Archive for the ‘Number Theory’ Category

The Pirate Tombstone

Monday, July 24th, 2017

It was a bright blue sunny Caribbean day with occasional fluffy clouds sailing by like galleons over an island with abundant jungle growth and a lively variety of bird chatter.

“Look there, Arby. Isn’t that a tombstone,” shouted Pascal, assistant to explorer Dr. Arbuthnot Smythe, as he ran to inspect a black rock hidden among foliage under some palm trees on the remote uninhabited Caribbean island Esmeralda.

“Well spotted,” Dr. Smythe followed Pascal to inspect the black rock. “Clean up the surface so we can see what’s written on it,” said Dr. Smythe, following close behind.

Pascal was already busy brushing the black rock free of debris and growth, revealing the large letters R I P with the name Jack Black Silver below.

“Could be treasure decoy work by Capt. Kidd, indicating that we are on the right track based on the parchment incrusted with an emerald and black seal we found in an old ship’s log at Boggs’ Antiquary in London,” said Dr. Smythe after examining the black tombstone.

“There is more text chiseled in under the name,” said Pascal enthusiastically. “Very strange, there are three large numbers with letters.” Pascal scratched his head with a puzzled look on his face.

“What are the numbers,” asked Dr. Smythe, standing ready with notebook and pen.

“Arby, they read as follows: 180XYZ34, 158XYZ and 982XYZ8, one number above the other,” replied a befuddled Pascal.

“I say, definitely not coordinates,” observed Dr. Arbuthnot Smythe, tapping his pen on the pad.

“Wait, I see some small writing below which reads:

By dyviding the dyvisor, ye pace 19 rest N, then pace 12 rest W.’

What in the world could all that gibberish mean?” exclaimed Pascal.

“Hmm,” said Dr. Smythe, scratching his van Dyke beard, concentrating on the numbers. “I would say we need to work out what the letters stand for. I believe Capt. Kidd’s treasure chest could be buried nearby,” said Dr. Smythe deep in thought.

“Maybe the word ‘rest’ means ‘remainder’,” commented Pascal.

“Inspired thought, Pascal, I think you are on the right track. Get your calculator out and let’s solve this riddle,” said Dr. Arbuthnot Smythe with enthusiasm.

About how many paces from the black tombstone would you say the treasure is located?

The Brimchester Football Bar

Friday, July 14th, 2017

Marco Slivovitz, owner of The Brimchester Football Bar – dedicated to all fans of the honorable soccer sport – had just installed a perfect number of spherical beer kegs, painted to resemble footballs, which were placed around a large circular bar room.

Enthusiastic  fans celebrating the most recent victory of their beloved team could freely tap and drink beer from any one of the perfect number of spherical football kegs, conveniently placed around the extensive bar area – instant payment being made by card, and large beer mugs provided by saucily dressed waitresses.

Marco had a problem.  As the bar guests would usually empty all the kegs during an evening’s merrymaking, he needed a delivery truck with the correct spherical capacity to fill his perfect number of spherical beer kegs so they would be ready for the next bout. The radius of each of his spherical beer kegs was 21 inches.

Being a practical man, he turned to the yellow pages of the phone book to check for a firm specializing in beer delivery trucks with spherical containers, keeping in harmony with his own containers.

Luckily, after a persistent search, Marcho Slivovitz found the right company for his purposes, Acme Spherical Deliveries.

After contacting Acme Spherical Deliveries and giving the friendly sales person the details regarding his spherical beer kegs, Marco was delighted to hear that two spherical delivery trucks could actually be made available on a regular basis for his purpose, however each with a different diameter.

With the situation happily resolved, being a practical man, Marco Slivovitz proceeded to have one of his waitresses help him test his spherical beer kegs for proper functionality.


Can you work out the radii of the spherical containers on the Acme Spherical Deliveries trucks that will  completely fill up Marco’s empty spherical beer mugs after a spirited football victory celebration?


The Special Number

Thursday, March 23rd, 2017

On a late June afternoon, Jill sat on the green grass by a small lake at the university with a concentrated look on her face. Some ducks were quacking nearby.

“Hi Jill, I see you are thinking about something, anything important?” said Jack, who had quietly walked up to her on the grass.

“I need a vacation and there’s this problem I’m trying to work out, Jack” said Jill in an exasperated tone of voice.

“Seems you are not progressing much. Tell me all about it,” said Jack, sitting down beside her.

