Problem: To Catch a Lion in the Sahara Desert.
1. General 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 perform 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 interior 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 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 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). Let q be the cage. Consider the integral
1 ó f(z)
--- ô------ dz
2p i õ(z-q)
C
where C represents the boundary of the desert. Its value is f(q), i.e. there is a lion in the cage.
1.9 The Wiener-Tauber method
We obtain a tame lion, L0, from the class L(-¥,¥), whose fourier transform vanishes nowhere. We put this lion somewhere in the desert. L0 then converges toward our cage. According to the general Wiener-Tauner theorem every other lion L will converge toward the same cage. (Alternatively we can approximate L arbitrarily close by translating L0 through the desert.)
1.10 The surgical method
A lion may be regarded as an orientable three-manifold with a non-empty boundary. It is known that by means of a sequence of surgical operations (known as spherical modifications in medical parlance) the lion can be rendered contactible. He then can be signed to a contract with Barnum and Bailey.
1.11 The logical method
A lion is a continuum. According to Cohen's theorem he is undecidable (especially when he must make choices). Let two men approach him simultaneously. The lion, unable to decide upon which man to attack, is then easily captured.
1.12 The functorial method
A lion is not dangerous unless he is somewhat gory. Thus the lion is a category. If he is a small category then he is a kittygory and certainly not to be feared. Thus we may assume, without loss of generality, that he is a proper class. But then he is not a member of the universe and is certainly not of any concern to us.
1.13 The method of differential topology
The lion is a three-manifold embedded in Euclidean 3-space. This implies that he is a handlebody. However a lion which can be handled is tame and will enter the cage on request.
1.14 The sheaf theoretic method
The lion is a cross-section of the sheaf of germs of lions on the Sahara Desert. Merely alter the topology of the Sahara, making it discrete. The stalks of the sheaf will then fall apart releasing the germs which attack the lion and kill it.
1.15 The method of transformation groups
Regard the lion as a surface. Represent each point of the lion as a coset of the group of homeomorphisms of the lion modulo the isotropy group of the nose (considered as a point). This represents the lion as a homogeneous space. That is, this representation homogenises the lion. A homogenised lion is in no shape to put up a fight.
1.16 The Postnikov method
A male lion is quite hairy and may be regarded as being made up of fibres. Thus we may regard the lion as a fibre space. We may then construct a Postnikov decomposition of the lion. This being done, the lion, being decomposed, is dead and in bad need of burial.
1.17 The Steenrod algebra method
Consider the mod q cohomology ring of the lion. We may regard this as a module over the mod q Steenrod algebra. Doing this requires the use of the table of Steenrod cohomology operations. Every element must be killed by some of these operations. Thus the lion will die on the operating table.
1.18 The homotopy method
The lion has the homotopy type of a one-dimensional complex and hence he is a K(p,1) space. If p is noncommutative then the lion is not a member of the international commutist conspiracy, and hence he must be friendly. If p is commutative then the lion has the homotopy type of the space of loops on a K(p,2) space. We hire a stunt pilot to loop the loops, thereby hopelessly entangling the lion and rendering him helpless.
1.19 The covering space method
Cover the lion by his simply connected covering space. In effect the decks the lion. Grab him while he is down.
1.20 The game theoretic method
If there are an even number of lions in the Sahara Desert we add a tame lion. Thus we may assume that the group of Sahara lions is of odd order. This renders the situation capable of solution according to the work of Thompson and Feit.
1.21 The Moore-Smith method
_
Letting A be the Sahara Desert, one can construct a net in A converging to any point in A. Now lions are unable to resist tuna fish, on account of the charged atoms found therein (see Galileo Galilei, Dialogues Concerning Tuna's Ionses). Place a tuna fish in a tavern, thus attracting a lion. As noted above, one can construct a net converging to any point in a bar; in this net enmesh the lion.
