chance of having a car behind it. He could switch or stay with door number 1.”
Mouse had a small but gracious coughing fit, which Tollie took to be a hint that he had answered too quickly. But was there more to this than he thought? Tollie thought it over.
It seemed that his logic was right. At first, with a choice between three doors, the odds of a car being behind door number 1 were one-in-three. With door number 3 revealed to have nothing behind it, the odds were now one-in-two, or fifty-fifty.
So it did not matter which door he picked, each would have a one-in-two chance of having a car behind it. He glanced at Mouse, wondering if Mouse had gone astray somehow.
Mouse admonished him, “Think of the first situation, where all three doors are closed. The odds of a car being behind either door number 2 or door number 3 would be two-in-three while the odds of it being behind door number 1 would be one-in-three.”
Tollie thought this over. It was right, of course, but what did it have to do with the problem at hand? Surely the situation changed when Monte Hall opened door number 3 and revealed that there was nothing behind it!
Or… did it?
If it didn’t, why not? Tollie was thinking now. Applying the principles of reasoning to the problem. It was something he had learned to do in the course of his travels in the World-Wide Web and he realized as he was doing it how wonderful it really was! He now understood better the meaning of Young Carl’s advice. The words came back to him:
“I always believe there are answers, it is just a matter of finding them. But believing in the reality of there being an answer is always the first step. Even if there is not an answer, you have to show why not. And you can think your way through it, by patiently applying logic and reason to the question at hand.”
This was indeed what Tollie was now doing and it seemed second nature to him. It had not been that way before he entered the World-Wide Web, met Mouse and embarked upon their adventures. The realization made him happy. It had turned out to be a wonderful birthday present.
Too, he realized something else. Just because Monty Hall had revealed that there was nothing behind door number 3 did not change the odds, the probabilities. There was still a two-in-three chance that a car would be behind door number 2 or door number 3, the only thing that had changed was that now he knew there was nothing behind door number 3. But that knowledge did not change the situation, which remained the same.
“Door number 1 has a one-in-three chance of having a car behind it,” he announced, a little proudly. “Door number 2 and door number 3 have a two-in-three chance of having a care behind them. Now that we know door number 3 does not have a car behind it, we also know that door number 2 now has a two-in-three chance of having a car behind it! Door number 1 still has only a one-in-three chance so we should switch our choice now to door number 2!”
Li’l Jumbo did a little dance now!
“One-foot, one-foot high am I,
One-foot high elephant,
One-foot high…”
He, too, had his song! Tollie wondered yet again about this, the song of the World-Wide Web… He found it natural, as it should be. The song of life, the celestial music of the spheres… Here in this brave new world.
“Oh, Tollie, you have done well!” said Li’l Jumbo, winding down his dance. “If you doubt it, think of this. You have ten doors rather than three and chose door number 1. Then eight of the other doors are opened, all with nothing behind them. You are left with door number 1 which you chose originally, and door number 2. Door number 2 still has a nine-in-ten chance of having a car behind it, but your choice, door number 1, still has only a one-in-ten chance!”
“Yes, that is true,” said Tollie. “You have extrapolated it right.”
Li’l Jumbo took up his dance and trumpeted mightily his satisfaction, producing a sound worthy of a full-sized elephant. He was clearly proud of himself as well and gave a bow, going down on one knee to do so.
Mouse smiled at the little elephant. Tollie thought it interesting in that Mouse was four-foot high and Li’l Jumbo only one-foot high. What a world it was… Yet, Mouse and Li’l Jumbo could get along here.
The three doctors now stepped forward.
Doctor Yes: “Come, come!”
Doctor No: “Yes, there is yet more here for you to see.”
Doctor Maybe: “Come, come!”
Tollie and Mouse took their leave of Li’l Jumbo, who trotted off happily to his personal water fountain. Tollie and Mouse left with the good doctors, as they did not want to wait too long. Li’l Jumbo might spray them again!
CHAPTER 20
REFLECTIONS ON THE CIGAR GAME
They entered the game room, or simply moved over to it, as it was really a part of the living room and library, something Tollie had not noticed before. It was a large L-space, with the library occupying the short arm of the L and the living room and game room taking up the long arm. The space was deceptive as each area could have been a large room by itself. Adding to this the high ceilings, one could be forgiven for not at once realizing it was one continuous space. The curved window took up the outer edge of the long arm of the L, discombobulating that honorable letter’s form by only a small warp.
It was a magnificent game room, awesome in content as well as dimension, having ample room for billiard and pool tables, a ping-pong table, air hockey table, foosball, a card table and a chess table, which were the same as Tollie had seen before, though he had not realized that he had wandered outside the library then and into part of the game room. The high bookshelves of the library had obscured the game area. Also, the game room then merged right into the living room, as he could see more fully now, from his vantage point at the corner of the L. This entire L space was geared to learning and fun, the library, the game area and the living area all of one piece and laid out for convenience, yet seeming separate somehow. You had only to walk from one area to the next yet it truly did feel like you were entering different rooms.