“The travel magazine ‘Your Perfect Vacation’ is holding a contest where the winner will get a two week, fully paid vacation for two at a bungalow in Curacao in the Caribbean, and I want to win the prize,” she said with a determined look.

“Sounds interesting. I enjoy basking in the sun and doing some snorkeling. Want some company on the trip if I help you?” offered Jack eagerly.

“You bet, Jack. Have a look at the problem, the deadline is tomorrow.” Jill handed him the magazine, pointing to the advertisement.

Jack began reading out loud: “There is a number that starts with a digit and is followed by four digits that are the same. Square this number and subtract one to give a ten digit number, none of whose digits are the same.”

Jill was looking expectantly at him, hoping to see a light turn on in his eyes.

Jack’s face brightened up. “Jill, this is no big problem. Some number theory that I know will make it easy,” smiled Jack.

“So glad to hear it, Jack. Let’s go over to my place, work it out and enter the solution on the magazine’s webpage,“ invited Jill.

“Curacao, here we come,” said Jack, nimbly pulling Jill up on her feet.

Jack and Jill gathered their things and walked off together down a tree-lined path.

What do you figure is the number Jack and Jill are looking for?

The Square Armies

Monday, March 13th, 2017

Emperor Manos was standing outside his tent on a hill overlooking his vast army camped in tents in the square formation normally used in battle. General Pathos and a Royal Guard were with the Emperor, as was his sorcerer Morgan.

A raven briskly landed on a large branch of an oak tree. “Guard, shoot the raven, Zanora must not know of our plans,” commanded Emperor Manos.

The guard rapidly and expertly pierced the raven with an arrow, and it fell croaking from the tree striking the ground.

“Well done, guard. It was a spy bird sent by the High Priestess, Zanora, who uses its eyes and ears to observe what we are discussing so she can report it to King Kali for his attack tomorrow,” explained Emperor Manos.

“How many men are left in my army, General Pathos,” asked the Emperor.

“Lord Manos, prior to our last battle we had counted twelve million men, but sadly we lost quite a few,” replied General Pathos, eyes downcast.

“What about the number of men in King Kali’s army. What can you tell me, Morgan?” requested Emperor Manos.

“Lord Manos, my divinations have revealed a maximum of 19  million men,” replied Morgan.

“General Pathos, blow the horn for attack at dawn. When the battle starts we will see how things develop, but as the moon is full, Morgan could invoke the Moon Goddess Iona and have her conjure up 18 auxiliary units of soldiers in square formation to reinforce our army, if needed,” the Emperor informed General Pathos.

“Lord Manos, I’m glad to hear this. We will be ready at dawn for this massive and decisive battle to save our people from the evils of King Kali’s black magic,” responded General Pathos.

At dawn, Emperor Manos’ army attacks, but, after seeing the great mass of soldiers in King Kali’s army storming forward, the Emperor soon realizes that his army is greatly outnumbered.

“Your conjuring rite for the 18 units is needed, Morgan,” shouted Emperor Manos.

Morgan climbed onto a small mound and raised his hands to the Moon goddess Iona, sounding an invocation. After a short while,  eighteen units of fully-armed soldiers materialized out of a large mist just ahead of Emperor Manus’ army, nine units on either side forming a bull’s horn attack configuration.

“I have done as commanded, my Lord. The Moon Goddess Iona informs me that our army and King Kali’s are now equally matched with the same number of men,” said Morgan.

“Thank you, Morgan, you have performed an invaluable service,” exclaimed Emperor Manos

Then King Kali’s army splits up into 16 equal square units forming a semi-circle and attacking Emperor Manos’ army with great intensity.

“My Lord Manos, if I knew how many men there are in any one of the 16 units of King Kali’s army, I could create a resonance spell resulting in a multitude of ghostly apparitions that sow great confusion and disorientation among King Kali’s soldiers,” said Morgan.

“Yes, this will definitely save the day for our island Ruta,” enthused Lord Manos.

“Guard, summon Leonoros, the Court Astrologer, immediately. We need to calculate the number of men in any one of these 16 units,” shouted Lord Manos.

The guard hurried off to fetch Leonoros, the Court Astrologer.

To help Lord Manos win the battle, how many men would you say are in one of King Kali’s 16 equal units?

Can you also work out the number of men in one of the eighteen conjured units?

And while you are at it, what about the size of the opposing armies?

Farmer Couples Out Shopping

Saturday, January 28th, 2017

In a late autumn afternoon, Jill was sitting under a tree on a campus hill looking despondent,  elbows on her knees and hands holding up her head. Sunlight was fading while yellowish and brown leaves floated erratically down from trees.

