======================================================================== from vol. 2 num. 9, "Do Nothing, Oscillate, or Blow Up: An Exploration of the Laplace Transform" -------------------------------------------------------------------- Now consider f(x) = sin(x). The derivative of a sine is a cosine, and the derivative of a cosine is a minus sine. So when a = 0, f(a) = 0 (sine of zero is zero), f'(a) = 1 (cosine of zero is one), f''(a) = 0 (minus sine of zero is also zero) and f'''(a) = 1 (minus minus cosine of zero is still one) and we're back where we started. So you can approximate sin(x) with: 0*x^0/0! + x^1/1! + 0*x^2/2! + x^3/3! + 0*x^4/4! + x^5/5! + ... ^ | SHOULD BE MINUS Those multiplications by zero cause the even terms to vanish and you're left with: 0 + x + 0 + x^3/6 + 0 + x^5/120 + ... ^ | SHOULD BE MINUS Likewise cos(x) works out to: x^0/0! + 0*x^1/1! + x^2/2! + 0*x^3/3! + x^4/4! + 0*x^5/5! + ... ^ | SHOULD BE MINUS or: 1 + 0 + x^2/2 + 0 + x^4/24 + 0 + ... ^ | SHOULD BE MINUS The odd terms have dropped out in this case. -------------------------------------------------------------------- ======================================================================== from vol. 9 num. 11 "War Games, Money Games, New Games and Meta Games" -------------------------------------------------------------------- It also reminds me of the the "Foxholes Game" described by Martin Gardner in "Mathematical Magic Show" (1965), chapter three. ( http://www.amazon.com/exec/obidos/ASIN/0394726235/hip-20 ) He describes it as a "simple, idealized war game that [Rufus] Isaacs uses to explain mixed military strategies to military personnel." The game is for a soldier to hide in one of five foxholes, labeled one through five, while a gunner fires at one of the gaps between foxholes, labeled A through D, like this: (1) A (2) B (3) C (4) D (5) If the gunner fires at a gap adjacent to the hole where the soldier hides, the gunner wins the round, otherwise the soldier wins the rounds. I recommend playing this game with someone -- it's less trivial than it appears. (Simply have each player write their move on a hidden piece of paper, then both reveal for each round. Record the score as hash marks. It makes a great travel game for kids.) The "guessing what your opponent is guessing about what you are guessing..." problem becomes particularly pronounced, leading to some nontrivial effects. Next month I will reveal the optimum strategy for each player. THE OPTIMUM STRATEGY IS: FOR THE SOLDIER: HIDE ONLY IN HOLES 1, 3 AND 5, SELECTING THE HOLE WITH A PROBABILITY OF 1/3 EACH. FOR THE GUNNER: ASSIGN PROBABILITIES 1/3 to A, 1/3 to D, AND ANY PAIR OF PROBABLILITES THAT ADD TO 1/3 TO B AND C. -------------------------------------------------------------------- ========================================================================