Mathematics' Nearly Century-Old Partitions Enigma Spawns Fractals Solution

Ramanujan’s statement concerned the deceptively simple concept of partitions—the different ways in which a whole number can be subdivided into smaller numbers. […]

[…]One way to think of partitions is to consider how a set of any (indistinguishable) objects can be divided into subsets. For example, if you need to store five boxes in your basement, you can pile them all into a single stack; lay them individually on the floor as five subsets containing one box apiece; put them in one pile, or subset, of three plus one pile of two; and so on—you have a total of 7 options:

5, 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+4, 1+2+2 or 2+3.

Mathematicians express this by saying p(5) = 7, where p is short for partition. For the number 6 there are 11 options: p(6) = 11. As the number n increases, p(n) soon starts to grow very fast, so that for example p(100) = 190,569,292 and p(1,000) is a 32-figure number. […]

∞ | 9:39 pm | 8Feb2011 | 7♥ math

Connect-Four is a tic-tac-toe-like two-player game in which players alternately place pieces on a vertical board 7 columns across and 6 rows high. Each player uses pieces of a particular color (commonly black and red, or sometimes yellow and red), and the object is to be the first to obtain four pieces in a horizontal, vertical, or diagonal line. Because the board is vertical, pieces inserted in a given column always drop to the lowest unoccupied row of that column. As soon as a column contains 6 pieces, it is full and no other piece can be placed in the column. Both players begin with 21 identical pieces, and the first player to achieve a line of four connected pieces wins the game. If all 42 men are played and no player has places four pieces in a row, the game is drawn. The game has been completely analyzed, so it is known that if both players play with optimal strategies, the first player can always win (Allis 1988). The numbers of possible positions after , 1, 2, … have been played is 1, 7, 56, 252, 1260, 4620, 18480, 59815, 206780, … . (via Connect-Four — from Wolfram MathWorld)

∞ | 9:14 pm | 8Feb2011 | 17♥ math game

A vortex street on the circle is unstable (an old result of Thomson). Here we see 31 vortices inicially alligned on the circle. Integrated with Runge-Kutta with a Mathematica code allowing to evolve an arbitrary number of vortices. (via abel.math.harvard.edu)

∞ | 6:41 pm | 8Feb2011 | 5♥ math