found this a while ago
Problem: To Catch a Lion in the Sahara Desert.
1. Mathematical Methods
1.1 The Hilbert (axiomatic) method
We place a locked cage onto a given point in the desert. After that
we introduce the following logical system:
Axiom 1: The set of lions in the Sahara is not empty.
Axiom 2: If there exists a lion in the Sahara, then there exists a
lion in the cage.
Procedure: If P is a theorem, and if the following is holds:
"P implies Q", then Q is a theorem.
Theorem 1: There exists a lion in the cage.
1.2 The geometrical inversion method
We place a spherical cage in the desert, enter it and lock it from
inside. We then performe an inversion with respect to the cage. Then
the lion is inside the cage, and we are outside.
1.3 The projective geometry method
Without loss of generality, we can view the desert as a plane surface.
We project the surface onto a line and afterwards the line onto an
interiour point of the cage. Thereby the lion is mapped onto that same
point.
1.4 The Bolzano-Weierstrass method
Divide the desert by a line running from north to south. The lion is
then either in the eastern or in the western part. Let's assume it is
in the eastern part. Divide this part by a line running from east to
west. The lion is either in the northern or in the southern part.
Let's assume it is in the northern part. We can continue this process
arbitrarily and thereby constructing with each step an increasingly
narrow fence around the selected area. The diameter of the chosen
partitions converges to zero so that the lion is caged into a fence of
arbitrarily small diameter.
1.5 The set theoretical method
We observe that the desert is a separable space. It therefore
contains an enumerable dense set of points which constitutes a
sequence with the lion as its limit. We silently approach the lion in
this sequence, carrying the proper equipment with us.
1.6 The Peano method
In the usual way construct a curve containing every point in the
desert. It has been proven [1] that such a curve can be traversed in
arbitrarily short time. Now we traverse the curve, carrying a spear,
in a time less than what it takes the lion to move a distance equal to
its own length.
1.7 A topological method
We observe that the lion possesses the topological gender of a torus.
We embed the desert in a four dimensional space. Then it is possible
to apply a deformation [2] of such a kind that the lion when returning
to the three dimensional space is all tied up in itself. It is then
completely helpless.
1.8 The Cauchy method
We examine a lion-valued function f(z). Be \zeta the cage. Consider
the integral
1 [ f(z)
------- I --------- dz
2 \pi i ] z - \zeta
C
where C represents the boundary of the desert. Its value is f(zeta),
i.e. there is a lion in the cage [3].
1.9 The Wiener-Tauber method
We obtain a tame lion, L_0, from the class L(-\infinity,\infinity),
whose fourier transform vanishes nowhere. We put this lion somewhere
in the desert. L_0 then converges toward our cage. According to the
general Wiener-Tauner theorem [4] every other lion L will converge
toward the same cage. (Alternatively we can approximate L arbitrarily
close by translating L_0 through the desert [5].)
i must say that is a nice looking ascii integral symbol if i've ever seen one
Topic: Math XP (Read 10317 times)
