Investor Jeremy Grantham of GMO illustrates this in his recent apocalyptic commodity analysis with an anecdote about ancient Egypt, which was one of the most successful civilizations in human history.
Grantham asked a group of mathematicians how big Ancient Eqypt would have gotten if its economy had growth 4.5% a year for the 3,000 years the civilization lasted. The mathematicians were directionally correct: Very big.
But not one of them came even remotely close to the actual number (which is mind-boggling).
It has been said that, "Life is like a box of chocolates—you never know what you're going to get." (Forrest Gump in Forrest Gump, 1994.) In this experiment you can test the "Forrest Gump Chaos Theory" by using M&M's, which are much cheaper than a box of chocolates. What if life is more like a bag of M&M's?
We start with two bags of M&Ms, one with 19 and one with 20.
In her turn, a player has to eat all the M&Ms in one of the bags, and then split the M&M from the other bag between the two, leaving at least one on each bag, and not necessarily evenly. The she gives the two bags to the other player. The player who receives two bags with one M&M each loses as he can no longer move.
Read: James Robert Brown’s Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures (Routledge, 2008).
There is a difference between general Platonism and the mathematical flavor. For Plato, each apple, say, is but an imperfect example of the absolute (and perfect) Idea of an apple. But as Aristotle quickly realized, Plato has it exactly backwards: we arrive at the general idea of ‘apple’ by mentally abstracting a set of characteristics we think common to all actual apples. It is we who conjure the ‘perfect’ idea from the world, not the world copying the concept.
But now contrast the idea of an apple with the idea of a circle. Here Aristotle’s approach becomes more problematic, as we don’t find any true circles in nature. No natural object has the precise geometric characteristics of a circle, and in a very strong sense we can also say that the circles we draw are but imperfect representations of the perfect idea of a circle. Ah – but whence does such a perfect idea come from?
Consider another way to put the problem. One major difference between science and technology is that science discovers things, while technology is about human inventions. We discover the law of gravity; but we invent airplanes to allow heavier-than-air flight despite the law of gravity. But where do mathematical objects, like circles and numbers, or mathematical theorems like the Pythagorean one, or Fermat’s Last one, come from? Are they inventions of the human mind, or are they discoveries?
On a game show there are 5 fortune tellers (A, B, C, D and E)
A has 81% chance of being correct B ..... 65% ......................................... C ..... 43% ......................................... D...... 35% ......................................... E ...... 8% ............................................
If you were trying to get the best prediction of your future, which fortune teller would you go to?
Suppose we have a circle inscribed within a square. Now suppose a rectangle extends from one corner of the square to a point on the circle (see diagram).
If this rectangle is 6 inches by 12 inches, what is the radius of the circle in inches?
In a 5 by 12 rectangle, one of the diagonals is drawn and circles are inscribed in both right triangles thus formed. Find the distance between the centers of the two circles.
UPDATE!
Triangle Formulas : Right Triangle C = A + B = Pi/2 radians = 90 degrees c^2 = a^2 + b^2 P = a + b + c s = (a+b+c)/2 K = ab/2
Given a positive integer k another positive integer x is said to be k-transposable if, when its leftmost digit is moved to the unit's place, the resulting integer is k*x
For example, the integer 142857 is 3-transposable since
Match each of the even numbers between 2 and 30 with a different odd number between 1 and 29 so that the fifteen resulting sums are pairwise relatively prime.
A cycloid is the curve generated by a point on a wheel as the wheel rolls. This diagram shows half of a loop of the standard cycloid, whose parametric form is (t - sin t, 1 - cos t):
Which is larger, the blue area or the orange area?
Consider two semicircles S and T that emanate from the same point with diameters along a common line, with S being the larger.
Draw an isosceles triangle whose base is the part of diameter of S that is outside T and whose apex is on S. Draw a circle inside S that is tangent to S, T, and the triangle.
Prove that the center of this circle is directly above the point common to the triangle and T
Can you get 88 using only the integers 1, 2, and 3, using each of them only once, and using the standard arithmetic operations, including factorials and repeating decimals?
Alice and Bob play a game. A positive integer starts the game and the players take turns changing the current value and passing the new number back to their opponent.
On each move, a player may subtract 1 from the integer, or halve it, rounding up if necessary. The person who first reaches 0 is the winner.
Alice goes first: she makes her choice of move on the starting value.
For example, starting at 15 a legal game (if not particularly well played) could be:
Alice .......................................15 → 8 Bob ......................................... 8 → 7 Alice ....................................... 7 → 4 Bob ......................................... 4 → 2 Alice ....................................... 2 → 1 Bob ......................................... 1 → 0 Bob wins.
