## dimanche 1 juillet 2018

### Bac

Among the 4 documents that you have prepared, the jury makes you pick 2 subjects out of 4, without looking, then you choose 1 out of 2, after reading them.

For the exercise, they'll make you choose 1 out of 2 subjects, on different themes, among probabilities, sequences, statistics and functions.

1) You'll have 20 min to prepare the exercise that they'll give you, (on probabilities, sequences, functions or statistics) and you'll have 10 min to present it on the white board. (The subjects of the exercises of Baccalauréat DNL of the last 6 years : ).

2) When you present the second part of the oral (the document), introduce your subject make a plan and announce it (1st part, 2nd part), then develop itand as a conclusion give your personal opinion about the document. You'll have 5 min to present your document and then 5 min of questions about it (or about more general questions).

Do not bring any Powerpoint or any documents from home.
Do it all from memory, without learning by heart.
Talk about it freely, but with a structure (introduction, development, conclusion).

Baccalaureat : how are the marks given during the oral exam ?

(T. S or T. ES)
2) Wealth inequality (T. S or T. ES)
3) Fractals (T. S or T. ES)
4) Four physics riddles (T. S.)
5) Sex and marriage theorems (T. S. or T. ES)
6) Kakeya's needle problem (T. S.)
7) Blob Infinity (T. S.)
8) Hilbert hotel (T. S. or T. ES)
9) Futurama theorem (T. S. or T. ES)
10) Rubik's cube and the Enigma machine (Alan Turing) (T. S.)
11) Magnus effect (Physics) (T. S.)
12) Anti-gravity wheel (Physics) (T. S.)
13) Computer Science Unplugged. (T. S. or T. ES)
14) Conditional probability (T. S. or T. ES)
15) Monty Hall (T. S. or T. ES)

## vendredi 8 septembre 2017

### Wealth inequality

The content of this video is explained here :

# Why Do Americans Tolerate Extreme Wealth Inequality?

New data shows Americans haven’t a clue how stunningly massive the wealth gap in their country really is. An excellent video blowing up on YouTube depicts this disparity and explains how badly Americans tend to underestimate it. No matter how many statistics you’ve read on the subject, the video deserves six minutes of your time.

To summarize the video's data:
• The richest 20 percent of Americans hold 84 percent of the nation’s wealth, while the bottom 40 percent have less than 1 percent.
• Americans believe their country is much less inegalitarian than it really is.
• The top 1 percent have a bigger slice of the pie (40 percent) than nine of out 10 Americans believe the top 20 percent should have.
• While the top 1 percent command 40 percent of the wealth, the bottom 80 percent have a measly 7 percent.
• The top 1 percent “own half of the country’s stocks, bonds and mutual funds. The bottom 50 percent of Americans own only half a percent of these investments.”
• “The average worker needs to work more than a month to earn what the CEO makes in one hour.”
Putting it all together, the wealth distribution in the U.S. looks like this(note that the top 1 percent is, literally, off the chart) :

The video concludes with this call to arms:
We certainly don’t have to go all the way to socialism to find something that is fair for hard-working Americans. We don’t even have to achieve what most of us consider might be ideal. All we need to do is wake up and realize that the reality in this country is not at all what we think it is.
There’s the rub. If the 2.2 million+ viewers of the video were to expand, Gangnam style, to all 311 million Americans and everyone finally saw with clear eyes just how vast wealth inequality truly is in their country, things would change, right?

Maybe, but I seriously doubt it. Even leaving aside Republican resistance to measures that could dent the wealth gap, many Americans would likely remain opposed to policies that would be necessary to seriously address the problem. The apparently most efficient mechanism for redistributing the pie is a tax on wealth like the “solidarity tax” (impôt de solidarité sur la fortune) in France: citizens with a net worth surpassing 1 million Euros are asked to fork over 0.25 percent of the value of their property every year; the annual levy rises to 0.5 percent for fortunes over 3 million. Such a measure cannot be implemented in the United States: Article 1, section 9, clause 4 of the U.S. Constitution explicitly prohibits “direct” taxation. Should we try to amend the Constitution to permit direct taxes? Well, we could, but if the American experience is anything like that of France, the wealth tax will not actually have a significant redistributive effect, and the 1 percenters would move to Canada to keep their assets intact. That would smooth out the curve at the top without improving the lives of poor and middle class Americans a bit.

Short of a wealth tax, then, how could we bring the American distribution more in line with what Americans would like? Judging by the tenor of his State of the Union address, President Obama seems committed to chipping away at the inequalities in American society by expanding opportunities for preschool and raising the minimum wage, among other proposals. Policies like these may help reduce wealth inequality to some limited extent, but the problem is much too big and too entrenched in American society to resolve with small-bore solutions. Without fundamental changes to the income tax code eliminating “stealth” subsidies for corporations, further raising rates on the wealthy and hiking the estate tax (something Obama has given up on), there is little on the horizon that has a good chance of bridging the rich-poor rift.

Maybe we really do need a socialist revolution.

## mercredi 17 mai 2017

### Fractals

1º) Introduce the subject with a summary of what you are going to present.
2º) Present it with a few steps (What is it ? What are they good for ?)
3º) Conclude and give your personal opinion

# Fractal

A Sierpinski triangle, after 7 iterations.
The Mandelbrot set is a famous example of a fractal.
fractal is any pattern, that when seen as an image, produces a picture, which when zoomed into will still make the same picture. It can be cut into parts which look like a smaller version of the picture that was started with. The word fractal was made by Benoît Mandelbrot in 1975 from the Latin word fractus, which means "broken" or "fractured". A simple example is a tree that branches into smaller branches, and those branches into smaller branches and so on. Fractals are not only beautiful, but also have many practical applications.

