## Why Is X The Unknown?

Terry Moore explains that the inventors of algebra were Arabic scientists. Their definition:

X = something, some undefined unknown thing = the unknown thing.

When the books were translated from Arabic to Spanish the Arabic unknown was “SH” that did not exist inSpanish. They replaced it by “X“.  It migrated to other European languages 600 years ago.

X is “X” because Spanish language does not have “SH“.

## Selfie Math

Jim is going on a tour around the world. He has 5 tops, 4 bottoms, and 3 pairs of footwear. He wants to post a selfie everyday to show different places and his different outfits to his girl-friend Mary, who stays at home. He does not want to wear the same outfit twice. ## For how many days does he have enough clothing?

Leslie Green gives the answer : 96 days.

Can you explain why it is 96? What is logic behind?

## Guesstimate Methods

Guesstimate is an estimate based on a mixture of guesswork and calculation.

“Everything should be made as simple as possible, but not simpler.” – Albert Einstein

“The purpose of computing is insight, not numbers.” – Richard Hamming

## Famous Math and Logic Paradoxes

Math and Logic are full of paradoxes.

1. Achilles and the Tortoise The Paradox of Achilles and the Tortoise was described by the Greek philosopher Zeno of Elea in the 5th century BC. The great hero Achilles challenges a tortoise to a footrace. He agrees to give the tortoise a head start of 100m. When the race begins, Achilles starts running, so that by the time he has reached the 100m mark, the tortoise has only walked 10m. But by the time Achilles has reached the 110m mark, the tortoise has walked another 1m. By the time he has reached the 111m mark, the tortoise has walked another 0.1m, then 0.01m, then 0.001m, and so on. The tortoise always moves forwards while Achilles always plays catch up.  Why is Achilles always behind the tortoise?

2. Bermuda Triangle Paradox. Why the sum of the interior angles of the Bermuda triangle is not 180 degrees?

3. Simpson Paradox. The average score for dance of boys and girls in class A are 16 and 21, respectively.  The average score of boys and girls in class B are 15 and 20, respectively.  Twenty percent of class A students are girls. Forty percent of class B students are girls.  Which class has a higher average score?

4.  Braess paradox. The diagram shows a road network. All cars drive in one direction from A to F. The numbers represent the maximum flow rate in vehicles per hour. Engineers want to construct a new road with a flow rate of 100 vehicles per hour. Drivers randomly choose the road at  crossroads. What new road decreases the capacity of the network (the number of vehicles at point F)?

5. Barber Paradox  In a city, the barber is the ‘one who shaves all those, and those only, who do not shave themselves.’
Who shaves the barber?

6. The Two Envelope Paradox. One envelope has twice as much money as the second one. Gerry does not know which envelope contains the larger  amount. He takes one of the envelopes, counts the money, and is offered the chance to switch the envelope. He thinks “If the amount of money in the chosen envelope is X dollars, then the other envelope contains either 2X of 0.5X dollars, with equal probability of 0.5. The expected value of switching is  0.5 (2X) + 0.5 (0.5X) = 1.25X. This is greater than the value in the initially chosen envelope.  It is better to switch.”  What is your advice?

7. Potato Paradox. I have 100kg of potatoes, which are 99 percent water. I dry them until they are 98 percent water.  How much do they weigh now?

8.  Leonard Euler’s Paradox.   Why the average of all  of the numbers is not a zero?

1, -1, 2, -2, 3, -3, . . .

10. Uninteresting Number Paradox. How many uninteresting numbers exist?

11. Gabriel’ Horn Paradox. The shape obtained from rotating the equation about x-axis resembles a trumpet. If we need an infinite volume of paint to paint the infinite horn, how much paint does the horn can contain inside itself?

12. Pop Quiz Paradox. A teacher announces that there will be a quiz one day during the next week. The teacher gives the definition that they would not when they come in to the class that the quiz was going to be given that day. The brightest student says that the quiz cannot be on Friday because they will know the day. With the same technique, she eliminates Thursday, Wednesday, Tuesday, and Monday. “You cannot give us a pop quiz next week” she says. When does the teacher give the pop quiz?  I know the paradox from Charles Carter Wald. Probably, Martin Gardner described it for the first time in The Colossal Book of Mathematics.

1. Achilles and the Tortoise

## Car Owner’s Puzzles

Can a specific subject provokes interest to Math?

For example, cars. How much does the gas cost in a month? What speed to choose? When do I need to change the tires? How to be on time? How much do I pay for a car during its life? . . .

What are other intersting subjects? Finance? Love? Sport?

## Fermi Problem

Often, a problem solver might estimate the required value without very much information. He/she might aim to get an order of magnitude estimate. The estimation technique is named after physicist Enrico Fermi, as he was known for his ability to make good approximate calculations with little or no actual data. Fermi problems typically involve making justified guesses about quantities and their variance or lower and upper bounds.

The classic Fermi problem, generally attributed to Fermi, is “How many piano tuners are there in Chicago?” – Wikipedia

Several simple examples of Aplusclick Fermi problems:

Tennis Balls in a Bus

English Language Words

Demographics

Security Control

## Dream Math

Everybody dreams.

The boy dreams of being an astronaut.

How long does he need to study and work until his dream can become reality?

Which job does not require math?

The photograph courtesy of Roland Sauter