Archive for the ‘Number Theory’ Category

The Road Crossing

Tuesday, October 23rd, 2018

Seymore, a green frog, wanted to cross the road to get to his favorite pond where frog mates were plentiful. However, the road was a dangerous place to cross, and Seymore was worried he might get run over by a passing motor vehicle.

Brer Rabbit had told Seymore that on the average about a 100 cars would pass per hour along this stretch of road, which was valuable information indeed.

Seymore needed one minute to hop across the road and any car passing by could be fatal.

So he looked up at his lucky star and got ready to jump.

What would you say is the probability that no car would pass while Seymore was crossing the road and destroy his froggy dream to arrive at his favorite mating pond?

The Chess Match

Monday, October 22nd, 2018

One Sunday afternoon, a chess match was being held at the King’s Hills Chess Club where bets were being made on Henry MacDuff versus Melvin Longspur. Henry MacDuff was rated as a three times better player than Melvin Longspur.

Judge Roy Bean decided that they would play ten rounds and whoever won three consecutive times would win a copy of the famous Royal Diamond Chess set.

How many rounds would have to be played for Henry MacDuff to have a good chance of winning three consecutive games?

The Emperor’s Triangular Array

Wednesday, October 17th, 2018

To emulate and surpass the famous emperor Qin’s necropolis achievement, emperor Wu Shu, who preferred to be known as “He Who Cannot Be Counted,” decided that after the end of his life he wanted to be buried at the head of an army of 10,440 terracotta soldiers standing at attention in a triangular array of rows.

“Chow Sao, I want to know the cumulative Grand Sum of the number of ways of arranging soldiers in each respective row,” said emperor Wu Shu to his Feng Shui advisor.

“This Grand Sum must be perfectly divisible by 12, the number of heavenly animals in the Shengxiao zodiac,” added emperor Wu Shu.

“Yes, my glorious emperor,” moaned Chow Sao, “your wish is my command,” starting to genuflect reversing himself out of the emperor’s lush palace quarters.

“If the Grand Sum is not perfectly divisible by 12, I want you to tell me exactly which rows of soldiers must be removed to obtain a zero remainder,” commanded emperor Wu Shu.

“As you wish, my illustrious emperor,” said Chow Sao, shuffling more rapidly to increase his reverse velocity as he saw empress Soo Lao enter the palace room.

“This grand sum must be perfectly divisible by 12. Empress Soo Lao has informed me that I shall suffer great misfortune in the heaven life unless this is so,” emphasized emperor Wu Shu, waving his scepter.

“This is so, Chow Sao,” said empress Soo Lao sternly, “any mistakes, and I will have sorceress Ba Fa turn you into stone and place you as the first soldier in the first row.”

“The Gods forbid,” said Chow Sao accelerating his backward velocity at a phenomenal rate and absconding from the emperor’s quarters in a great hurry to begin his calculating task.

Would you say that this Grand Sum was divisible by 12, and if not, would any rows of soldiers have to be removed from the triangular array of soldiers envisioned?

 

Intriguing Puzzles Book 1, containing 50 fascinating puzzles selected from Ken’s Blog, is now available in the e-book formats mobi and epub.

Click here for more details.

 

A Basket of Fruit

Monday, September 24th, 2018

“I hear the jingle of the fruit vendor Zoltan passing by, Lennie,” said mommy. “Won’t you please run out and buy some fruit for your lunch box next week, and take this basket.”

“Sure, mommy.” Lennie ran out with the basked to catch Zoltan before he rode away on his bicycle stand.

“Mommy wants me to buy some fruits for my lunch box,” said Lennie eagerly as he arrived breathing heavily at the fruit stand.

“Fruits are good for you, young man,” replied Zoltan, “I have five apples, four pears and three plums. What would you like for your lunch box.”

“I’ll take three apples, two pears and one plum, please,” said Lennie.

“That will be one dollar and twenty cents, my little friend,” said Zoltan.

“Ooops, I forgot to bring some money,” said a flustered Lennie.

“Young man, if you can tell me how many different baskets you could fill with apples, pears and plums, taking none or any number of each of these fruit types each time, you won’t have to pay,” laughed Zoltan, who had taken a shine to the bright boy.

“Ok, sir, I’ll think about it,” said Lennie, scratching his head full of hair…

“Mommy, here are some fruits for my lunch box next week,” said Lennie, handing over the basket with the fruits.

“How much did it cost,” asked mommy.

“I didn’t have to pay because I gave the right answer,” said Lennie.

Can you work out how many baskets Lennie told Zoltan could be filled with the fruits on display?

 

The Chinese Round Table

Friday, August 31st, 2018

One fine evening, Jack and Jill were dining at the Fu Manchu Magic Noodle House, their favorite Chinese restaurant. They sat at a round table served by their regular waitress, Wang Shu.

A number of dishes containing delicious tidbits for plucking off with their chopsticks were situated in front of them on a rotatable round tray.

Jack and Jill sampled the many delectable Chinese morsels to their heart’s content.

“Pick a fortune cookie, Jack,” said Jill, pointing at the decorated cookie jar in the center of the table.

“The cookie says: ‘Good fortune. Soon get special dessert’,” said Jack.

“Hmm,” said Jill, “what does it mean, Wang Shu?”

“Now, special desert,” announced Wang Shu while clearing up after dinner.

Wang Shu placed four dishes containing fried ice cream balls and spicy chocolate truffle balls – delicious Chinese desserts.

In each of the first three dishes there were two fried ice cream balls and eight spicy chocolate truffle balls. The fourth dish contained six fried ice cream balls and four spicy chocolate truffle balls.

“Fortune cookie offer special dessert,” said Wang Shu.

“How generous,” replied Jack.