## Contents

• 1Examples
• 1.1The Koch Curve
• 1.2Similarity dimension
• 1.3Koch snowflake
• 2Uses

## Examples

There are many types of fractals, made in a large variety of ways. One example is the Sierpinski triangle, where there are an infinite number of small triangles inside the large one. Another example is the Mandelbrot set, named for Benoît Mandelbrot. The Sierpinksi triangle is constructed using patterns, but the Mandelbrot set is based on an equation.
There are also many natural examples of fractals in nature including trees, snowflakes, some vegetables and coastlines.

### The Koch Curve

How to make the Koch Curve
The Koch Curve is a simple example of a fractal. First, start with part of a straight line - called a straight line segment. Cut the line into 3 same-sized pieces. Get rid of the middle of those pieces, and put in the top part of a triangle with sides which are the same length as the bit to cut out. We now have 4 line segments which are touching at the ends. We can now do what we just did to the first segment to each of the 4 bits. We can now do the same thing again and again to all the bits we end up with. We now do this forever and look at what we end up with.
The length of the Koch Curve is infinity, and the area of the Koch Curve is zero. This is quite strange. A line segment (with dimension 1) could have a length of 1, but it has an area of 0. A square of length 1 and width 1 (with dimension 2) will have area 1 and length of infinity.

### Similarity dimension

So, the Koch Curve seems to be bigger than something of dimension 1, and smaller than something of dimension 2. The idea of the similarity dimension is to give a dimension which gives a better idea of length or area for fractals. So, for a Koch Curve, we want a dimension between 1 and 2.
The Koch Curve can be cut into 4 pieces, each of which are ${\displaystyle {\frac {1}{3}}}$ of the size of the original.
The Koch Curve is one of the simplest fractal shapes, and so its dimension is easy to work out. Its similarity dimension and Hausdorff dimension are both the same. This is not true for more complex fractals.

### Koch snowflake

The Koch snowflake (or Koch star) is the same as the Koch curve, except it starts with an equilateral triangle instead of a line segment.

## Uses

Fractals have many applications e.g. in biology (lung, kidneys, heart rate variability, etc...), in earthquakes, in finance where it is related to the so called heavy tail distributions and in physics. This indicates that fractals should be studied to understand why fractals are so frequent in nature.
Some fractals exist only for artistic reasons, but others are very useful. Fractals are very efficient shapes for radio antennas and are used in computer chips to efficiently connect all the components. Also, coastlines can be thought of as fractals

## lundi 24 avril 2017

### Four physics riddles

1) Introduce the subject.
2) Summarize it shortly in 4 minutes (only what you can explain).
3) Conclude and give your personal opinion on the subject of open problems.

## lundi 20 mars 2017

### Hilbert hotel

1) Introduce the subject.
2) Summarize it shortly in 4 minutes. (Only what you can explain).

The sequence u(n) = n + 1 proves you can always find space for a new guest.
The sequence v(n) = 2n proves that there are as many even numbers than integers.

3) Conclude for example, by explaining why infinity is a surprising subject full of paradoxes. (Part of an infinite set can have the same number of elements than the whole set).

David Hilbert (1862-1943)

Imagine an ordinary hotel of 500 rooms, all of them occupied by some clients. You arrive there one evening and the receptionist tells you that there's no vacancy. You walk away sadly. There's no paradox about that.

Now, imagine a hotel in which there are infinitely many rooms, all of them occupied. Although the hotel is full, the receptionist is able to give you a room.How is it possible ? Later, on the same day, infinitely many persons arrive, and the receptionist is able to make room for all of them, (earning a fortune in doing so).

The german mathematician David Hilbert invented these paradoxes in the 1920s to illustrate the mysterious properties of infinity.

Lets see how we can obtain a room in Hilbert's hotel infinity. When only you arrive and the hotel is full, the receptionist is able to give you a room by asking the client who was in room 1 to go to room 2, the one who was in room 2 to go to room 3, and so on forever, each client moving to the next room. Thus, room 1 is now empty for you.

To be able to receive an infinity of new clients, all the previous ones move to the rooms of even numbers, the one who was in room 1 moving to room 2, the one who was in room 2 moving to room 4, the one who was in room 3 moving to room 6, and so on, every client moving to the room having the number double from his previous room. The receptionist is now able to move all the new clients to the rooms of odd numbers.

Hilbert's hotel infinity paradox may be understood with Cantor's theory of transfinite numbers. Thus, while in a normal hotel, the number of odd numbered rooms is inferior to the total number of rooms, in an infinite hotel, the "number" of odd numbered rooms is not inferior to the total "number" of rooms. (The mathematicians use the term of "cardinality" to refer themselves to the size of these sets of rooms).

The receptionist can even receive infinitely many buses containing infinitely many travelers each, by using all the powers of 2 for the previous clients, all the powers of 3 for the first bus, and all the powers of each different prime numbers for each different bus.

See also Zeno's paradox (445 B.C.), Cantor's transfinite numbers (1874), Peano's axioms (1889) and Hilbert's twenty three problems (1900).