Tolliver's Travels . . . with Mr. Mouse in the Worldwide Web

by Fred Stang

  “We’ll just take a short tour,” Tollie said. “We have to see one of the villages – the Village of Truth, I would think. That would be better for us. Who knows how calamitously we could be misled in the Village of Lies?”

  Mouse set out with Tollie. “I know of this island. It is not that easy to get to the Village of Truth, since we must first figure out which one it is.”

  “Hmm… I didn’t think of that. How would we know which one is which? We couldn’t just ask, for we would not know whether we were speaking to a Truth-teller or a Liar.”

  “Think of the Apples and Oranges problem we faced with Young Chef,” Mouse suggested. “Since the labels referred to their respective boxes and since they were all wrong, we could figure out which was which by pulling out only one fruit. Perhaps there will be a question that would let us know which way to go to get to the Village of Truth, no matter which villager we encounter, whether he be Truth-teller or Liar.”

  Tollie looked doubtful.

  Sure enough, they came upon a fork in the road, and there, where the path diverged into two, stood a villager. There was nothing in his appearance or demeanor to betray his status as a truth-teller or liar. You simply could not tell.

  “Suppose,” Mouse said, “that all people from Crete are liars. You meet a Cretan and he says to you ‘I am lying.’ Is he telling the truth or is he lying?”

  Tollie thought this over, his face scrunching up a bit. “How can I answer that? If he is telling the truth, then he is lying. And, if he is lying, then he is telling the truth, and if he is telling the truth…”

  “Yes, exactly,” Mouse said, peering at Tollie through his bright eyes. “It is known as the self-referential paradox. The Cretan is referring to himself, but he is not simply describing himself, he is also describing his own statement within his statement. That is, the statement is about the statement itself. In a similar manner, if you can get the villager to have to answer self-referentially, you might be able to get the correct answer, no matter which villager it is.”

  The villager addressed them at this point.

  “Hello there. You seem strangers here. Can I help you?”

  The villager was friendly enough, but Tollie did not see how he could help them if they did not know which he was, truth-teller or liar. How could you ask him anything and be sure of getting a helpful answer?

  “In the case of the boxes of oranges and apples,” Tollie mused, “since all the labels were wrong, there was one thing you knew – that whatever the label on the box said, that was NOT what was in the box.”

  “True, true,” Mouse said encouragingly. “But in this case, IF it is a truth-teller, then he will be saying which way the village is.”

  “And the liar would tell you the way to the wrong village, the Village of Lies, say, since we seek the Village of Truth…” Tollie thought out loud.

  The actual villager watched them curiously. Tollie looked back at him. “I could just ask you if you are a Liar or a Truth-teller,” he said, “but, then, if you were a Liar you would still say you were a Truth-teller, and I still would not know which you are.”

  The Villager smiled apologetically. Not very helpful but not harmful either. Tollie realized he would have to outwit the Villager, truth-teller or liar, either way. And he would have to do it in one question, as he already realized. If it could not be done in one question, it could not be done in two! So far, he had reasoned correctly, but he needed the one question to be the right one, the one that would work. He had simplified the problem, but he had yet to come up with that one question.

  Something struck him about what he had said before, about asking the villager directly if he was a truth-teller. The truth-teller would say “yes” but so would the liar. You would get the same answer from both, but, even so, you would not know the truth. Was there a question about the road to the village that would get the same answer from both the truth-teller and the liar AND still tell you which road to take?

  If you asked “Is this the road to the village of the truth-tellers?” and it was the road, the truth-teller would say “yes” and the liar would say “no.” BUT, if it was instead the road to the village of the liars, the truth-teller would say “no” while the liar would say “yes.” You don’t know which villager you are speaking with, so the answers give you no information. A better question was needed.

  Tollie pondered on this. Mouse looked down each road curiously. The villager stood expectantly waiting. Then Tollie asked the question.

  “Is this the road to your village?” He indicated the left road.

