Archive for the ‘Number Theory’ Category

The Boring Homework

Thursday, March 29th, 2018

“Daddy, give me an interesting problem to think about. I’m bored with these homework problems, they’re all too easy,” said Junior laying down his pen on his notebook in disgust.

“I’ll give you one Euler offered to his pupils,” said daddy.

“Tell me, tell me,” said Junior enthusiastically.

“Divide 100 into two summands so that one is divisible by 7 and the other by 11,” said daddy.

Junior got busy with his pen on the notebook…

“I liked that one, daddy,” said Junior, “tomorrow I’ll give it to my teacher in class to see if she knows the answer. I’m sure she’ll be very happy,” laughed Junior, clapping his hands.

What do you say are the two summands?

The Fruit Basket

Wednesday, March 28th, 2018

One sunny afternoon, Mrs. Cuddleworthy was sitting in the shade of a tree on a small hill in the Rolling Hills Village Park observing children play below her. She had a basket of fruits which contained four apples, three oranges, two pears and a plum.

Mrs. Cuddleworthy was in doubt about how to distribute these fruits to two imp-faced youngsters – a boy and a girl – who had climbed up to her with eager eyes attracted by the fruit basket decorated with a red cloth.

I wonder in how many ways could I distribute the fruits to these two little urchins? …

After some moments of confused thought, Mrs. Cuddleworthy decided to hand out the fruits in a random manner. Later she would ask her son Leonard to work out the answer to her puzzle.

What would you say are the number of ways Mrs. Cuddleworthy could hand out the fruits, ensuring that none of the children would receive nothing?

The Strange Number Pair

Wednesday, February 21st, 2018

One evening, Jack found Jill sitting at a café table with her attention riveted on a small, well-worn and water-logged ship’s logbook in her hand.

“What’s up, Jill,” he said cheerfully. “You look puzzled.”

“I really am, Jack. My great uncle drowned recently when his sailboat sank in the Aegean Sea near Skiathos, and I’m trying to make sense of a message he wrote in his logbook before sinking regarding my inheritance. They found it floating with him in the sea,” said Jill.

She gestured for Jack to sit down, calling for the waitress to bring a coffee.

“Sorry about your great uncle, Jill. What’s the message?” asked Jack, sitting down.

“It says I have to go to the SchwitzerliBank in Zurich to obtain my inheritance, but I need to give them a special code number to receive it,” said Jill in frustration, pulling on a black hair lock.

“And what’s the number?” asked Jack, preparing his coffee with a spoon.

“Well, that’s the problem. It’s in some kind of code,” said Jill with a baffled look on her face.

“Explain, please,” said Jack, taking a careful sip of his hot coffee.

“On this page my great uncle says:

‘Move the last to the first of the six, then find the four greater twin and use it,’”

said Jill, “which doesn’t make any sense to me,” she groaned, randomly flipping the pages of the logbook.

“Hmm, that really is a mouthful,” said Jack. “Seems your great uncle didn’t want to advertise the number.”

“No joke. Got any ideas?” Jill looked up at Jack with big green eyes full of hope.

After thinking a while on the strange statement, Jack suddenly got an idea.

“You have to give a number to the SchwitzerliBank in Zurich to get your inheritance, right? Maybe this is a six-digit number,” said Jack.

“Could be you are right, but what about ‘…move the last to the first of the six’?”

“Hey, maybe the word ‘twin’ means another number derived from the first,” said Jack enthusiastically.

“In this case you get it by moving the last digit to the first position.”

“Yea, that really makes sense. But what about the ‘four greater’ part?”

“Seems the second number is four times greater than the first, and since it says ‘use the twin’ I would guess this is the number to present to the bank,” said Jack.

“Good thinking, Jack. But how do we work out what the twin number is?”

“Well, I’m taking a course in Number Theory and I’m sure I can work it out. So, what will you give me in return,” said Jack with a roguish grin.

‘How about dinner at your favorite Hindu restaurant – followed by some dessert,” said Jill demurely.

“Deal! Give me a pen and a piece of paper, Jill.”

What would you say the number Jill had to give to the SchwitzerliBank in Zurich was?

The Egg Dozens

Monday, January 22nd, 2018

Mr. Jones, the math teacher, was told by his wife Molly to go to farmer Elmer Sudd’s grocery store to buy some eggs as she wanted to bake a cake for their son Seymour’s birthday.

