”We have removed two squares,” Doctor Maybe offered in explanation, “so why has it now become so difficult to cover the board with the dominoes?”
Tollie looked at the board again, assessing the situation. Two corners had been removed, that was all. Otherwise, it was the same. Why wouldn’t 31 dominoes cover it?
“The diagonally opposite corners are the same color,” Mouse offered, giving Tollie a hint, as was his job.
Doctor Yes: “And we can see that any domino placed on the chessboard will cover exactly two squares.”
Doctor No: “And the squares on a chessboard alternate between black and white.”
Doctor Maybe: “Soooo…each domino must cover one black and one white square.”
“I see,” Tollie said, though he was only beginning to see. “By removing diagonally opposite squares, we are removing two squares of the same color, say two white squares for example, leaving 30 white squares and the original 32 black squares and 62 squares in all…”
He left the problem hanging for a while, mulling it over. Booker Quirm, finished with his lunch, squirmed into the room oozing his gustatory satisfaction, glanced at the chessboard and gave Tollie a snidish grin. Mr Quirm had a small bit of malice in his heart, Tollie thought.
He sighed and returned to his contemplation of the chessboard. The “Aha” reaction dawned upon him. “By removing two white squares you are leaving two black squares unpaired, they don’t have corresponding white squares to form the rectangle that would be covered by one domino. On a chessboard, remember, the domino can only be placed so that it covers one white and one black square. So there will be not one, but two dominoes that cannot be placed on the board, because they will not have corresponding pairs of squares of alternating color to be placed upon.”
Booker Quirm slithered off, disappointed that again Tollie had handily solved the new puzzle. His mind churned over until he had yet a new challenge for Tollie. The same snidish grin stole over his round face. This one would stump him.
Doctor Yes: “Very good, Tollie, well done!”
Doctor No: “Indeed, by removing two squares of the same color you have made it so the board can no longer be fully covered by the dominoes.”
Doctor Maybe: “We have many such puzzles here, it is our game room.”
Mouse winked at Tollie and Tollie winked back. The others had not noticed Mouse’s key contribution to the solution, but Tollie had. The estimable Mr Mouse was certainly well-traveled and there was not much that he had not encountered while serving as first-line guide in the World-Wide Web.
Booker Quirm now sidled over. “We like to play bridge,” he announced portentously.
“I never learned the game,” Tollie confessed.
“The doctors and I play often,” Mr Quirm said, smiling at Tollie and Mouse. The doctors, all three, nodded the affirmative.
“How do you play?” Tollie asked. “I would think it would be difficult for you to hold the cards.”
Booker Quirm now unveiled one of his surprises. From his sides, he revealed two tiny hands. “They are enough to hold the cards,” he assured them. “And now, I have another little puzzle for you…”
CHAPTER 17
NO BIG DEAL
At this moment, in gamboled Li’l Jumbo, trumpeting loudly through his little trunk. The doctors regarded him kindly as he stood in front of them, his baleful demeanor demanding attention. “He wants his water,” Doctor Yes explained.
Doctor No went to the corner of the room and turned a spigot. Tollie and Mouse now saw that Li’l Jumbo had his own water fountain, which let out into a small pool that he could dip his trunk in and draw out as much water as he wanted. They watched as the tiny elephant had his fill. Li’l Jumbo used the bucket for his water only when the doctors had not turned on the fountain, Tollie noted to himself. He thought it smart of Li’l Jumbo to have a backup source of water. But, naturally, Li’l Jumbo had all the intelligence of a full-sized elephant.
If Tollie had any questions about how a worm could have hands even though that was a biological impossibility on the face of it, given that Mr Quirm had no feet, he now realized that it was perfectly possible here, if not in his own world. There were one-foot high elephants here!
The worthy gentleworm, Mr Quirm, cleared his throat for attention and Tollie and Mr Mouse turned from watching Li’l Jumbo to paying attention to the self-important worm. After all, they were guests in the home of the doctors and if the doctors had Mr Quirm in their employ, it was their duty to be courteous to him.