“Bring it on, Wang Shu,” said Jill.

“Close eyes,” said Wang Shu, “you too,” addressing Jill. Wang Shu spun the dessert tray with the four dishes.

“Please, keep eyes closed and pick one dessert from dish,” said Wang Shu to Jack. Jack picked a dessert ball, which Wang Shu quickly replaced from a large pot containing both dessert types.

Jack opened his eyes and saw a fried ice cream ball melting in his hand, which he promptly ate to his great satisfaction.

“You guess which dish ball from, you no pay. Boss say special promotion,” informed Wang Shu.

“We like that,” said Jill.

Jack scratched his head and then wrote some figures on a napkin, Jill peeping on. Then Jack pointed to a dish.

“Correct dish you choose. Boss say, this time free, next time you pay plenty,” said Wang Shu with a big smile.

Jill was already busy munching away at the balls on the dessert dishes.

“Doggie bag, please,” said Jack.

Was Jack just lucky or had he worked out a probable dish to pick?

Which of the dishes do you think Jack chose?

Straight vs. Full House

Tuesday, August 14th, 2018

On another fine evening, there was a heated discussion at the Royal Flush Card Club as to how many occurrences of Full House and Straight would occur in the course of 30 throws of five poker dice.

Bing Jones III was betting $100 against Franz Fingerflitzer that he would get at least two instances of a Full House, the latter claiming at least two Straights during the trial – ties leading to a rematch until a winner emerged.

Lots of paper was expended and the whisky flowed freely while members of the Royal Flush Card Club busied themselves calculating the probabilities involved for throwing a Full House and a Straight.

However, accuracy and interest faded rapidly as whisky was poured, and this task was abandoned in favor of registering bets and proposing exuberant toasts, expectations rising to a high pitch, as if prior to a championship horse race.

Who do you think won the bet, Bing Jones III or Franz Fingerflitzer?

The Deck Split

Sunday, July 29th, 2018

At the Royal Flush card club one evening, there was a lively discussion among its members about splitting a deck of cards into two equal piles after a thorough shuffle, in which, respectively, there would be exactly ten red cards in one pile and exactly ten black cards in the other pile.

Bets were taken for twelve trials to check whether this event would occur at least once.

How much would you bet on this event occurring in the course of twelve such splits?

The Date Lineup

Wednesday, July 25th, 2018

Wilbur went to a dating club one Saturday evening to pick up a date for dinner.

As he entered The Quick Date Club, he saw what he wanted: a tall brunette with a shapely figure. She had the name Alice displayed on a badge dangling from a well-filled blouse.

The system at the The Quick Date Club was to seat an equal number of men and women at random in a row with eight numbered chairs, the numbers and names to be displayed on a screen on the wall.

The rule was that any female sitting next to a male would be eligible for a date request, unmatched persons to be recycled for the next round.

What would you say was the probability that Wilbur sat next to Alice so he could invite her for dinner?

Playing “Dragon Slayer”

Saturday, June 30th, 2018

Iggy was playing “Dragon Slayer” on the screen. On his quest for the Holy Grail, Sir Lancelot mounted on his white charger was nearing the bridge of the Grand Dragon who posed a riddle, and the eerie scream of the Knight who answered incorrectly as he was hurled into the dark abyss with his steed could be heard fading away.

The Grand Dragon breathed fire and spoke in a hollow, deep voice: “Answer my riddle Knight or perish in the abyss.”

“Pose your riddle, Grand Dragon and be quick about it,” replied Sir Lancelot defiantly.

“As you wish, bold Knight: The generators of a right triangle are the numbers of the Beast and of Man. The area will be a number that attracts all vibrations from zero to nine. What is the number my brave Knight?” boomed the Grand Dragon.

Iggy put the game on hold to be able to work out the answer to the riddle, which puzzled him no end. He called his friend Leonard to ask what a triangle generator was, which Leonard quickly explained.

With a calculator, Iggy was able to work out the answer fairly rapidly, which he entered for the Grand Dragon’s approval.

“You may cross the bridge Knight,” boomed the Grand Dragon, “and go in peace.”

Can you work out the area of the triangle?

 

The Winning Seats

Thursday, June 28th, 2018

The McDuffy Travel Agency was offering special prizes for vacationers to improve its turnover. The advertising stated that the prizes would be determined on the airport bus.

One day there were fifteen vacationers going to the Caribbean island of Buena Siesta for a two week holiday at the Balmy Breezes Beach Resort. Before boarding the airport bus, which seated twenty persons, they were addressed by their charming tour guide Melanie Goodbottom.

“Two of you could win some awesome prizes if you are lucky enough to sit in the two seats with pre-determined prize seat numbers,” explained Melanie Goodbottom waving a clipboard with the passenger list at the entrance to the airport bus, skirt flapping in the wind.

“The winners will receive a week of free dinners for two at any of the best restaurants in Morgan Town, their choice of sightseeing tours for two all over the island and/or scuba diving lessons for two – for those of you who are of the sporty type, of course,” said Melanie Goodbottom enthusiastically, brushing some windblown locks out of her eyes.

The vacationers eagerly thronged towards the airport bus door, climbing over each other to mount the steps of the bus. Those who entered scurried to seat themselves at random, leaving five seats unoccupied for staff.

The bus driver grunted, sitting down at the wheel, and the porter got busy loading luggage into the lower compartments of the airport bus, while Melanie Goodbottom helped pull a number of the more portly passengers up the steps leading to the seating level of the bus.

When all the passengers were finally settled into their lush seats eagerly anticipating the results, Melanie Goodbottom picked up a microphone to announce any winners via the loudspeaker system, and the bus driver switched gears to get the show rolling towards the airport.

What would you say is the probability of two passengers winning these prizes?