Distracted, she did not notice that Jack was approaching from below.

“Hi Jill, you look unhappy,” said Jack. “Want to tell me about it?” He sat nimbly down beside her.

“I think my math professor, Mr. Torquemada, doesn’t like me,” moaned Jill. “He gave me an assignment I have to turn in tomorrow and explain in class, and I can’t figure out head or tails of the problem,” she said shaking her head with a forlorn expression on her face.

“Why don’t you explain the problem to me, maybe I can think of something,” offered Jack with a sympathetic smile, placing his hand lightly on her back.

Jill sighed deeply. “Ok, Jack, here goes:”

“Eight young couples went to the Texas Combo Farm Equipment and Livestock Market to make purchases for the farms they were establishing to form an ecological farming enclave.

The husbands’ names were Henry, Archie, Guiseppe, Pavel, Heinz, Juan, Erik and Shu Wu. The names of their wives were Elfrida, Gunnhilde, Lola, Mary, Teresa, Sally, Jasmine and Lulu.

They all bought the same number of an item as the price of the item in dollars.

It turned out that, respectively, all of the husbands spent 945 dollars more than their wives.

Guiseppe bought 35 more chickens than Lola. Pavel spent 2673 dollars more than Elfrida. Erik bought 95 more pigs than Lulu. Henry spent about one fifth as much as Heinz. Juan bought just about twice as many cows as Sally. Jasmine bought more items than any of the other wives. Mary bought thrice the number of seed bags as Lola. Erik spent about three times as much as Gunnhilde, and relatively a little bit more than Teresa.

 How much did Shu Wu spend, and what was the name of his wife?”

“What a muddle, Jill. What did you do to upset him?”

“Professor Torquemada invited me to a friendly Saturday night dance at his fraternity’s house, where he is a supervisor, and I turned him down, somewhat abruptly I’m afraid,“ explained Jill.

“Oh, I see,” replied Jack, suppressing a laugh.

“Jill, I am taking a course in computer programming and figure that this is an easy problem to solve with a few lines of code,” said Jack encouragingly.

“Please  help me with this, Jack,” pleaded Jill. “I would naturally compensate you somehow,” offering a demure smile.

“Deal,” said Jack. “Come with me. I know a place where we can write some code. I think a double loop will do the job,” he added.

“What’s a double loop?” wondered Jill.

“I’ll show you later.”

Jill jumped up, gathered her books and followed Jack down the hill.

Shopping for Wine

Monday, January 23rd, 2017

Fred, Marcus and Seamour and their wives went to a winery to buy bottles of wine for a party they would be holding for family and friends. The wives’ names were Gracie, Mabel and Lucy, not respectively.

Both the husbands and the wives bought the same number of bottles as the price of the bottles in each case.

The men each bought 72 dollars worth of wine more than their respective wives.

Marcus bought 6 fewer bottles than Lucy and Fred bought 16 more bottles than Mabel.

What do you say were the names of the wives of Fred, Marcus and Seamour, respectively?

The Size of the Persian Army

Monday, January 9th, 2017

The king of Persia, Darius – self-proclaimed as the “King of Kings”– jerked aside the flap of his elaborately decorated campaign tent and strutted out in splendid war regalia. King Darius’ soothsayer Macumba had just informed him after inspecting some dots in the sand that the number of soldiers in King Darius’ army would tell whether his coming battle against the Greeks would be victorious or not.

Outside the campaign tent stood Zopyrus, his chief general, looking over the vast field of soldiers who were getting ready for battle.

“General Zopyrus, how many men do we have in my army,” King Darius demanded to know.

“I don’t know exactly, O King of Kings,” replied general Zopyrus, but I will surely find out.

“Line them all up in columns of two,” said King Darius.

General Zopyrus instructed his aide Mortius to carry out the order. Mortius returned, reporting: “O King of kings, there are many columns of two, but alas one soldier remains unpaired.”

“Then line them up in columns of three,” said King Darius. Mortius ran off to comply, almost tripping over his dangling sword.

Mortius returned, reporting: “O King of kings, there are many columns of three, but two soldiers remain in excess.”

“Do I have to do all the thinking,” shouted King Darius. “Continue with four, five, six, etc. and let me know when there are no soldiers in excess,” barked King Darius.

“Yes, O King of kings,” Mortius ran off again to comply, general Zopyrus shuffling his feet nervously.