He looked around at the walls, which were covered with illustrations of puzzles and enigmas of various sorts, geometrical and logic puzzles, optical illusions, art trickery, mazes, and metal and wood interlocking puzzles which could be disassembled and then reassembled if you could. There was a larger than usual Rubik’s Cube constructed of different shades of natural wood that made a wonderful decorative piece, yet could still be manipulated into its millions of possible permutations.
Tollie could see now that an arch led from the living area into an ample dining room, capable of seating twelve comfortably and yet still a cozy space. “Eat, drink, and be merry,” Tollie thought. And read and play games and dance as well, there being room for all manner of activity.
Having already seen the chess table and card table and realizing their situation between the library and the game area, Tollie now looked beyond the billiard and pool tables and saw a man sitting in the living area, on a couch in front of a rectangular coffee table. He was bulky and wore a pin-striped three-piece suit that contributed to the impression that he might have been a banker, but it was the gold pocket-watch with gold chain curving elegantly from his vest that completed the characterization. Cigars were scattered all around him, but he was not smoking, though it would not have seemed out of character.
He stood and walked briskly toward them, moving gracefully given his weight, and extended his hand.
“Hello, PJ Nagrom here, and who might you be?”
“I’m Tollie and this is Mr Mouse,” Tollie said, extending his hand in return. “We are travelers and guests here, guests of the good doctors.”
“Well, so am I,” said PJ. “The good doctors are kind enough to be my hosts. And I, in return, am indeed their banker. You can bank on me!”
“They are our hosts as well and I take it you handle their financial affairs with great discretion,” Mouse said, extending his “hand.”
“Indeed, I am
good at it.” Ceremonies being observed and his role here clarified, with a certain pride of his own, PJ waved them over to the couch and the coffee table.
“I was just considering a new game,” he began. “This excellent setting inspired me. I thought I would invent a new game. Would you like to play?”
“We would like to,” said Mouse. “But I must let you know, my good friend here, Tollie, has become an accomplished player.”
Now it was Tollie’s turn to blush. He let it go, as there really was nothing for him to say. He had met a few challenges by now and was curious to see what PJ would propose.
“I have been thinking about a new game, a cigar game, on this very coffee table.” PJ gestured expansively, encompassing the coffee table, the couch, and all his cigars strewn about. But I have a problem. It seems to me that it is possible that one player will win while the other must surely lose. Either the one who moves first or the one who moves second. I am not sure.”
“What do you mean?” Tollie asked, his curiosity piqued. “Please tell us the rules of your new game.” He now saw a glimmer of a reason for all those cigars surrounding the heavy banker.
“The players take turns laying cigars on the coffee table. You notice that the cigars are standard, they have a rounded end and a flat end. So the players can place the cigar by lying it down or they can stand it up on the flat end. The object of the game is to be the last player to lay a cigar on the table before it runs out of space. Is there a move that the first player can make or the second player can make that will guarantee victory for that player?”
“Wouldn’t they have to play it out to see?” Tollie asked. “It is a normal-sized coffee table, a lot of cigars would fit on it. How could you tell the outcome in advance?”
“Perhaps,” Mouse suggested, “there is a strategy that one or the other player can use that will win the game. An algorithm, if you will.”
“There is also the fact that there are two ways of placing the cigars, one flat and one standing on end, balanced on the flat end of the cigar,” Tollie reflected. “So there are a lot of combinations of these up-and-down and flat placements. You are asking for a strategy, a method of placement, an algorithm as you call it, that, if followed throughout the game, will guarantee victory to one or the other player.”
“That’s right, Tollie,” said PJ Nagrom. “I have a feeling there is such a strategy, but I cannot figure it out.” He paced up and down, moving gracefully given his bulk, deep in thought.
“There is only one position on the table that cannot be duplicated on the other side of the table,” observed Mouse quietly. “That would be the center. Any other position would have a corresponding opposite position.”
Tollie and PJ thought this over, its relevance becoming slowly clear. It was to be a hint to Tollie, but it really revealed the whole thing, something unusual for Mouse, who usually gave small hints. Tollie was working out a strategy based on Mouse’s observation.
As PJ paced back and forth, Tollie gave voice to his thoughts on the solution. “I believe one player could win if he made the right first move. That move would be to place the cigar on end, standing up, in the center of the table, since the center is the only position that cannot be duplicated on the opposite side of the table. The second player would place his cigar anywhere on the table. Now the first player has only to duplicate this move on the opposite side of the table, that is, make the reverse move.”
“Yes, yes,” said PJ Nagrom excitedly. “The first player can always play a cigar on the table, since the second player’s move will always have its equal and opposite position. When the second player plays the last move on one side of the table, there will be a corresponding last move on the opposite side, so player one will always have a position to place his cigar and will be the last one to have a space to place it on. He wins the game.”