“We were playing bridge here at this very table,” Quirm proceeded, satisfied that he had their undivided attention. “I was the dealer.” He waved his small arms and wiggled his yet smaller fingers, showing that he could, in fact, play cards.
“I dealt out about half of the deck when the telephone rang. I left the table to answer.”
Tollie blinked. He could not for the life of him picture Mr Quirm handling a telephone. But it must have been so. The good doctors were not fazed at all at the idea of a worm answering the phone. Quite the opposite, Tollie got the impression that it was natural to them that this was one of Quirm’s duties here. And why should not the resident librarian answer telephone inquiries?
“When I returned to complete the deal, I found that I had only dealt half the cards, and I could not remember to whom I dealt the last card!”
Why this was so monumental a discovery escaped Tollie’s understanding, but he remained, as was his way, courteous, and waited for Quirm to finish.
“I could have reshuffled, then deal the cards out as usual. But, I realized that there was a way I could continue with the deck as dealt, and deal the rest of the cards out so that everyone would get the same cards they would have if I had not been interrupted by the telephone call. Now… How did I do this?”
He puffed up his chest, clearly proud of his accomplishment in logical reasoning and his timesaving solution. He was not a worm to waste time. He was an early worm.
Tollie scrunched up his face a little, something Mouse had not seen for a while, but the thing was that he did not quite realize what the challenge here was. “He could not remember to whom he had dealt the last card,” Mouse said, hoping to help Tollie see the question posed by the worm. “Then he was able to continue dealing, without having to start over, but guaranteeing that each player would receive the same hand as they would have if the deal had not been interrupted.”
“So he came back to the table, not knowing who got the last card, then dealt out the rest of the cards so that everyone got the same cards they would have gotten…” Tollie said, considered this. “Bridge is a game of four, there are 52 cards in a deck. 52 is divisible by four. In other words, in a regular deal, the dealer would get the last card.”
Mouse danced a bit, humming under his breath, and Tollie took this to mean he was on the right track. Indeed, he now saw the solution. He also saw Quirm squirm, as sure sign that even he could tell that Tollie would ultimately solve his riddle.
“The dealer would get the last card, which is the bottom card! Of course! He deals out a few cards, goes and answers the phone and comes back. Then he deals from the bottom of the deck in reverse, dealing himself what would have been the last card first. The second to last card would go to the second to last player, as it would have if the deal had gone through in a forward direction as usual. By extrapolation, the third to last card would go to the third to last player and so on through the deck until the last card went to the player who would have received it anyway if the deal had gone through forward from the top instead of backward from the bottom.”
Quirm slinked away a distance, crestfallen. Again, Tollie had solved his challenge.
But Tollie smiled at him. “Mr Quirm, it was a wonderful puzzle. I was completely stumped by it at first. I didn’t even see it as a problem that could have a solution. You should not be disappointed.
Mollified, Quirm smiled a bit, almost shyly, then grinned broadly. “You did great, Tollie, w
ith both puzzles. I was just being hard with you, maybe because you were a stranger here. But now I see that you belong and are welcome here. You are one of us.”
“Well, thank you very much, Mr Quirm. I’m happy to be here. And I did enjoy your puzzles very much. I will remember them and they will bring joy to others as well.”
Quirm blushed. This was something none of them had ever seen, a blushing worm! It was more curious than seeing Mouse blush. Returning from his fountain, having drank his fill, Li’l Jumbo snorted. Quirm was almost as purple as he was! Worms look purple when they blush, Tollie thought, making a mental note to remember this, too. It could come in handy in a trivia quiz. Li’l Jumbo came as close to laughing out loud as an elephant, even a tiny one, could.
CHAPTER 18
PICK A NUMBER, ANY NUMBER
Doctor Yes, Doctor No, and Doctor Maybe joined in the laughter.
Doctor Yes: “We have one for you.”
Doctor No: “All you have to solve is how it is done.”
Doctor Maybe: “Yes, No is right, just figure out how we do it.”