Mortius returned: “O King of kings, columns of four left three soldiers in excess, columns of five left four, columns of six left five, columns of seven left six, columns of eight left seven, columns of nine left eight, columns of ten left nine, columns of eleven left ten, columns of twelve left eleven, and columns of thirteen left zero in excess, ” he reported breathlessly.

“Well then, that should be sufficient information to determine the total number of soldiers in my army,” said King Darius. “General Zopyrus, have your aide get busy at once. This information is vital.”

“O King of kings, I will ask the court astrologer, Leonardis,” said general Zopyrus. “He is good with numbers.”

“Report the result to me immediately,” the King of kings Darius swept aside the tent flap and returned into his campaign tent, awaiting the result of the battle.

How many soldiers do you think Darius had in his army?

Did King Darius win the battle with the Greeks?

The Magic Number

Saturday, November 26th, 2016

Manu, a king in a long-forgotten land called Lemuria, that flourished prior to recorded history, had a nubile daughter named Lailai he decided to marry off to someone brave and intelligent to ensure his lineage would continue with prosperity.

So he had a proclamation posted on all trees and buildings in the kingdom that whosoever could work out the magic number would marry Princess Lailai. Those who failed would have their head chopped off.

According to the court astrologer Bolbol, the magic number was the test that would endow the person who could work it out with the power of dominion of Lemuria. The number was the total area of the kingdom, expressed in mets, the area unit used in Lemuria. Three digits were missing in the number.

The future prince would have to work out what these three digits were to qualify for marrying Princess Lailai and inheriting the kingdom after King Manu decided to leave his body for other regions.

The threat of death deterred all but three applicants, the first two of which were rapidly despatched by Zok, the executioner. The last candidate was Manas, the son of a blacksmith of a small town in Lemuria.

He had always had a thing with numbers and would dream about them all day long – even while working in his father’s smithy. Manas would write down his discoveries in a notebook during his free time.

Manas saw the proclamation on a tree he often conversed with, and was his special friend.  He remembered having seen Princess Lailai in a procession, when a glow suddenly appeared around her head.

Next day, Manas, with his notebook, presented himself as a candidate before King Manu in the presence of Princess Lailai, who looked at him with large eyes.

Manas was led to a room in a turret by Bolbol the court astrologer – accompanied by a leering Zok with his axe – given writing materials and told to submit the answer to Bolbol the next day after breakfast.

The magic number 8*45*64*9 was written on a board on the wall of the turret room. Bobol said it was divisible by 9 and 11, and was a square. Then he left, locking the door behind him.

Initially, Manas had some difficulty in working out the number, but then he remembered something he had discovered about the endings of squares and written down in his notebook, which made the whole thing much simpler.

After breakfast the next day, Manas submitted his answer to Bolbol, the court astrologer, who confirmed that it was correct.

Soon thereafter, Manas and Lailai rode off on two royal stallions of the time to see the countryside. Standing on a balcony, King Manu watched them briskly ride off with a smile.

What do you think was the magic number Manas worked out so he could marry Princess Lailai?

The Economics Problem

Saturday, November 19th, 2016

In the late afternoon of a sunny day, Jack sat under a spreading, umbrella-like tree on a campus hill studying his economics textbook and pondering on a problem assigned to him. Jack scratched his head and began to yawn and gaze into the distance.

Then, Jill walked barefoot up the hill on the verdant grass. “You look puzzled, Jack. What’s up,” she said with a big dimpled smile.

“I can’t figure out this economics problem professor Thrombastus von Vogelkopf assigned me,” Jack wailed.

“Tell me about it,” offered Jill and sat down, pulling up the legs of her tight jeans to get comfortable.

“Well, in this problem there is a Maya Insecurities Corp. which sells papers it calls insecurities, and last year they had a turnover of one type of insecurity to the value of 23,887.09 dollars. This year, sales of the  insecurity rose to 43,445.89 dollars. So the professor wants me to figure out how much the insecurity concerned costs, and how many of it were sold each year,” Jack explained in frustration.

What do you think about that, Jill? Any ideas?” he moaned.

“Doesn’t the professor like you, Jack,” said Jill with a sympathetic tone. “What did you do or say to him, Jack?” she said with a knowing look.

“Well, I did kind of say that ever since the gold standard was dropped modern economic theory, practice and structure can be compared to a house of cards and a Ponzi scheme on a large scale promoted and taught by unscrupulous types with little concern for the human race, and that it would eventually lead to a big crash causing a grand depression and large-scale suffering – even another world war,” said Jack sincerely.