Mr. Jones puttered out of the house in his loafers and headed for Elmer Sudd’s store by the farm.

“Mr. Sudd, do you have some eggs for sale,” inquired Mr. Jones politely, adjusting his thick-lensed glasses on his bulky nose.

“Mr. Jones, you flunked my son Homer in math. That’s why he didn’t graduate from high school this year,” growled farmer Sudd, fixing Mr. Jones with beady eyes.

Mr. Jones shuffled uncomfortably in front of the shop counter and began polishing his glasses with great concentration, checking the level of polish against the ceiling light every now and then.

“Yes, well… that was very unfortunate,” stammered Mr. Jones, “but your son Homer got all the problems wrong on the final exam, and spilled ink on my shirt – so I had no choice,” he apologized.

“But, please, do you have any eggs for sale?” repeated Mr. Jones. “My wife will be very angry if I show up without any eggs for the cake which she is baking for my son Seymour’s birthday,” appealed Mr. Jones.

“Is that so, Mr. Jones. I do have some eggs and you can have them if you can solve a problem I read in Reader’s Digest that should be simple for a math teacher,” said farmer Sudd with malicious grin.

“Very well, what’s this problem?” inquired Mr. Jones timidly.

“The product of the eight divisors of the number of eggs I have is 331776,” said farmer Sudd with a smile twisted halfway to his ear.

“If you can tell me how many dozen eggs I have, I will sell them to you for a premium price, Mr. Jones, so you can make your wife happy.”

Mr. Jones stopped polishing his glasses, pulled out a Casio calculator and tapped the keys ten times, after which he announced the correct dozen of eggs to farmer Sudd.

“Hmmm… Homer, get ready, I’m sending you to summer school to learn math,” announced farmer Sudd to Homer skulking in a corner.

Can you work out how many dozen eggs farmer Sudd had?

The Fibonacci Demo

Saturday, December 30th, 2017

“You lose again, Lenny“ sneered Snipe after his loaded dice turned up another seven on the green felt table mat.

“Now you owe me a hundred dollars. Hand it over now,” demanded Snipe, hand extended.

“Well. . . seems you have had an extraordinary streak of luck, Snipe,” replied Lenny, raising an eyebrow while handing over a Franklin note.

“To show I’m not a bad loser, how about a bet for two hundred bucks this time?” suggested Lenny.

“What kind of a bet,” said Snipe suspiciously, eyes narrowing while he pocketed the Franklin note.

“Well, it goes like this, Snipe. You think of two numbers. Then I ask you to manipulate the numbers some times, after which I tell you the result,” explained Lenny.

“You mean, I don’t tell you what the numbers I think of and manipulate are? said an incredulous Snipe.

“That’s right,” said Lenny with a mischievous smile.

“You’re on, and you’ll lose again,” laughed Snipe.

Lenny pulled out a pocket calculator, a yellow notepad and a pen from his backpack and placed them on the table in front of a surprised Snipe.

“Here you go, Snipe. You just think of two numbers and get ready to write down all the results, so we can check them later, if necessary,” explained Lenny.

“I’ve already got two numbers in mind,” said Snipe, eyeing some cars passing by.

“Then add them together,” instructed Lenny.

“No problem, “ replied Snipe, noting the sum with his pad well hidden from view.

“Now add one of the two numbers you first thought of to the sum you just got,” continued Lenny.

“Done, no sweat.” Snipe was already anticipating the two hundred dollars he would soon win.

“To the new sum, add the last sum.”

“Ok.”

“To the new sum, add the last sum,” said Lenny.

“Ok, how long will this continue,” complained Snipe with a bored expression on his face.

“To the new sum, add the last sum,” said Lenny.

“You sound like a parrot.”

“To the new sum, add the last sum,” said Lenny.

“Come on. I don’t have all day.” Snipe was getting irritated.

“To the new sum, add the last sum,” said Lenny.

“Ok, the last time or the bet’s off,” threatened Snipe.

“Now divide the last sum you have by the previous sum,” said Lenny.

“Good enough,” said Snipe.

“Multiply the result by ten and strike out the fractional part,” said Lenny.

“Ok, and now what?”

“Now I’ll tell you what the number you have left is, and if I am right, you lose the bet,” said Lenny.

“No way you can know it. Go ahead, tell me the wrong number.”

“The number you have left is sixteen. Show me your notepad results,” demanded Lenny.

A surprised Snipe handed over the notepad.

“Rats, I don’t believe it!! Tell me how it works.” said Snipe mouth agape.

“Pay up the two hundred dollars and I’ll give you some advice,” said Lenny.

Snipe handed over the two hundred dollars. “So what’s your advice,” said Snipe eagerly awaiting the advice.

“I won’t tell you how it works, because you’ll use it to make money from people ignorant of mathematics. My advice is to get some education and learn a useful profession,” said Lenny.

Snipe threw down the pen and stomped off in disgust.

Can you work out how Lenny knew the final result?

The Pentagon Estate

Monday, December 18th, 2017

Five-star general Chester T. Hopscotch (the T standing for Tecumseh, a Shawnee chief he admired very much) decided to retire at the age of 75.

The reason general Chester T. Hopscotch  waited so long to retire was that he had been looking to purchase a pentagon-shaped estate to live on – for nostalgic reasons.

His adjutant, Major Janice Crackshaw, had located a property shaped like a pentagon enclosed by rows of palms on a balmy island in the Caribbean. The property Major Janice Crackshaw found was owned by a certain Cantus Sperarum Pythagorean Society which wanted to sell the property to finance a special educational project.

Adjoining the pentagon estate, were five properties in the shape of right triangles, each respectively contiguous to a side of the pentagon estate. The triangular properties were owned and occupied by five elders of the Cantus Sperarum Pythagorean Society, their leader, the Hierophant, living in the smallest triangular property as an indication of humility.

The Cantus Sperarum Pythagorean Society was willing to sell the pentagon property to a buyer who could demonstrate signs of mathematical appreciation and perspicacity, as a neighbor without these admirable qualities would be anathema.

The buyer would have to demonstrate this mathematical appreciation and perspicacity by determining the correct price for the pentagon property – this price being five million dollars times the ratio of the area of the pentagon property to the total area of the five adjacent triangular properties belonging to the Cantus Sperarum Pythagorean Society.

The correct payment offer was to be delivered within 15 days.

Major Janice Crackshaw informed General Chester T. Hopscotch of the conditions of the purchase, who, being of a mathematical bent, was glad to also have found neighbors with a similar inclination, and immediately set about to calculate the payment offer with a pen, a slide rule and a block of yellow paper.

General Chester T. Hopscotch worked out the purchase amount in a short while.

The offer made by Major Janice Crackshaw was gladly accepted by the Pythagorean Society, which looked forward to having an erudite neighbor.

Can you work out how much General Chester T. Hopscotch offered to pay for the pentagon estate rounded to the nearest dollar?

The Boxes of Wine

Thursday, November 30th, 2017

A wealthy merchant decided to send 63 boxes containing bottles of wine plus seven individual bottles to 23 relatives in New York for their New Year’s celebration.

The bottles were to be shared equally among the relatives.

Any relative who could guess how many bottles of wine there were in each box would win another box full of bottles of champagne.

How many bottles would you say there were in each box of wine?

The Globe Balls Quiz Show

Saturday, September 16th, 2017

The host of the show, Barnstorm McDoodle, stood on an elevated podium studded with multi-colored lights. Barnstorm was wearing a jacket splattered with bright small reflectors, flashing a wide Hollywood smile under a pencil thin moustache – his thick, abundant black hair extending backwards like freshly plowed furrows in a field.

A large, transparent plastic globe full of thousands of small colored plastic balls stood in the middle of the floor between Barnstorm McDoodle and the opposing consoles of team A and B. From the globe a tube led into a small tray.

“Tonight, the contestants – Jack and Jill on my left, Mabel and Seymour on my right – can win a lot of money by guessing how many balls there are in this large globe,” announced an animated Barnstorm McDoodle, waving his arms regally to contestants and audience.

“But…if they fail…they win nothing, and the pot will be doubled for next week’s contestants,” added Barnstorm McDoodle.

“To investigate the contents of the globe, the contestants will enter numbers at their panel, to be repeated on this large screen. The number will divide all the balls in the globe into equal groups, ungrouped balls rolling out from the globe through this tube into the tray – which balls my lovely assistant Jane will fetch and replace in the globe.”

