Archive for March, 2017

The Seating Arrangement

Friday, March 31st, 2017

A dating club arranges meetings at concerts for its members. The club will buy a certain number of random adjacent concert seats for some male and female members. If a man sits next to a woman, the club considers this a date opportunity.

For the coming concert, the club bought 13 adjacent seats for 7 men and 6 women. Some lucky ladies could give a card with their phone number to the man or men sitting next to them.

As an average estimate, about how many cards with phone numbers do you think were given by the ladies at the concert?

A Medieval Tournament

Wednesday, March 29th, 2017

Lord Highcastle of the Moors was holding a tournament in which the first event would be jousting between the three knights Godfrey, Hotspur and the Black Knight. The winner of this event would receive the hand of Lady Marion and a mansion with a large estate.

The knight winning two jousts in a row would get the prize.

Godfrey and Hotspur were to do battle in the first joust, the winner jousting against the Black Knight. If the winner of this match had not already won twice, he would joust against the knight awaiting his turn, and so forth. This to continue until one knight won twice in succession.

Heavy bets were being made in favor of the Black Knight winning within five jousts.

What do you say is the probability that the Black Knight wins within five jousts?

What is the probability that Hotspur rides off with the Lady Marion?

What is the maximum number of jousts needed to arrive at a final winner?

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?