“I told him that incorruptible gold is the metal of the powerful Sun, imaginary values are the stock in trade of the deceptive Moon, where justice is a forgotten concept,” explained Jack. “From the look on his face, I believe he didn’t appreciate my point of view,” he said, shrugging his shoulders.

“Ok, I get the picture,” said Jill, and burst out laughing. “I am taking a class in number theory, Jack, where I have learned things that can help you with this problem. And what will you give me in exchange for this, Jack?” She briefly pursed her lips and arched an eyebrow.


Can you help Jill work out Jack’s economics problem and find the value and amount of the insecurity sold for each of these years?

The Aureus Coffers

Saturday, October 29th, 2016

The famous explorer Arbuthnot Smythe and his assistant Pascal stood before the stone door of an inaccessible ancient Roman underground tomb over which their torches showed was engraved “ET REQVIESCENT IBI TRES THESAURIS” in large letters.

Beside the door they saw three niches, each containing the bust of a lovely female aged from young to mature.

“Such exquisite sculpture work,” said Pascal full of admiration. “I wonder who the women were.”

“They appear to be related,” commented Arbuthnot, on an impulse pushing on the brooch sculpted on the older woman’s bust with his thumb and pulling her head forwards. This caused the stone door to open slowly on creaky hinges.

“Clever intuitive move,” exclaimed Pascal as they both entered and to their surprise saw that the interior of the tomb was illuminated by seven flaming lamps fixed around the walls.

They saw three exquisitely ornamented rectangular pools in each of which floated a perfectly preserved female body in serene repose, immersed in a transparent liquid. The older woman’s pool was placed on the left.

Arbuthnot Smythe and his assistant Pascal were speechless, eyes popping out and mouth dropping in wonder at this impossible spectacle.

After standing there for a while in a daze, Arbuthnot recovered himself sufficiently to speak. “I have read about this type of tomb in Blavatsky’s Isis book. It was for Cicero’s daughter,” he said in wonder. “Blavatsky even gives the formula for making these eternal flames. The tomb was found in the 19th century, news of which was inevitably suppressed by religious authorities,” he related.

“What shall we do,” asked an astounded Pascal.

“Nothing,“ replied Arbuthnot sadly. “What we see here has been accomplished with knowledge of advanced spiritual technology. Science and western religions are not ready for this, least of all so-called modern medicine.”

“We will make a record of what we find for some future generation, when religion has evolved to become scientific and science to become religious,” he said quietly.

They switched their attention to the foot of each stone tomb, where they saw a marble coffer with the inscriptions “SUMMAM FACIAT QUADRATUM – DIFFERENTIA EST AEQUALIS.”

Arbuthnot forced open the first marble coffer with his sturdy knife. As the lid fell back, they saw that the coffer was full of gold coins.

Arbuthnot picked up a gold coin and studied it with a magnifying glass. “This is an aureus coin of the type minted by Julius Caesar,” he shouted jubilantly. “It contains about eight grams of gold. What a treasure we have found, Pascal.”

“Really unbelievable. I wonder what’s in the other marble coffers,” said Pascal, unsheathing his knife to pry open the next marble coffer lid. Again they found it was full of aureus coins, although this marble coffer was smaller. The third, even smaller marble coffer revealed the same: full of aureus coins.

“I wonder how many aureus coins there are in these marble coffers,” said Pascal scratching his head. “Based on their size, each one contains a smaller number of coins.”

“Judging from the engraved Roman text, I would say that the sum of the number of coins in any two marble coffers, taken pairwise, adds up to a square.” replied Arbuthnot, replacing his knife in its scabbard.

“Brilliant, then the difference between the number of coins in one marble coffer and the next descending one must be the same, judging from the second inscription,” added Pascal with a big smile.

“Yes indeed, that must be so,” exclaimed Arbuthnot. “Then we only have to do the math to determine how many coins there are in each of the marble coffers.”

“Well,  being a purist, you do the math, Arbuthnot. To avoid a headache, I will use a computer,” laughed Pascal. Arbuthnot had already pulled out a notebook and was writing down the details in equation form.

“Yes, I wonder who the women were,” said Arbuthnot Smythe. “My guess is they were the wife and daughters of a nobleman who met with a sudden unfortunate fate. So he left his treasure with them.”

Can you help Arbuthnot Smythe and Pascal work out how many gold coins each marble coffer contained?