The Math King’s New ClothesBy Clement Fung

Once upon a time, in a land far, far away, there existed a wonderful kingdom known as Mathalot. Mathalot was a place like no other; everyone who lived there was a math genius!

The King of Mathalot, King Fraschetti, was the biggest genius of them all. He was very confident in hisabilities and would show them off whenever he could.

Anybody in here think they can challenge

me?

If there was one thing King Fraschetti loved more than math, it was his clothes. He had the finest collection of clothes from all over the world. One morning, King Fraschetti woke up and opened his closet, trying to see what he would wear that day.

There were too many choices to describe: King Fraschetti had 314 different tops, and 159 different bottoms. Along with this, he also had 265 different pairs of socks, and 359 different pairs of shoes. “What a glorious collection!” King Fraschetti thought. “How many possible combinations of outfits could I wear today?”

The Answer

314 x 159 x 265 x 359 = 4,749,710,010 combinations

King Fraschetti spent the next hour and a half trying on different combinations of clothes. He was not satisfied. “These outfits are boring! I want to try something new! I shall hold a contest. Whoever presents me with the finest robe will win a large cash sum!” And so, King Fraschetti called upon all his servants.

Come, my servants!

King Fraschetti had three primary jobs for his servants: Cooks, cleaners and messengers. Servants were not restricted to only having one of these jobs, and some servants had jobs outside of these three main types. Quickly in his mind he began to construct a Venn diagram of his 200 servants.

Cooks

MessengersCleaners

52

27

7

31

43

17 12

11

King Fraschetti only wanted his messengers, and didn’t want any cooks or cleaners to be taken off their duties. Thus, he only took the 17 messengers who weren’t cooks or cleaners and sent them off, with flyers explaining the details of his contest.

However, these messengers wereextremely lazy and didn’t really want to go all around town posting up flyers. They knew that there were 5 main math buildings where people were most likely to see the flyers. They could simply post the flyers there. In how many ways could they post their flyers? They knew they had to post at least one flyer.

The Answer

Number of Combinations

= 25 – 1

= 31 possible combinations

These messengers were REALLY lazy. They simply went to the closest math building and posted a flyer there. Then they went inside and spent the rest of the day finding derivatives. Two scoundrels, named Ryan and Peter, saw the flyers as a special opportunity to embarrass King Fraschetti.

The next morning, Ryan and Peter appeared at the palace gates. There were 8 other tailors with them. The group was led inside, and ordered to stand in a straight line, facing King Fraschetti. How many possible arrangements are there for all 10 of the tailors, if Ryan and Peter must stand next to each other?

The Answer

Number of Arrangements

= 2 x 9!

= 725760 arrangements

Ryan and Peter decided to stand at the far right end, so they would present their idea last. Once all the other tailors had shown their designs and left, Ryan and Peter went to King Fraschetti with their proposal.

Next!

“This robe is like no other robe you have ever seen!” exclaimed Ryan, “The fabric and design will be extraordinary!” “But that’s not all!” continued Peter, “This robe will also have a special property!” Intrigued, King Fraschetti leaned forward.

“Only people who possess adequate math skills can see the robe!” said Peter. “To everybody else, the robe is invisible!” Peter explained. “Yes, in fact, it is rumored that only 7% of all men and 9% of all women in Mathalot can see it!” Ryan added.

King Fraschetti was sold. He gave Ryan and Peter the money to begin work on this new robe immediately. Not only would he have a new beautiful robe, he could also discover who else possessed math skills that could rival his own.

*snicker* *snicker*Get it done

by tomorrow!

He began to form a diagram in his mind concerning the citizens and his servants. He was interested to see who would be within the intersection set of those who could see the robe, and those who were employed under his rule.

Let’s see who REALLY can

challenge me...

Those who can see the robe

Servants under King Fraschetti

Intersection of the two sets (people who can see the robe and are servants of King Fraschetti)

Ryan and Peter took the money and left; the first part of their plan was complete. While they were busy working on King Fraschetti’s new clothes, King Fraschetti himself began to work on some special math problems. If there are 900 men and 800 women in Mathalot, what is the probability that a citizen chosen at random would be able to see the robe?

If you can’t see the robe, you’re an

amateur!

Male

Female

9/17

8/17

Can See

Can’t See

Can See

Can’t See

7/100

93/100

9/100

91/100

P (Can See)

=(9/17)(7/100) + (8/17)(9/100)

= 63/1700 + 72/1700

= 135/1700

= 27/340 or 7.9%

At last, the robe was completed. King Fraschetti sent his prime minister, Lord Gatti to go and retrieve it. Lord Gatti went through town, looking for Ryan and Peter’s address, but he could not find it. He decided to ask a nearby man for help.