1.22 The Mittag-Leffler method
The number of lions in the Sahara Desert is finite, so the collection of such lions has no cluster point. Use Mittag-Leffler's theorem to construct a meromorphic function with a pole at each lion. Being a tropical animal a lion will freeze if placed at a pole, and may then be easily taken.
1.23 The method of natural functions
The lion, having spent his life under the Saharan sun, will surely have a tan. Induce him to lie on his back; he can then, by virtue of his reciprocal tan, be cot.
1.24 The boundary value method
As Dr Morphy has pointed out, Brouwer's theorem on the invariance of domain makes the location of the hunt irrelevant. The present method is designed for use in North America. Assemble the requisite equipment in Kentucky, and await inclement weather. Catching the lion then readily becomes a Storm-Louisville problem.
1.25 The complex method
(A solution to solve this classic problem without being cagey). We assert, as is clear, that the Skin, SL, of the lion, L, is a compact Riemann surface. Consider a meromorphic function, f(z), defined on SL, and apply the Removable Singularities Theorem. After removing the singularities, we take scissors in hand and cut along the dotted lion.
1.26 The field theory method
Irrigate the desert and plant grass so that it becomes a field. A zero lion is trivial to capture, so we may assume the lion ¹ 0. The element 1 may be located just to the right of 0 in the prime subfield. Prize it apart into LL-1 and discard L-1. (Remark: the Greeks used the convention that the product of two lions is a rectangle, not a lion; the product of three lions is a solid, and so on. It follows that every lion is transcendental. Modern mathematics permits algebraic lions.)
1.27 The kittygory method
Form the category whose objects are the lions in the desert, with trivial morphisms. This is a small category (even if lions are big cats) and so can be embedded in a concrete category. There is a forgetful functor from this to the category of sets: this sets the concrete and traps the embedded lions.
1.28 The backward induction method
We prove by backward induction the statement L(n): "It is possible to capture n lions". This is true for sufficiently large n since the lions will be packed like sardines and have no room to escape. But trivially L(n+1) implies L(n) since, having captured n+1 lions, we can release one. Hence L(1) is true.
1.29 Another topological method
Give the desert the leonine topology, in which a subset is closed if it is the whole desert, or contains no lions. The set of lions is now dense. Put an open cage in the desert. By density it contains a lion. Shut it quickly!
1.30 The sequences method
The real lion is non-compact and so contains non-convergent subsequences. To overcome this let W be the first uncountable ordinal and insert a copy of the given lion between a and a+1 for all ordinals a < W. You now have a long lion in which all sequences converge. Proceed as in 1.5 above.
1.31 The group ring method
Let G be the free group on the set G of lions, and let ZG be its group ring. The lions now belong to a ring, so are circus lions, hence tame.
1.32 The Bourbaki method
The capture of a lion in a desert is a special case of a far more general problem. Formulate this problem and find necessary and sufficient conditions for its solution. The capture of a lion is now a trivial corollary of the general theory, which on no account should be written down explicitly.
1.33 The Hasse-Minkowski method
Consider the lion catching problem modulo p for all primes p. There being only finitely many possibilities, this can be solved. Hence the original problem can be solved.
1.34 The PL method
The lion is a 3-manifold with a non-empty boundary. Triangulate it to get a PL manifold. The can be collared, which is what we wish to achieve.
1.35 The singularity method
Consider a lion in the plane If it is a regular lion its regular habits render it easy to catch (e.g. dig a pit). without loss of generality it is a singular lion. Stable singularities are dense, so without loss of generality the lion is stable. The singularity is not a self-intersection (since a self-intersecting lion is absurd) so it must be a cusp. Complexify and intersect with a sphere to get a trefoil knot. As in 1.7 above, the problem then becomes trivial.
1.36 The measure-theoretic method
Assume for a contradiction that no lions can be captured. Since capturable lions are imaginary, all lions are real. On any real lion there exists a non-trivial invariant measure m, namely Haar or Lebesgue measure. Then m x m is a Baire measure on LxL. Since a product of lions cannot be a bear, the Baire measure on LxL is zero. Hence m = 0, a contradiction. Thus all lions can be captured.
