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How To Catch A Lion
in the Desert? |
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Problem: To Catch a Lion in the Sahara Desert.
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| Mathematical Methods [click on each method to see how it
works!]
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| The Hilbert (axiomatic) method
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| 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.
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| The geometrical inversion method
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| 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. |
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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. |
| The Bolzano-Weierstraß method
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| 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. |
| 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. |
| 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. |
| 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. |
| The Cauchy method |
We examine a lion-valued function f(z). Be \zeta the
cage. Consider the integral
1 [ f(z)
------- | --------- 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].
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| 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-Tauber 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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| Theoretical Physics Methods
[click on each method to see how it
works!]
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| The Dirac method |
| We assert that wild lions can ipso facto not be observed in
the Sahara desert. Therefore, if there are any lions at all in
the desert, they are tame. We leave catching a tame lion as an
exercise to the reader. |
| The Schrödinger method
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| At every instant there is a non-zero probability of the
lion being in the cage. Sit and wait. |
| The nuclear physics method
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| Insert a tame lion into the cage and apply a Majorana
exchange operator [6] on it and a wild lion.
As a variant let us assume that we would like to catch (for
argument's sake) a male lion. We insert a tame female lion
into the cage and apply the Heisenberg exchange operator [7],
exchanging spins.
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| A relativistic method |
| All over the desert we distribute lion bait containing
large amounts of the companion star of Sirius. After enough of
the bait has been eaten we send a beam of light through the
desert. This will curl around the lion so it gets all confused
and can be approached without danger. |
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| Experimental Physics Methods [click on each method to see how it
works!]
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| The thermodynamics method
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| We construct a semi-permeable membrane which lets
everything but lions pass through. This we drag across the
desert. |
| The atomic fission method
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| We irradiate the desert with slow neutrons. The lion
becomes radioactive and starts to disintegrate. Once the
disintegration process is progressed far enough the lion will
be unable to resist. |
| The magneto-optical method
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| We plant a large, lense shaped field with cat mint (nepeta
cataria) such that its axis is parallel to the direction of
the horizontal component of the earth's magnetic field. We put
the cage in one of the field's foci. Throughout the desert we
distribute large amounts of magnetized spinach (spinacia
oleracea) which has, as everybody knows, a high iron content.
The spinach is eaten by vegetarian desert inhabitants which in
turn are eaten by the lions. Afterwards the lions are oriented
parallel to the earth's magnetic field and the resulting lion
beam is focussed on the cage by the cat mint lense. |
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| Contributions from Computer Science
[click on each method to see how it
works!]
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| The search method |
| We assume that the lion is most likely to be found in the
direction to the north of the point where we are standing.
Therefore the REAL problem we have is that of speed, since we
are only using a PC to solve the problem. |
| The parallel search method
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| By using parallelism we will be able to search in the
direction to the north much faster than earlier. |
| The Monte-Carlo method
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| We pick a random number indexing the space we search. By
excluding neighboring points in the search, we can drastically
reduce the number of points we need to consider. The lion will
according to probability appear sooner or later. |
| The practical approach
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| We see a rabbit very close to us. Since it is already dead,
it is particularly easy to catch. We therefore catch it and
call it a lion. |
| The common language approach
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| If only everyone used ADA/Common Lisp/Prolog, this problem
would be trivial to solve. |
| The standard approach |
| We know what a Lion is from ISO 4711/X.123. Since CCITT
have specified a Lion to be a particular option of a cat we
will have to wait for a harmonized standard to appear.
$20,000,000 have been funded for initial investigations into
this standard development. |
| Linear search |
| Stand in the top left hand corner of the Sahara Desert.
Take one step east. Repeat until you have found the lion, or
you reach the right hand edge. If you reach the right hand
edge, take one step southwards, and proceed towards the left
hand edge. When you finally reach the lion, put it the cage.
If the lion should happen to eat you before you manage to get
it in the cage, press the reset button, and try again. |
| The Dijkstra approach |
| The way the problem reached me was: catch a wild lion in
the Sahara Desert. Another way of stating the problem is:
Axiom 1: Sahara elem deserts
Axiom 2: Lion elem Sahara
Axiom 3: NOT(Lion elem cage)
We observe the following invariant:
P1: C(L) v not(C(L))
where C(L) means: the value of "L" is in the cage.
Establishing C initially is trivially accomplished with the
statement
;cage := {}
Note 0:
This is easily implemented by opening the door to the cage and
shaking out any lions that happen to be there initially.
(End of note 0.)
The obvious program structure is then:
;cage := {}
;do NOT (C(L)) ->
;"approach lion under invariance of P1"
;if P(L) ->
;"insert lion in cage"
[] not P(L) ->
;skip
;fi
;od
where P(L) means: the value of L is within arm's reach.
Note 1:
Axiom 2 ensures that the loop terminates.
(End of note 1.)
Exercise 0:
Refine the step "Approach lion under invariance of P1".
(End of exercise 0.)
Note 2:
The program is robust in the sense that it will lead to
abortion if the value of L is "lioness".
(End of note 2.)
Remark 0:
This may be a new sense of the word "robust" for you.
(End of remark 0.)
Note 3:
From observation we can see that the above program leads to
the desired goal. It goes without saying that we therefore do
not have to run it.
(End of note 3.)
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(End of approach.)
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