  Of course, if it was the road to the truth-teller’s village, the truth-teller would say “yes (it is the road to my village).” And, as it was not the road to the village of the liars, the liar would also have to say “yes (it is the road to my village).” Thus, a “yes” answer would tell you the road led to the Village of Truth.

  Likewise, if it was the road to the liar’s village, the truth-teller would say “no (it is not the road to my village).” The liar, too, would say “no (it is not the road to my village).” Because it was the road to the liar’s village, he would be forced to lie and say it was not the road to his village. A “no” answer would tell you it was the road to the Village of Lies.

  “Well,” Mouse said, “that question will do it!” Then Mouse chuckled quietly, and Tollie looked at him quizzically.

  Oh,” Mouse went on, “I was just thinking about the time I was the guide for an anthropologist. He had quite a different question he used to find out which road led to the village.” And he laughed at his recollection.

  “What other question could there be?” Tollie asked, doubting there was any other question that would do the trick.

  Mouse laughed merrily. “He asked, ‘Is it true they are serving free beer in the Village of Truth?’” Then he simply followed the villager.

  Tollie laughed with Mouse and they set out for the Village of Truth. Of their adventures there, we will hear in a future volume, but, suffice it to say, they learned much and had a wonderful time. And when they were done, they returned to Maddie and her helicopter.

  CHAPTER 13

  BETWEEN THE LIONS

  Upon seeing them, Maddie did a little dance and sang, in the same tune as Mouse’s little song:

  “I fly, fly a whirl ybird,

  Fly a whirlybird, bird,

  Whirlybird…”

  Mouse did his little jig and joined in.

  “Happy, happy are the mice,

  happy are the nice, nice,

  nice, nice mice…”

  Tollie wondered if everyone here sang the same tune! He recalled Ancient Egyptian’s song. But, thinking there was nothing better to do at the moment, he joined in with the fun.

  “We are, living in a dream,

  living in a dream, dream,

  streaming dream…”

  And the three of them danced and sang happily together, forming an improbable but merry trio in a circle under the beamish sun.

  When they were aloft in their “whirlybird” they looked back at the island, the two villages and the expanse of ocean that grew between, until the island was but a speck on the horizon.

  Maddie called their attention forward, to see a little island full of grass, lions and one sheep, a curious place. “This island is populated by 100 lions, one sheep and much grass,” Maddie told them.

  Mouse looked and visibly shuddered. “Those are cats, Tollie, nothing but big cats!”

  “We are safe, Mr Mouse, we are up here and they are down there,” Tollie assured him.

  “I still don’t like it,” Mouse grumbled. “Cats!”

  “We won’t land on the island, Mr Mouse,” Maddie said, offering her own assurance. “They can’t hurt you.”

  “There are so many of them, running all over the place. That poor sheep!”

  “Now that you bring it, up, Mr Mouse, that is an interesting question about this island. Why don’t the lions eat the sheep? They don’t have to, they have
enough grass to eat to live on. But they don’t eat the sheep, which surely would be tastier for them. Why not?”

  Mouse was not much mollified, but Tollie took an interest now. He wondered why, indeed, the lions did not simply eat the sheep.

  Maddie read his expression. “There is a special condition that exists on this island. If a lion eats a sheep, he turns into a sheep, and then he could be eaten by another lion. The problem is why they cannot eat the sheep. Since the sheep is there and alive, why not just eat it? Perhaps the lion that ate it and turned into a sheep would be eaten as well, but perhaps not. How do they know whether or not they can eat that sheep and get away with it? There are a hundred lions and they are not eating that sheep.”

  “Yes,” Tollie said, seeing the reasoning behind the question. “How do they know whether or not they would be safe if they ate the sheep? Since they are not eating the sheep they must have figured out it would not be safe to do so. But why?”

  “Exactly,” Maddie said.

  Mouse still looked miserable, barely able to look at the island with all those big cats roaming over it. “I’m just glad they’re not eating that poor little sheep! But why not?”

  Tollie now recalled young Carl Friedrich Gauss and his solution of the addition problem. There might be a sequence that would explain why the lions did not eat the sheep. There were one hundred lions and one sheep… “What if there were one lion and one sheep?” he asked out loud.

  Mouse groaned. “Then he would eat the sheep,” he said plaintively. “He would turn into a sheep but that would not matter, because there would be no other lion that would eat him in turn.”

  Maddie was realizing something, too. “If there were two lions and one sheep…” she ventured, leaving the thought hanging.

  “They would not eat the sheep,” Tollie concluded for her. “If one of them ate the sheep and turned into a sheep, the other lion would eat him in his turn!”

  Maddie smiled. ”If there were three sheep, then a lion could eat the sheep and be safe, because you would be back at the case where there are two lions and one sheep and the sheep is safe!”

  Mouse seemed heartened by this, but he was still too disturbed by the mere sight of the lions, the big cats below, to fret much about following the argument through.

  Tollie took mercy. “Let’s leave this area,” he suggested to Maddie. “I think we have seen all we need to see and we have our answer to why the sheep is safe.”

  To Mouse’s vast relief, they headed away from the island, and he relaxed as the distance from it grew, until he was comfortable again.

  Gathering his wits again, he took on the completion of the reasoning.

  “One lion, one sheep, the sheep is not safe. Two lions, one sheep, the sheep is safe, because if a lion ate it, he would be back at case one, one lion and one sheep, and the sheep (which he has now turned into) is not safe.

  “If you extrapolate that you will find a pattern.”

  “Yes,” Tollie agreed. “That is how Young Carl would have done it. “Extrapolation is extending the sequence out. One lion and one sheep, the sheep is not safe. Two lions and one sheep, the sheep is safe. Three lions and one sheep, the sheep is not safe. Four lions and one sheep, the sheep is safe because eating the sheep would take it back down to three lions and one sheep – the lion that ate the sheep and turned into a sheep would not be safe… Take it out to one hundred lions and one sheep.”

  “The sheep is safe,” Mouse concluded. “It is the same case as two lions and one sheep. Given any even number of lions and one sheep, by extrapolation the sheep is safe.”

  “That is why,” Maddie agreed. “If a lion ate the sheep and turned into a sheep, there would be 99 lions and one sheep, and he would not be safe. If there is an odd number of lions and one sheep, the sheep is not safe. Because one hundred is an even number, the sheep is safe.”

  Mouse smiled at this. He felt much sympathy for the poor sheep, alone among all those big cats. He was still grateful he was not the sheep. Even though he would be safe, he just would not be comfortable on that island full of cats!

  CHAPTER 14

  WHICH DOCTOR?

  The Island of Doctor Yes, Doctor No, and Doctor Maybe

  “What is that island, the one in the distance that glitters so?” Tollie asked, having noticed something of interest on the horizon.

  “That is the Island of Doctor Yes, Doctor No, and Doctor Maybe,” Maddie told him. “They are brothers, triplets in fact.”

  Tollie thought it odd that they should encounter yet another set of triplets!

  “They live together because they were born with certain conditions that would make it hard for them to be separated,” Maddie went on. “First, they must speak in turn, the same doctor cannot speak twice in a row. Doctor Yes, Doctor No, and Doctor Maybe must each speak before Doctor Yes can speak again. In each case, each of them can say only one sentence.

  “Then, they can say what they want except when they are asked a ‘yes-or-no’ question. In that case, Doctor Yes is only allowed to say ‘Yes’ or ‘Maybe,’ Doctor No is only allowed to say ‘No’ or ‘Maybe,’ and Doctor Maybe is only allowed to say ‘Yes’ or ‘No.’ Since they were born in the Village of Truth, which you visited just a while ago, they must still tell the truth insofar as they can, given these conditions.”

  “Interesting,” Tollie said thoughtfully. “A very interesting situation. And, apart from when they are answering a yes-or-no question, they can say whatever they want, as long as it is true, since they are truth-tellers from the Village of Truth. But they must speak in turn, one sentence at a time. If they are to communicate at all with the outside world, they must stay together.”

  “That is the nub of it, and the rub of it,” Maddie confirmed. “Now I am going to drop you off on their island. I’m afraid that I will have to leave you there. I have to take my whirlybird back home now, for routine maintenance. But you should have a good time with the doctors. They are entertaining fellows.”

  “Oh, I see,” Tollie said. “Well, Maddie, thank you for the ride. I thoroughly enjoyed it!”

  “It was a pleasure,” she said. “And I will miss both of you, but I expect I will see you again another time.”

  “You will see me, I am sure,” piped in Mouse. “I am always around.”

  “Of course you are,” Maddie said. “I see you all the time and you are the absolute best!”

  Mouse blushed, an odd thing considering he was a gray mouse.

  Maddie landed her helicopter and took her leave. “You just have to walk up the hill, to the house of the three doctors. They will be gracious hosts and you will enjoy yourselves there.”

  “Thank you, Maddie,” Tollie said, always polite. “I hope to see you again.”

  Mouse, still blushing, just smiled at her.

  They watched as the helicopter whooshed into the air and headed off over the vast ocean, becoming just a wisp of motion in the air. Tollie glanced at his watch and saw that it was noon. Time was surely relative here. He knew it must still be nighttime in the “real” world.

  Turning, they headed toward the path leading up the hill to the impressive mansion of the three doctors. Now that they looked at it more thoroughly, they noted how modern and spacious it was. The glass walls glinted in the sunlight. Facing them and the ocean was the broad, curved window that they guessed would have been for the living room, as it would have the best view from the hill. As they walked up the path and drew nearer, they could see that the curved glass windows were darkened, no doubt to shield the interior from the midday sun. Mouse and Tollie had started up the hill at noon, so the overhead sun would be quite bright viewed from the home of the doctors, especially that one wall, situated as it was to give the best view. Later on, they learned that the glass would lighten up, much like a pair of eyeglasses that would be shaded when the light was bright, but clear when it was darker.

  Making their way up the path, they could see that a lot of attention had been paid to its design as we
ll. A waterfall paralleled the path, cascading down in sections, divided by pools formed on the way down, each spilling over to the next waterfall and thence to the next pool. The pools were surrounded by landscaped small gardens with inviting resting areas, so one could pause on the way up to enjoy moments of tranquility. One could tell that the doctors were the kind of men who paid attention to the details that added together to make a world of their own, one created to induce a calm sense of wonder and enjoyment that made time seem to slow so that these simple elements were experienced in a leisurely pace.

  About halfway up, Tollie and Mouse stopped at a particularly beautiful pool and rested on the bench placed there for that purpose. They listened to the water falling into the pool and watched it flow off at the other end. It was an idyllic interlude for them and they felt fully refreshed when they started again up the path.

  Arriving at the house, they were greeted by the three doctors at the top of the path. They were tall, slim, elegant men, wearing crisp white cotton smocks, as befitted doctors, and white cotton slacks, white shoes, and no socks. Since they were outside, they also wore panama hats, perched jauntily atop their heads at an angle. Practical garb for the lush tropical isle they inhabited. They smiled pleasantly and gestured welcomingly to their home. Tollie thought they were quite a contrast to their encounters with the other triplets. One would take these men more seriously, but, still, they were very pleasant and well-mannered and one instinctively liked and trusted them.

  “We have been expecting you!” said the first doctor, “Doctor 1.”

  “Yes,” said Doctor 2, “and you are very welcome.”

  “We have been watching your travels in the World-Wide Web,” said Doctor 3.

  Tollie remembered what Maddie had said. They had to speak in turn, one sentence each.

  “How so?” he asked. He wondered how they had been watching him and Mouse as they went from adventure to adventure.

  It being Doctor 1’s turn, he answered. “We monitor the World-Wide Web to make sure that the places you go to are safe for you.”