1.37 The method of parallels
Select a point in the desert and introduce a tame lion not passing through that point. There are three cases:
The geometry is Euclidean. There is then a unique parallel lion passing through the selected point. Grab it as it passes.
The geometry is hyperbolic. The same method will now catch infinitely many lions.
The geometry is elliptic. There are no parallel lions, so every lion meets every other lion. Follow a tame lion and catch all the lions it meets: in this way every lion in the desert will be captured.
1.38 The Thom-Zeeman method
A lion loose in the desert is an obvious catastrophe. It has three dimensions of control (two for position, one for time) and one dimension of behaviour (being parameterized by a lion). Hence by Thom's Classification Theorem it is a swallowtail. A lion that has swallowed its tail is in no state to avoid capture.
1.39 The Australian method
Lions are very varied creatures, so there is a variety of lions in the desert. This variety contains free lions which satisfy no non-trivial identities. Select a lion and register it as "Fred Lion" at the local Register Office: it now has a non-trivial identity, hence cannot be free. If it is not free it must be captive. (Note: if "Fred Lion is thought to be a trivial identity, then call it "Albert Einstein").
1.40 Good Method
Let Q be an operator that encloses a word in quotation marks. Its square Q2 encloses a word in double quotes. The operator clearly satisfies the law of indices. QmQn=Qm+n. Write down the word "lion", without quotation marks. Apply to it the operator Q-1. Then a lion will appear on the page. It is advisable to enclose the page in a cage before applying the operator.
1.41 Category Method
Let L denote the category whose objects are lions, with "ancestor" as the only nontrivial morphism. Let p be the category of caged lions. The subcategory p is clearly complete, is nonempty (by inspection), and has both a generator and cogenerator1. Let F: L-->p be the forgetful functor, which forgets the cage. By the Adjoint Functor Theorem the functor F has a coadjoint C: L-->p, which reflects each lion into a cage. We remark that this method is superior to the Good method above, which only guarantees the capture of one lion, and which requires an application of the Weiekäfig Preparation Theorem.
[1] Moses, The Book of Genesis, vii, 15-16
2 Mathematical Logic Methods
2.1 Nonstandard Analysis Method
In a nonstandard universe (namely, the land of Oz), lions are cowardly and may be caught easily. By the transfer principle, this likewise holds in our (standard) universe.
2.2 Set Theory Method 1
If the set of lions is bounded, you can simply build a cage around the boundary. So assume that the set of lions is unbounded. It will then have an element in common with a stationery set. But a stationery lion is trivial to capture.
2.3 Set Theory Method 2
Assume V=L. Since the lion is in the universe, it is constructible. So just carry out its construction within a cage in the first place.
2.4 Set Theory Method 3
Assume AC. Perform a Tarski-Banach decomposition on the lion to halve its size. Repeat until the lion is small enough to be captured easily.
2.5 Recursion Theory Method
Assume you can capture a lion. Having done so, you can easily bring it to a standstill, and you would thus have a solution to the halting problem. Since the halting problem is unsolvable, you cannot capture a lion after all. In conjunction with previous results, we have: COROLLARY. Mathematics is inconsistent.
3 Theoretical Physics Methods
3.1 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.
3.2 The Schroedinger method
At every instant there is a non-zero probability of the lion being in the cage. Sit and wait.
3.3 The Quantum Measurement Method
We assume that the sex of the lion is ab initio indeterminate. The wave function for the lion is hence a superposition of the gender eigenstate for a lion and that for a lioness. We lay these eigenstates out flat on the ground and orthogonal to each other. Since the (male) lion has a distinctive mane, the measurement of sex can safely be made from a distance, using binoculars. The lion then collapses into one of the eigenstates, which is rolled up and placed inside the cage.
3.4 The nuclear physics method
Insert a tame lion into the cage and apply a Majorana exchange operator 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, exchanging spins.
3.5 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.
