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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.
    --------------------------------------------------------------------

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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.
    --------------------------------------------------------------------
    
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