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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/hip20 )
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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