“The first team to correctly state how many balls the globe contains will win an amount in dollars equal to the number of balls,” proclaimed Barnstorm McDoodle.

“Any team can request a five minute break to work out a guess, but if it’s wrong they’ll be disqualified, so let’s get on with the show,” shouted Barnstorm McDoodle.

“But first a word from our sponsor, Ace Dogfoods.”

After Ace Dogfoods’ prolonged message, Jack and Jill entered “2” at their panel, which then appeared on the large screen. One ball rolled out into the tray and Jane stepped forth briskly on long legs to put the ball back into the globe.

“Write it all down, Jill,” said Jack.

Mabel and Seymour entered “5” and four balls rolled out into the tray, rapidly picked up and replaced by a smiling Jane.

Jack and Jill entered “6” and five balls rolled  out into the tray, Jane wiggling into action again.

“I see where this is going,” said Jack. “Let’s find a zero.”

Then, Mabel and Seymour entered an “8,” which resulted in seven balls in the tray, recovered in lithe strides by Jane.

Jack and Jill decided to take a chance and entered “11” at the panel. This time no balls came out of the globe.

“Yes!!! We request a break,” said Jack quickly, pulled out his calculator, a pen and notepad.

“Ok, folks, let’s see if Jack and Jill can guess the right number while we hear a word from our sponsor, Ace Dogfoods,” said Barnstorm McDoodle cheerfully.

“Come on Jill, use what you learned in the number theory course to figure this out rapidly before Mabel and Seymour do,” said Jack urgently. “I want to buy that motorcycle.”

“Ok, Jack, I think I see the pattern,” said Jill with a dimpled smile. “Write down what I say and get your calculator ready to win the Caribbean cruise I want.”

At the end of the five-minute word from Ace Dogfoods, Jack announced the result Jill had worked out – which, happily, turned out to be the correct number of balls in the globe.

“People, we have a winner for xxxxx dollars,” shouted Barnstorm McDoodle jubilantly, giving Jane a big hug.

What would you say was the least possible number of balls in the large globe?

The Pirate Gold

Tuesday, August 29th, 2017

A crew of pirates robbed a Spanish galleon off the coast of Crooked Island. Among the booty they found a chest full of gold coins.

The captain was killed during the battle, so to ensure a fair share the crew decided to immediately divide up the gold coins on a nearby uninhabited island before a new captain was chosen.

The 17 surviving pirates began distributing the coins into 17 piles on the beach, but three coins were left over. No agreement could be reached as to who would get the three coins, and, after a large amount of rum was consumed, fighting followed in which one of the pirates was killed.

The pirates then divided up the coins into new piles, but, alas, this time there were as much as ten coins left. With further violent discussions, more rum and fighting another pirate was killed.

The remaining crew set about to divide the gold coins of the chest into equal piles and this time they succeeded in obtaining an equal distribution, which, after opening another barrel of rum, they celebrated until they were spread about on the beach in a drunken stupor.

It turns out they had been observed by a small Dutch crew which had been marooned on the island after a terrible storm. This crew carried away the booty and sailed off while the pirates were sleeping off the rum.

Can you work out what the least number of gold coins in the treasure chest was?

The Family Ages

Monday, August 14th, 2017

Jack asked Fred “How old are your family members?”

“Ten years ago my father was twice my age and my mother’s age was a perfect number,” said Fred.

“That helps with your mother’s age, but I need more information about your father,” said Jack.

“No problem. The square of my father’s age is equal to the year he was born,” replied Fred.

“Ok, got it,” said Jack. “What about your brother and sister?”

“Ten years ago my sister Lara was the same age as my brother Nick is now. In two years my father will be half the age his father was when he died,” said Fred.

“Very interesting,” said Jack.

“My great grandfather told my father he would be reborn in a year equal to one quarter of a perfect number, which is when Nick’s turtle – that my great grandfather gave to my father newly hatched ten years before he died – will probably die,” said Fred.

“How nice, any more info?” said Jack.

“My sister Lara’s age is a divisor of the cross sum of the turtle’s present age,” said Fred.

“And what about your grandfather’s age?” asked Jack.

“When my father was born my grandfather was as old as half the sum of my mother’s age and that of my father’s ten years ago,” said Fred.

“Gee thanks, I’ll go and think about it,” said Jack and walked off.

What do you say are the ages of these family members, including the turtle.