“But,” Tollie added, “player one will also have to reverse the cigar itself. If it is laid down along its length with the rounded end pointed in one direction, he will have to lay down his cigar with the rounded end pointing in the other direction. Otherwise, it is not properly mirrored and player two will be able to introduce asymmetries that will make the outcome of the game unpredictable. He could lay a cigar at an angle to the previous cigar, cutting off the small space allowed by the rounded shape of the end, which could not be duplicated in reverse unless the cigar itself had been placed in reverse as I described. If it had not been so reversed, it is possible that a cigar could be so placed that it would cut off the space for another cigar, by spanning two cigars for example.”
“I see,” said PJ. “The cigar must be symmetrical but reversed. Round end north, then next cigar placed on the diagonally opposite side with round end south.”
“That would cover it,” said Tollie.
Mouse hummed a little. He was proud of himself, for he had in fact seen the solution before anyone else, but pride was unbecoming in a Mouse, so he refrained from his little dance. In fact, only Tollie noticed the humming. PJ was already looking at them with a mischievous gleam in his eye. Mouse and Tollie could tell that another puzzler was coming…
CHAPTER 21
PING-PONG BALL… AND PINS GALORE
PJ Nagrom paced for a while, chomping on one of his cigars. Tollie wondered if he ever really smoked them.
“Here, here,” he said, “what you have is a ping-pong ball.” PJ fell silent, as if this were enough.
Tollie and Mouse watched him. If there was a train of thought going on, they did not want to interrupt it.
“You have also a length of pipe, fastened to the floor at one end, the open end facing up, one foot high. The diameter is just such that you can drop a ping-pong ball into the pipe and it will rest on the floor. You can visualize this?
Mouse and Tollie nodded duly and affirmatively.
“Well, then, you also have a sharp pin such as a tailor uses to fasten hems, a baker’s rolling pin, and a bowling pin, three various pins in all. How can you get the ping-pong ball out of the pipe?”
PJ Nagrom looked around at them triumphantly. He was sure this was the puzzle that would stump them.
Tollie indeed felt stumped! “I can see that you could perhaps spear the ping-pong ball with the sharp pin, but how would you get it down there? The rolling pin and the bowling pin don’t seem to help.”
Mouse looked dejected. He could not see any use for the pins at all. “If you used the sharp pin to spear the ping-pong ball, the ball would be damaged. This violates the premise that you are to get the ping-pong ball out of the pipe intact. I know it is not stated outright, but certainly there would be no use in ending up with anything less than a usable ping-pong ball. Suppose you had to crush the ball to get it out. That would seem to violate basic principles. You would not really be getting the ping-pong ball out of the pipe fairly, and we have only been working with fair problems, that have fair solutions.”
“Well said!” Tollie agreed, and they both looked at PJ.
“I might be heavy, but I am not unfair!” protested PJ Nagrom. “You can get the ping-pong ball out intact and with no piercings or crinkles of any kind marring its spherical perfection.” He huffed and chomped vigorously on his cigar, evidently offended at even a mere suggestion that he might not be posing a fully honest problem.
“Okay, okay, we did not mean to suggest that you were not being aboveboard or that your problem is not a legitimate one, with a legitimate solution,” Tollie said to mollify the big man’s displeasure. “We just cannot see how the pins can be of any help in getting the ping-pong ball out of the pipe!”
“I never said they would be,” PJ said airily. “All I said was that you have them.” He looked around at the ceiling, perhaps seeing things that were beyond the perceptive abilities of Tollie and Mouse to detect.
Said Mr Mouse at this point, “You say then that the pins might not have anything to do with the removal of the ping-pong ball from the pipe?”
“I have presented you with the problem at ha
nd. It can be solved.”
“If the pins are of no use, why did you mention them?” Tollie asked.
“Because, of course, you do have them, as the challenge is given.”
“If we have to use other means to remove the ball,” Tollie protested further, “surely you have to tell us what other means are available for us?”
“Not necessarily,” PJ answered mysteriously. “There are some things that you might be said to have about you all the time.” He looked at them, his posture challenging, arms folded, as though daring them to claim that this statement was not self-evident.
“I can think of no way,” Mouse announced after a long pause, during which all had fallen into a meditative silence.
Tollie’s scrunched-up face showed no sign of impending illumination of the solution to their newest challenge.
PJ waited patiently, seeming to believe they could indeed solve his puzzle, if only they had the right flash of insight.
But the silence dragged on, and PJ, Mouse, and Tollie all paced around in deep thought. Tollie was beginning to think that PJ’s thinking was somehow mistaken. He could not bring himself to believe PJ was being deceptive, as it did not seem in line with what they had learned of his character generally.
But, finally, both he and Mouse had to acknowledge they could not find a way to get that ping-pong ball from the ground to the top opening of the pipe.
PJ laughed uproariously, his entire bulk quivering with mirth.
“Why,” he said, “you just pee in the pipe and the ping-pong ball will float to the top.”
The two traveling companions looked at him questioningly, plainly feeling there was something unfair about this answer, as it did not involve the use of even one of the various pins posed in the original challenge.