Ken's blog

Pierre, the First Mate, was sitting in the mess room of the Nautilus flipping cards from a Tarot deck. He would periodically stop and write some numbers in a small notebook near the cards.

“What are you doing, First Mate?” said Captain Nemo, who had stopped to look as he was walking by on his way to the bridge.

“Sir, I am checking to see how many cards on the average I need to draw from a shuffled Tarot pack before I get an Ace,” said Pierre with a bright smile as he drew The Hanged Man.

“Why don’t you just calculate it? It’s much faster,” grinned Captain Nemo.

“I thought about doing that, but there are so many combinations on how to get an Ace,” moaned Pierre scratching his head as he pulled The Fool.

“Why don’t you think about how not to get an Ace instead,” suggested Captain Nemo, continuing on his way to the bridge.

“Hmm, that’s not a bad idea, Sir,” the First Mate replied and started to write some symbols in his notebook.

What do you figure is the average number of Tarot cards that need to be drawn from a well-shuffled Tarot pack before an Ace appears?

The Wild Weed Band had just completed its concert on a fine day in August and was reaping enthusiastic and noisy applause from the five thousand fans extending far beyond the band stage.

“As it’s my birthday today, I feel generous,” shouted the band’s singer Magno Solo jubilantly.

“All of you who have your birthday today are invited to come up here with your concert ticket and receive a 5,000 dollar check as a token of my great appreciation of my fans,” Magno Solo held his glittering arms out in a welcoming gesture.

Rapidly a group of people with smiling anticipative faces materialized on the stage.

Bongo King, the drummer, stood ready with a bundle of checks, handing them out one by one to the overjoyed people in the group. He gave each one a big hug and a flower.

“For an additional special prize of 20,000 dollars, give your ticket to our guitarist Marco, who will convert your ticket number to a day of the month number with his little calculator. You’ll win if your number converts to today’s date,” exclaimed Magno Solo with a regal bow.

The group on the stage rushed to Marco, who checked each ticket and converted the ticket number to a day of the month number, announcing any winner with a fanfare from the band.

The concert ended with enraptured winners loudly singing the song “For he’s a jolly good fellow,“ accompanied by the Wild Weed Band to appreciative shouts and whistles from the audience.

About how many persons had the same birthday as Magno Solo?

How many persons do you think won the final prize of 20,000 dollars?

It was a sunny Caribbean day with fluffy clouds in a blue sky. A pirate ship was smoothly gliding along on a mild breeze driving its white sails.

The ship was Captain Hook’s and Jacko was walking the plank for insubordination. Jacko was blindfolded and had already drunk a jug of grog given him to send him happily on his way to the wet kingdom of Neptune. Jacko was standing on the very edge of the long plank, about to plunge into the waves.

Peg-leg, the bosun, was busy taking bets from the crew while his blue-winged, yellow-bellied parrot screeched “Take your bets, take your bets…” at full volume from his shoulder.

From previous experience, Peg-leg, an amateur mathematician, had worked out that after downing this amount of grog plank walkers would at any given moment step forward with a probability of 33%, or step backward with a probability of 66%. The length of the step was somehow constant. All this seemed to be induced by the grog mixture.

The basis for betting had been established by Captain Hook, who, being a humane type, had decreed that plank walkers who managed to step their way back onto the ship after drinking a jug of grog would be pardoned.

So, Peg-leg was busy taking bets from the crew as to whether Jacko would make it back on board the ship, or plunge into the laps of mermaids.

Jacko was about to take his next step.

What do you figure is the probability that Jacko joins the mermaids?

Indiana Jones faced a large stone gate deep down in a Yucatan cave. He thought it would lead to the inner Earth – to Agartha, the mysterious land discovered by Admiral Richard Byrd after accidentally flying into it through an opening in the North Pole.

Junior held the torch to illuminate the gate, heavily decorated with symbols. The gate had five ornate wheels set into it. Indiana Jones was busy deciphering the Mayan glyphs along with their horizontal lines and dots.

“These glyphs have a very strange message,” growled Indiana Jones, wiping his brow of sweat while staring fixedly at the engraved symbols. “They seem to indicated a code for opening the gate.”

“What do you think the gate leads to,” asked Junior.

“You know, Junior, many of the Maya disappeared suddenly in the tenth century. I think they passed through this gate on their way to the inner Earth,” said Indiana Jones. “And I wonder if the Inca had their own,” he mused.

“Interesting idea. If so, we’ll need lots of provisions. What do the glyphs say?” asked Junior, getting ready to take notes.

“They say: ‘Behold Imix, the whole divides in two or three or four or five or six or seven or eight or nine, always leaving one, but with eleven the whole leaves none.’”

“What kind of gibberish is that, pops,” exclaimed Junior.

