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?
Jack and Jill were sitting in the shade of a large tree, relaxing after attending some university lectures.
“What about flipping some coins, Jack,” suggested Jill.
“Good idea, I have a silver dollar,” offered Jack.
“Professor Wiseman suggested a coin flipping game,” said Jill. “We alternately flip a coin and the one who gets the same face as the last face to appear wins. Sound good to you?” asked Jill.
“Sure,” said Jack,“so long as I go first.”
“No problem, Jack, as always, you go first, even after I win,” smiled Jill cheerfully.
They proceeded to play quite a number of games, in which to Jack’s dismay Jill always remained on top.
What is the probability of a game finishing before the coin is flipped six times?
Why did Jill win twice as many times as Jack?