4 Experimental Physics Methods
4.1 The thermodynamics method
We construct a semi-permeable membrane which lets everything but lions pass through. This we drag across the desert.
4.2 The atomic fission method
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.
4.3 The magneto-optical method
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.
4.4 The method of analytical mechanics
Since the lion has nonzero mass, it has moments of inertia. Grab it during one of them.
5 Computer Science Methods
5.1 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.
5.2 The parallel search method.
By using parallelism we will be able to search in the direction to the north much faster than earlier.
5.3 The Monte-Carlo method.
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.
5.4 The practical approach.
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.
5.5 The common language approach.
If only everyone used ADA/Common Lisp/Prolog, this problem would be trivial to solve.
5.6 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.
5.7 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.
5.8 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.
6 Social Science Methods
6.1 The pricing method
Raise the price of a lion hunting licence to £1,000,000. Anyone who can pay this fee can afford to hire a hunter skilled in the mathematical theory of big game hunting.
6.2 The cost-benefit method
For some lions the cost of avoiding capture (all that running, dodging spears, etc.) is greater than the benefits of remaining free. These lions will give themselves up voluntarily.
6.3 The utility calculation method
For some lions, the utility of roaming around free will be less than the utility of being in a cage with (...fill in number) lionesses and a guaranteed (...fill in number) pounds of meat per day. These lions are easily persuaded to give themselves up. Note that with this method the supply of captured lions can be closely controlled. Also, an obvious variant leads to the capture of lionesses. Warning - non-discrimination laws require the keeping of tired, old, albino, handicapped or gay lions that find their way into the cage.
6.4 The theoretical political method
Define the following: P, the probability of the lion remaining free if it resists; B1, the benefit to the lion of remaining free; B2, the benefit of being caught (e.g. free food); C, the cost to the lion of resisting capture (all that running again). Then, assuming the lion is sure to be captured if it doesn't resist, E(U)=E(resisting) - E(not resisting) = (PB1 + (1-P)B2-C) - B2. If E(U)<0, the lion, being rational, will not resist. Note that there is no way of knowing ahead of time for which lions E(U)<0. But then political theorists usually have difficulty dealing with reality.
6.5 The method of international diplomacy
I would explain this method, but it never works anyway.
6.6 The method of democratic prides
Wait for the election of the cowardly lion. Capturing this lion should be easy. Warning: election methods do not always produce the anticpated consquences.
6.7 The method of monarchial prides
Where there is a king of the beasts there are pretenders also. Use them as allies.
6.8 The method of dictatorial prides
If one is not too choosy, the strongest lion will help liquidate certain of the species.
6.9 The survey research method
Interview a random sample of lions, asking them how strongly they would resist capture. For those who are suspicious of verbal responses, psychophysical methods are available. Note that hunters using this methods will be accused by theorists of not knowing what they are doing.
6.10 The socialization method
While they are young, train the desired number of lions not to attack men with guns. The rest is easy. For a more sporting variant, train the young lions not to attack men with bows and arrows. For the still more daring, train the lions to attack, but with their front claws retracted. The most daring of all will try to resocialize older lions.
6.11 The cliometrics method
Construct a large computer file containg records of the number of lions bagged, their location, size, hunter, etc... and use it to show that Big Game Hunting never was a problem.
6.12 The intellectual history method
Simply outwit the lion. (It must be added that this method is considered unsporting by true social scientists).
7 General Methods
7.1 The biological method
Obtain a number of planarians and subject them to repeated recorded statements saying "You are a planarian". The worms should shortly learn this fact since they must have some suspicions to this effect to start with. Now feed the worms to the lion in question. The knowledge of the planarians is then transferred to the lion. The lion, now thinking he is a planarian, will proceed to subdivide. This process, while natural to planarians, is disastrous to the lion.
7.2 The method of moral philosophy
Construct a corral in the Sahara and wait until autumn. At that time the corral will contain a large number of lions, for it is well known that a pride cometh before the fall.