“Hmm,” said Indiana Jones, adjusting his hat, “I am beginning to get an idea. Quick, Junior, give me your calculator.”

Can you help Indiana Jones find the gate code indicated by the glyphs?

In ye legendary olden times in Sherwood Forest, Robin Hood, Little John and Friar Tuck set out to bag a great deer for a large celebration to be held on the occasion of King Richard’s return to England from the Crusades.

Robin’s probability of bagging a deer with his bow and arrow is 0.9, Little John’s and Friar Tuck’s 0.7 and 0.5, respectively.  The feathers of their arrows are green, blue and red, respectively. They will fire simultaneously.

Shortly they track down and bag a large deer and haul it back to the camp on a wagon to be received by a multitude of cheering band members preparing to make merry.

What is the probability of the cook and his helpers finding a green, blue and red arrow in the deer?
And the probability of only a red arrow?
What about just a green and a blue arrow?
Or just a green arrow?
Or at least one arrow?

At a Shanghai dart tournament, Henry asked a referee about the rounds played by Jack and Jill.

“I don’t know exactly how many rounds they played before they left, but they flipped a coin each time to see who would go first, and I know that Jack won the toss four times,” said the referee.

“Thanks,” said Henry and pulled out a small notebook and a pen from his shoulder bag. “I’ll work it out.”

How many rounds did Jack and Jill play?

An eccentric billionaire decided to build two tall pointed towers consisting of a series of stacked cubes. They would serve decorative purposes being placed in parallel at a large futuristic gate in his theme park. The material to be used would be a high-strength, transparent nanomaterial. The billionaire stated that rainwater and some kind of energy would be collected for undisclosed purposes.

The first cube was to have a side length of 1 cm, the subsequent cube sides 1 cm greater than the previous, up to the last cube with a side length of 100 cm.

The architect scratched his head thinking how to add up the volumes. Can you help him greatly simplify the calculations?

What would be the height of the towers, and their combined volume?

“Master, what are relative primes,” asked Lanoo.

“Lanoo, they are two integers that cannot be divided by any one prime number. The only common divisor of the two numbers is 1,” replied the Master.

“Does that mean that if I have two different integers, no prime number will divide both of them,” repeated Lanoo.

“Yes, indeed,” smiled the Master. “You might try working out the probability that any two random integers less than 100 will be relatively prime,” challenged the Master.

“I will certainly give it a try,” said Lanoo eagerly, pulling out a notebook and a pencil.

Can you help Lanoo work this out? Could the probability you obtain also apply in some way to any two integers greater than 100 picked at random?  

King Midas suspected that his minter of gold coins, Horatio, was stealing from the mint by inserting false coins.

As he had 1000 boxes, each containing 100 coins, ready for shipment to King Neptune to pay for construction of some luxury ships, he wondered how he best could check on his suspicions.

So he called on the court mathematician, Leonard, for advice.

“I want to make 99% sure that there is a maximum of 1% of counterfeit coins in the shipment. How can this be done,” he demanded of Leonard.

“Sire, I can have a number of the boxes checked, selecting a sample of one coin at random until we find a false coin,” said Leonard.

“How many boxes will have to be checked,” asked King Midas.

“It depends on the percentage of false coins in them,” replied Leonard. “I will give you a number of boxes to be checked for the case of 1% false coins and a number for 10% false coins.”

After some thought, Leonard gave King Midas the respective number of boxes.

“Very well, proceed. If Horatio has been greedy, he will very quickly lose his head,” thundered King Midas.

How many boxes would have to be checked to ensure that 99% of the shipment had less than one false coin per hundred coins?

If there actually were 10% of false coins, after how many random boxes checked on the average would this be discovered?

John was out with his three children, when Albert, the youngest, spotted “Guiseppe’s Italian Ice Cream Parlor.”

“Daddy, I want an ice cream,” whooped Albert, pulling his dad towards the ice cream shop with great enthusiasm and force.

“Yes, so do we,” shrieked Jenny and Mary, helping Albert drag their dad along.

John shrugged patiently. There was nothing for it but to follow and go see what Guiseppe had to offer.

“I can offer you seven flavors: chocolate, vanilla, strawberry, pistachio, stratiachiella, lemon and banana. Delicious home-made Italian ice cream. You will love it,” promoted Guiseppe cheerfully.

“If you choose a different combination each time you come, I give you one ice cream cone for free,” offered Guiseppe.

“Dad, I just figured it out. We can come here lots of times to get different combinations of flavors,” shouted Albert enthusiastically.

John groaned, a budding mathematician.

How many different combinations of flavors, one per ice cream cone, were possible for the four of them, even if one or more of them chose the same flavor?

And how many combinations if Mary always wanted the same flavor as Jenny, so long as nobody else wanted the same flavor?