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Tuesday, December 11, 2012

How the pirate share the gold

  The following is a interesting question, it seems very hard at first thought but if you grab the right method you can solve it easily, and get some interesting conclusions.

  Suppose there are 5 pirates,  P1, P2, P3, P4, P5, and they have 100 gold. They have strict social statues:  
  P1< P2 < P3 < P4 < P5
  Their principle to share these gold is that: the pirate in the highest take up a way to share these gold, then all the pirate vote for the method. If the method the proposer will be killed. If the votes of agree and disagree is equal then the proposer have the right to decide.
  So there three basic rules to make any decision: 
     1. Keep alive 
     2. Get gold as much as possible
     3. Try to kill others
  And all the pirate is clever enough. So if you are the proposer, P5. What will you do?
  If you share most of the gold to the others and make yourself alive, clearly you are not a clever pirate.
  It's hard to solve the problem in the normal way. So let's think about what will happen when only two pirates left,P1, P2.
  Then P2 will give all 100 gold to himself for he can vote for himself and he is the proposer he can make the decision whether P1 agree or not. So the solution is:
(P2, P1)→(100,0)
  If P1, P2, P3 left. P1 know if P3 is killed, then P2 will be the proposer, the situation above will happen and he will get nothing. P3 knows that too, so he just need to share 1 gold to P1 then P1 will support him. 
Solution: (P3, P2, P1)→(99,0, 1)
  It's the same for P4, P3, P2, P1 is the same: (P4, P3, P2, P1)→(99,0, 1, 0)
  Well for P5, P4, P3, P2, P1 there is a little different, for P5 has to get another 2 votes. So the solution is (P5, P4, P3, P2, P1)→(98,0, 1, 0, 1)
  Is that amazing that you just need to share a few gold and you can get support and get almost all the gold?

The prince of math: Gaussian

  As you know, there are so many talent people in math, they have solved so many problems that almost any of these problem can suffer you all your life. Then Gaussian is called the prince of math. You can imagine how great he is.
  Johann Carl Friedrich Gauss (April 30, 1777 - February 23, 1855) was a German mathematician, astronomer, and physicist. Although Gauss made many contributions to science and to the understanding of the nature of electricity and magnetism, his true passion was mathematics.
He referred to math as the “queen of sciences” and his influence on the field of mathematics was extraordinary. Gauss was, for example, the first mathematician to prove the fundamental theorem of algebra, and he proved it four different ways over the course of his lifetime. Gauss is widely celebrated as one of the greatest mathematicians in history.
When he was a child, he could add from 1 to 100 very quickly, he added 1 and 100 to 101, 2 and 99 to 101… At last times 101 with 50, then he got the answer 5050.
On his own as a teenager he began to discover advanced mathematic principles, and in 1795 – at the age of 18 – Gauss became the first person to prove the Law of Quadratic Reciprocity, a theory of math that allows us to determine whether quadratic equations can be solved. The same year he entered Gottingen University. While at the university, he made one of his most important discoveries. Using a ruler and compass, he constructed a regular 17-sided polygon.
He has so many achievement in his life: Primzahlverteilung und Methode der kleinsten Quadrate, Einführung der elliptischen Funktionen, Fundamentalsatz der Algebra, Beiträge zur Verwendung komplexer Zahlen, Beiträge zur Zahlentheorie...
Any of these work can make you the best mathematician all over the world.


Fantastic theorem in math

  Who tells math is hard and boring(unfortunately sometimes it is), look at these theorem below, you will find how interesting math can be.
  1.A drunk people can always finds his home but a drunk bird can never find its way. This has been proved easily, suppose there is enough time, the drunk man walk in the city by random, the chance he return his home is 100%. Well, it's a little pity for a drunk bird for the chance it fly back to its home is only about 34%, for the bird can fly south, north, west, east, up and down.
  In higher dimension, the chance you can come back is become lower and lower as the dimension goes up, 19.3% in 4-dimision space, and 7.3% in 8-dimision space.
  2.Hairy all theorem: There is no nonvanishing continuous tangent vector field on even dimensional n-spheres. Simply to say, you can never smooth all the hair on a ball.
  3.ham sandwich theorem: Give a sandwich,you can half the bread, the cheese and the ham with just one cut. More general,given n measurable "objects" in n-dimensional space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single (n − 1)-dimensional hyperplane.
 These theorems are just what we can learn from school, the only difference is we have found these interesting applications. Maybe the reason why some theorem in math is boring just because we have not found the interest in it, these fantastic word is just waiting for you!!!


God love Golden ratio

  Let’s see a normal formula first:
  This little formula lead to the Golden ratio:
  The Golden ratio is much more than a ratio(At least it's golden ^-^). The ration somehow fits the concept of beauty of human. It has been used almost all over the world.
  Look at the five-pointed star, each point near the middle is the Golden ratio point of the line it belongs to.
  Further more,The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram inside a circle, implying a relationship to the golden ratio.
This is regard as the most beautiful ratio of human
  The artist found when the ratio of length of leg to the stature the people looks most beautiful, actually many famous statues follow this principle.

  Most legs are short than the Golden ratio, that may be why women likes wearing 
high-heeled shoes, and dancer always lift their heels when dancing to reach the Golden ratio.
  In nature,Adolf Zeising found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals...
  And there are so many applications of the Golden ratio, if you want you can find these application everywhere.


Sunday, December 9, 2012

The legend of e and π

 The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159.

 The wonder of π is much more than the definition.

There are much more interesting formula about this constant, and maybe the only constant which can be equal to π is e.

This formula is called the most beautiful formula, you can know how e is important in math.
One piece of the beauty of math is connected together, two totally different definition can be connected inside. 


Can all the 8-puzzle be solved?

   Do you ever play the game below?
   At first, the order of these numbers is in a mess, you can move the number to the blank. If you can place the order you win the game. So if you are given a 8-puzzle can you solve it? At first thought it seems can be solved easily. But it's not true!!!
   Here is the proof:
   Give the note [2 3 1 4 5 6 7 8 9] for the following 8-puzzle:
   Define the inverse order of the sequence. The inverse order of [2 3 1 4 5 6 7 8 9]  is two, for [2,1],[3,1] is disorder in the sequence.
      The movement when you solve the puzzle never change the even or odd of the inverse order.
      The inverse order of the target order [1 2 3 4 5 6 7 8] is even, Which means if you are given a puzzle, which disorder is odd, you can never solve it.
      You can play the trick on your friend, you already have the magic of math. Math can be the interesting part of life, right?

Math is the best epitaph of mathematician

   Many mathematicians regard math as their whole. They devoted everything to math. Here are some epitaphs of mathematician. We can see a little part of their amazing miracles. 
   Archimedes wanted no other epitaph than a sphere inscribed within a cylinder -- he had determined the sphere's relative volume and considered this his greatest achievement.

   Henry Perigal's tomb in Essex displays his graphic proof of the Pythagorean theorem.
     Gauss wanted to be buried under a heptadecagon, which he'd shown can be constructed with compass and straightedge. 

I can't dream about being someone like one these great mathematicians. But I want to get to know these great work done by these great men. If you can design your epitaph now, what do you expect on your epitaph?