How To Catch a Lion

Ξ February 14th, 2006 | → 0 Comments | ∇ Humour |


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].)"

 

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