For which values of n is there a winning strategy for Alice?
The original problem: The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens attack each other. http://en.wikipedia.org/wiki/Eight_queens
In this version, we need to place as many queens as possible on an NxN board, such that each queen will threaten at most ONE other queen. We ask to prove an upper bound and give a solution matching it for the standard 8x8 board as well as for a 30x30 board.
Prove: if we remove two opposite corners from the chessboard, the board cannot be covered by dominoes. (Each dominoe covers two neighboring cells of the chessboard.)
Look for an “Ah-ha” proof: clear, convincing, no cases to distinguish.
Let S be a set of n + 1 integers from {1, 2, . . . , 2n}. Prove that two of them are relatively prime. First show that this can be avoided if S has only n numbers
Suppose we have 13 real numbers with the following property: if we remove any one of the numbers, the remaining 12 can be split into two sets of 6 numbers each with equal sum.
Rubik's Clock Objective: Get all clock's at both sides at 12 o'clock. By pushing or pussing one of the four buttons different clocks will move at the same time.
Rubik's Dice Object: Re-arrange the metal plates inside the cube in such a way that the dice has only white spots. If red is shown anywhere on the dice even through the small controll holes, the puzzle is not complete. The number of possible combinations is 7! x 4^7=82,575,360. There is only ony one correct solution
Question: What is the relationship between game theory and poker?
Barry Nalebuff: There’s a great example of game theory and poker that is from an economist colleague of mine named Randy Heeb. He teaches strategy and sometimes teaches with me at Yale.
It was I think 2002 when the World Championship, he was playing a fellow named Thomas Preston, who was probably better known as “Amarillo Slim” and they were playing the World Championship Heads Up, which is a form of Texas HoldEm.
It was €40,000 pot, which this day may seem small, but he was assistant professor, so it’s real money. And it turns out the top two people were Slim and my friend Heeb. Heeb have a pair of Knights, and based on the betting and the initial hands, he was convinced that Slim either had a King or an Ace on his hand, but not both.
Now, what happens next is three cards, or the flop, are presented, and then there is another round of betting. And Heeb had decided that if either a King or an Ace came out in the flop, he was going to fold, because basically the chances of the pair of Kings or Aces, which is too high. And clearly if Slim had a pair of King or Aces, Heeb stay in, he was going to leave with that kind of a hand.
The cards came out and there’s a 2 of Clubs and 3 of Diamonds and a King of hearts. And at that point, Heeb sort of realized, damn it, I’ve got big pot, I’m going to lose it, and his eyes just gave it away. He realized that he had not done a good job hiding his emotions, and at the same time, he also realized that Slim had looked into his eyes seeing that dejection and went all in.
At that point, Heeb said, “Ah, Slim wasn’t betting based on his cards. He was betting based on the fact that I was giving in, and therefore it means he had the Ace not the King, so he doesn’t have the pair and so he went all in and won.”
And it was this great analysis of, okay, what is he thinking, because if you truly have the pair then what he is happy about is the cards, not about my rejection and that wasn’t what Slim was betting on.
But, typically, poker players spend a lot more time watching the other players than they do watching their own cards and also that’s why a lot of people wear sunglasses.
Stephen Miles: When you interview people and you ask them about their careers retrospectively what most people will say is stuff just happened to them. “I was here.” “I was lucky.” They’ll attribute it to all sorts of things
If you think about your career and think about it broadly in terms of what am I learning, what is this position doing for me, how am I going to get to the next one, what is the most important next move for me and you think two or three moves ahead you can start to put yourself in advantageous positions to accelerate your career disproportionately. As opposed to sort of creeping along the floor you can start to sprint.
Before you set out on your own personal game theory you have to assess your own playing field and who is on the playing field:
Your Boss:
Does your boss want to be known for creating talent and being a net contributor of talent inside your company or organization or does your boss want to be known for really delivering outstanding results because although subtle it really matters for you as an employee because if they want to be a net contributor of talent they’re willing to rotate people through positions, coach mentor them and then if you will, set them free in other parts of the organization. If they’re solely focused on delivering results then what they’re going to do is over hire people for the job and keep them in the job for as long as they possibly can because that is how you’re going to deliver results. You’re not going to take any risk on people. So you as a person playing on that field needs to think about what are your boss’s motives and what are they known for.