Doctor Yes: “Pick a number, any number.”
Doctor No: “Both of you pick a number.”
Doctor Maybe: “That way you will see that it truly works for any number.”
Tollie and Mouse thought up numbers, each on their own. “I have a number in mind,” Tollie announced.
“And so do I,” Mouse added.
Doctor Yes: “Multiply your number by seven.”
Tollie and Mouse duly multiplied their numbers. It took Mouse a little longer, so you can guess he had originally picked a larger number.
Doctor No: “Now add 11 to it.”
That was easy enough.
Doctor No: “Remember the number you now have, file it away so you can recall it later.”
“Done,” said Tollie, and Mouse nodded as well.
“Doctor Yes: “Now, add the digits in that number together.”
Doctor No: “Yes, for example, if your number is 31, you get four when you add the digits together.”
Doctor Yes: “Subtract the number you now have from the number No told you to remember.”
“Yes?” Tollie had done the math, but he wondered where this was going.
“Well, well…” said Mouse.
Doctor No: “Okay, now add 23 to the number you now have.”
They did so, taking a little longer than before. They were working with larger numbers now.
Doctor Maybe: “Add the digits in that number together.”
“Yes,” Tollie said and Mouse said, “Ditto.”
Doctor Yes: “Do you have a one-digit number.”
“No,” said Tollie. “Yes,” said Mouse.
Doctor No: “Tollie, add the digits in your number together again.”
Doctor Maybe: “And you, Mr Mouse, can just work with the number you have, since it is a one-digit number.”
Tollie added the digits together again. “I have a one-digit number now,” he offered.
Doctor Yes: “Excellent!”
Doctor No said with a flourish, “The number you now have is five!”
Tollie and Mouse looked at each other for confirmation. Of course, it was so, they both had five!
Doctor Maybe: “Do you think No is a mind reader?”
“No,” Tollie said. “I think it is a number trick, but I don’t see how it is done.”
The doctors laughed, happy that they had pulled off their trick to the mystification of Tollie and Mouse. But they did not realize they had underestimated Mouse…
“I have noticed,” said that worthy, “that when you take any two-digit number, add its digits together and then subtract the result from the number, you end up with a number that is a multiple of nine.”
Tollie did some quick mental math and realized that this was, in fact, true. He even went beyond this and saw that it held for three-digit numbers as well, and he suspected it would hold true for any multiple-digit number.
“A two-digit number has two columns, the tens and the ones,” he ventured forth in his effort to explain the number trick. “If it is 34, for example, and you subtract 4, you are taking the ones away, leaving it at 30. If you now take away 3, representing 3 tens, you will get 27, a multiple of nine. 34 minus the sum of its digits, 7, will give you 27, a multiple of nine.”
Doctor Yes: “That is correct, Tollie.”
Doctor No: “Another way of representing that is that 30 is three times ten, and three times nine would be three less than three times ten, since there is a difference of one between nine and ten.”
Doctor Maybe: “Let me write that down so we can see it.”
He wrote:
3 x 10 = (3 x 9) + 3, or
3 x 10 = 3 x (9 + 1) = (3 x 9) + (3 x 1) = (3 x 9) +3
“So,” Mouse took up the explanation, thinking at least he could say more than one sentence at a time, and Tollie might appreciate that at this point, having listened at length to the three doctors alternating their sentences, “a number times nine will become that same number times ten, when you add that number again. Four times nine plus four equals four times ten. Four nines plus four equals four tens. Seven times nine plus seven will become seven times ten, and so on, again by that method we have learned is called ‘extrapolation.’ Then, you add to that again the number of ones that exceed the multiple of ten.”
“Yes, I understand,” Tollie said. “It sounds harder than it is, but if you write down some numbers, it becomes clear and obvious. Take any two-digit number, let’s say 67 as an example.” Tollie wrote it down, to illustrate.
67 = (6 x 10) + 7
“Adding the digits together and subtracting, gives you this.”
6 + 7 = 13 and 67 - 13 = 54
“54 is a multiple of nine, as it has to be, given what we have already observed. Six tens plus seven ones gives you the difference between 67 and 54, the multiple of nine. This is true of any two-digit number.”
“That’s perfect,” Mouse said. “It will also hold true for higher numbers than two-digit numbers. If it were a three-digit number, you would be adding the number of hundreds, plus the number of tens, plus the number of ones, and you can extrapolate this to numbers of many digits – the number of thousands, plus the number of hundreds, plus the number of tens, plus the number of ones, all the way to the very largest numbers.”
Tollie considered this. “To complete the doctors’ number trick, you just reverse the process, once you have a multiple of nine. That is, a multiple of nine plus five will give you a number whose digits add up to five. Take 27 as an example. 27 plus 5 equals 32, and adding the 3 and the two gives you 5.
“But the doctors added another wrinkle. They added 23 instead of 5. But the digits of 23 add up themselves to 5, so it is the same as adding 5.”
“That’s right,” said Mouse. “And that is because 23 is itself a multiple of 9, that is 18 in this case, plus 5. So all you are doing is removing the multiple of nine, giving still a multiple of nine, after the 5 is accounted for.”
“Yes,” Tollie said, seeing the whole thing now. “Take any multiple of 9. Let’s say 36 – add 5 and you get 41 and 4 plus 1 add up to 5. But, add 23 (which itself adds up to 5) to 36 and you get 59, wherein 5 plus 9 adds up to 14, and, since you do not have a one-digit number yet, you add the digits together again, and the 1 plus the 4 again add up to 5. No matter what number you chose to start with, the doctors can get you to 5, or, indeed, to any number they wish.”
Mouse hummed his little tune. The trick was revealed.
The doctors smiled. They knew Tollie could figure it out eventually, but they had not realized that Tollie and Mouse together could do it that much faster.
Now Li’l Jumbo lumbered into the room, took up station in the middle, and regarded them sideways with a friendly elephant eye. They all looked at him and he nodded his head up and down to show that he knew he had their attention.
CHAPTER 19
THE MONTE HALL PARADOX
“I ha
ve a teasing little problem for you,” announced Li’l Jumbo.
Tollie and Mouse jumped. They had thought that Li’l Jumbo could not speak. Neither could have explained why they thought that, perhaps it was just because Li’l Jumbo had not spoken before. But they had encountered all kinds of animals that had spoken to them, so there was really no reason the little elephant could not also speak.
Evidently, Li’l Jumbo was over his fear of mice now, as he addressed Mouse along with everyone else perfectly naturally.
“I like to watch old TV,” Li’l Jumbo told them. “Even old game shows. There was one called ‘Let’s Make a Deal,” which was hosted by Monte Hall. I watch that one a lot and I have heard of a paradox related to it. You might know the show.”
Mouse nodded but Tollie had to admit he had never seen it. It was before his time.
“It is a show,” Li’l Jumbo explained, “in which contestants were presented with deals by the host, Monte Hall. These deals usually involved a choice between something they knew, and knew what its value was, against something in a box or behind a door, that could be nothing, something of lesser value, or something of greater value. They could stay with what they had, but they had a chance to get something even better.”
“I see,” Tollie said. “They could lose it all or they could improve on what they had.”
“Yes,” said Li’l Jumbo. “Now, the paradox I heard is about a contestant who has a toaster oven already, but is offered a prize of considerably greater value that is behind one of three doors, perhaps an automobile. He has a one-in-three chance of getting a car, so he goes for it and picks door number 1.”
“I would probably do the same. A toaster oven is nice, but having a one-in-three chance of getting a car is tempting.”
“Yes. So, he has picked door number 1, but… Monte Hall now opens door number 3 and reveals that there is nothing behind it. He asks the contestant if he would like to change his choice from door number 1 or stick with door number 1. The question is: Should the contestant stick with door number 1 or switch his choice to door number 2. He now knows there is nothing behind door number 3.”