I am SO lost….

This old man; however, would only help on one condition. “I’ll help you, if you can beat me at my game.” Lord Gatti was intrigued and decided to accept. “In this bag I have 7 red rocks and 2 yellow rocks. You will pull out four rocks one at a time, with replacement. You must pull at least one yellow rock to win.” What was the probability of Lord Gatti winning the game?

The Answer

Probability of Winning

= 1 – P (Lose)

= 1 – (5/7)(5/7)(5/7)(5/7)

= 1 – 625 / 2401

= 1776 / 2401 or 74 %

With a 74% chance of winning, Lord Gatti pulled a yellow rock on his first try. The old man kept his word and led Lord Gatti to an old, rundown house. Inside he found Ryan and Peter, standing over an empty table. “Look at the robe, Sir!” said Ryan. “Isn’t it amazing?”

This was part of Ryan and Peter’s plan; in reality, there was no robe at all. It was just a trick! However, Lord Gatti did not know this. He froze in fear. He could not see anything! Was he a fraud? Were his math skills truly inept? If word had gotten out, he could lose his position.

This can’t be!

He could not risk it. “What an elegant design!” he exclaimed. “I must show this fabric to King Fraschetti immediately!” And so, Lord Gatti took the “fabric” back to King Fraschetti, with Ryan and Peter following.

I don’t believe this!

“Emperor!” Lord Gatti announced, “the robe has arrived for you to try on.” When King Fraschetti looked into Lord Gatti’s hands, he almost fainted in shock. He could not see the robe!

Suddenly King Fraschetti realized that nobody else knew this. As long as he pretended he could see the robe, nobody else would know. He stripped naked and Lord Gatti draped the robe over his head. King Fraschetti looked in the mirror, wearing absolutely nothing.

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Still looks good to me!

“You boys have done an excellent job!” King Fraschetti said. Ryan and Peter held in their laughter. Their plan was working perfectly! The joke could not end now. “You must show everyone in the kingdom this wonderful robe!” said Peter. “Yes, you must!” shouted Ryan. “Everyone must see King Fraschetti’s new clothes!”

Although King Fraschetti was naked, he could not be afraid. He began to plan a parade route. If King Fraschetti had to pass by each Math building in Mathalot, how many possible pathways would take him from one end of town to the other?

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Why am I doing this

again?

*

*

*

*

*

1

1

1

1 1

2 3

3 6 6 6

6

6

12

18

18

36 36 36

36 72

36 108

108

216

216

216

216 216

432

648

648

1296

* - Math Building

There are 1296 possible pathways through the city, if

you must visit each Math Building

After a short while, this was the pathway that King Fraschetti chose. He really wanted to avoid a certain area of town, where he knew his old friends from the

AP days would mock him terribly.

Special Area

The next day, the parade path had been chosen. King Fraschetti walked out into the street naked, and marched his way around town. All the citizens of Mathalot took the day off from math problems to view this robe. Word had gotten around of its beauty, and everyone wanted to see.

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Many people also wanted to discover whether or not they possessed superior math skills. Of the 1700 citizens of Mathalot, 600 men and 700 women had gathered to watch. Based on the probabilities given by Ryan and Peter, what was the probability someone in the crowd was female, given they could see the robe?

Male

Female

6/13

7/13

Can See

Can’t See

Can See

Can’t See

7/100

93/100

9/100

91/100

P (Female / Can See)

= (7/13)(9/100) / [(7/13)(9/100) + (6/13)(7/100)]

= 63/1300 / [63/1300 + 42/1300]

= (63/1300) / (105/1300)

= (63/105)

= 3/5 = 60%

However, the real probability of this happening was 0. There was no robe to be seen. As King Fraschetti passed by, murmurs could be heard in the crowd. Was it true? Did they not possess superior math skills? A little boy in the crowd who did not know about the robe’s reputation saw King Fraschetti and shouted, “King Fraschetti is naked! Look everybody! King Fraschetti is naked!”

Look everybody! Look!

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Everybody in the crowd began to speak. “It’s true! King Fraschetti is naked! The robe is a fake!” The only thing worse than being naked in public was admitting he had been tricked. Ryan and Peter hadstolen his money, and were long gone. King Fraschetti ordered his servants to continue onwards,while the citizens pointed and laughed, as King Fraschetti proudly marched on.

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Amateurs! All of you!HA! HA! HA! HA!