Your Peers:
There is only one job, so you’re competing for that job with your peers and you have to assess what are their strengths, what are they doing that is disproportionate to what you’re doing and how do you position yourself to be the best candidate for the next job, so you have to assess horizontally if you will, the peer playing field.
Your Subordinates:
If they choose you to go to the next job, do they have somebody to backfill? Have you created a successor for yourself because if you haven’t, if you’ve made yourself indispensible, guess what, you’re probably going to be indispensible and, therefore, not moved. So you have to think about below are you creating the conditions to make it easy for you to move and allow your boss to make that move with you?
Take two identical discs and slot them together along diameters so that the discs are perpendicular. There is one separation of centres for which the compound shape has a centre of mass which remains a constant height above the surface of a table. The resulting shape rolls along in a most intriguing way.
Historically, the first straight line linkage was described by Sarrus in 1853. It differs in that its parts move in three dimensions. It is applied widely in jacks, elevating platforms and similar devices.
Prof. John Conway proved that there are uni-stable polyhedra. That is to say, polyhedra which are stable on only one face. More information is available from http://www.howround.com/
The width of a smooth shape is the distance between parallel tangents. When a shape is not smooth we must talk about parallel support lines instead. If we have a circle then the width is constant. If the width is constant, then is the shape a circle? These solids have constant width.
You have been captured by an evil wizard. However the wizard also likes maths, and sets you a challenge. You have two identical vases, and 100 white beads and 100 black beads. You must arrange the beads however you like between the two vases. The only condition is that no vase can be empty. And all the beads must be in the vases.
The wizard will then get his assistant to choose a single bead from one of the vases. The assistant will pick purely at random, and will not peek! If the assistant picks a black bead, you will go free. If the assistant picks a white bead... well, let's not go there. Obviously you would like the assistant to pick a black bead.
The question is this. How do you arrange the beads so as to give yourself the best chance of freedom?
For example, 5, 14, 17, and 23 give a collection of four positive integers. Now, form the differences of pairs of these numbers by subtracting the smaller number from the larger in each pair (although this won’t really matter). In our case, this process would give us the following:
Now, form the product of all these differences. This gives us 9 x 12 x 18 x 3 x 9 x 6 = 314,928, in our case. Let’s call the result the Prod-Dif of the four integers. So, the Prod-Dif of the collection {5, 14, 17, 23} is 314,928. We can form the Prod-Dif of any collection of four positive integers. Then we have a very interesting problem to ponder:
What is the largest integer that divides evenly into all Prod-Difs? That is, what is the greatest common factor of the collection of all Prod-Difs? And of course, WHY?
Question : Can the board be set such that no legal move is possible?
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Bejeweled Blitz is a game played on an 8x8 space with 6 different color gems. See the picture below.
The object of the game is to move the gems so that combinations are formed, hence gain points.
1) The board is set such that the same color gem does not occupy 3 consecutive spaces in any column or row.
2) A legal move consists of exchanging 2 neighboring gems (horizontally or vertically NOT diagonally) such that combinations of 3 or more are formed.
In the picture above, the highlighted yellow gem can exchange positions with the purple above it to form a combination of 3 horizontal yellows. There are other possible moves e.g. moving white to the bottom right position to form a combination of 3 vertical whites
What is the smallest number that can be expressed as a sum of six different squares but cannot be expressed as the sum of five different square numbers?
UPDATE!
Take,
1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91
Could you express 91 as a sum of five different square numbers?
n^4 + 4 is composite (i.e., not prime) for all n > 1
Well, n^4 + 4 = (n^2 - 2n + 2) (n^2 + 2n + 2) is a proper factorization, since the smaller of the two factors is greater than 1 when n > 1.
This type of factorization is often called Aurifeuillian (sometimes also spelled Aurifeuillean) in honor of the French mathematician [Léon François] Antoine Aurifeuille.
Rearrange the key caps of the 1 through 9 on a numeric keypad so that no cap is on its correct key, in such a manner that each of the three rows forms a 3-digit perfect square.
Now an ugly issue: what happens if in the prime decomposition of the numbers involved in the Ruth Aaron pairs, one or several of these prime are powered? Will you take all the prime factors involved (with repetitions) or will you take only the distinct primes involved (without repetition)?
As a matter of fact you can do it in one way or another, with the result that you will generate two kind of sequences: Ruth-Aaron pairs, prime factors
Ruth-Aaron triplets do exist. That is to say, three consecutive numbers, n, n+1 and n+2 such that the sum of the prime factors of each number adds up to the same quantity a) without repetition or b) with repetition.
The first example, prime factors with repetition, is the triplet 417162, 417163 and 417164:
