The following is the famous, seminal paper by H. Petard*, which appeared in the American Mathematical Monthly in 1938 and introduced to the academic community at large the mathematical theory of big game hunting. As is evident from the other articles that follow, Petard’s work prompted a good many others to add to this literature. This, together the subsequent articles we include, do not form a complete compendium, but they do provide the interested reader a solid introduction into this exciting branch of mathematics.
This first appeared in Amer. Math. Monthly 45 (1938) p446-447. In fact, the Method 10 included here did not actually appear in the original Monthly version, but in a slightly expanded version of that appeared in Eureka. [ link ]
A contribution to the mathematical theory of big game hunting [1938]
– H. Petard
Princeton, New Jersey
(c) Amer. Math. Monthly 45 (1938) p446-447
This little known mathematical discipline has not, of recent years, received in the literature the attention which, in our opinion, it deserves. In the present paper we present some algorithms which, it is hoped, may be of interest to other workers in the field. Neglecting the more obviously trivial methods, we shall confine our attention to those which involve significant applications of ideas familiar to mathematicians and physicists.
The present time is particularly fitting for the preparation of an account of the subject, since recent advances both in pure mathematics and theoretical physics have made available powerful tools whose very existence was unsuspected by earlier investigators. At the same time, some of the more elegant classical methods acquire new significance in the light of modern discoveries. Like many other branches of knowledge to which mathematical techniques have been applied in recent years, the Mathematical Theory of Big Game Hunting has a singularly unifying effect on the most diverse branches of the exact sciences.
For the sake of simplicity of statement, we shall confine our attention to Lions (Felis leo) whose habitat is the Sahara Desert. The methods which we shall enumerate will easily be seen to be applicable, with obvious formal modifications, to other carnivores and to other portions of the globe. The paper is divided into three parts, which draw their material respectively from mathematics, theoretical physics, and experimental physics.
The author desires to acknowledge his indebtedness to the Trivial Club of St. John’s College, Cambridge, England; to the MIT chapter of the Society for Useless Research; to the F o P, of Princeton University; and to numerous individual contributors, known and unknown, conscious and unconscious.
I. Mathematical methods
1. The Hilbert, or axiomatic, method. We place a locked cage onto a given point in the desert. After that we introduce the following logical system:
- Axiom I. The set of lions in the Sahara is not empty.
- Axiom II. If there exists a lion in the Sahara, then there exists a lion in the cage.
- Rule of 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.
2. The method of inversive geometry. 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.
3. The method of projective geometry. Without loss of generality, we can view the desert as a plane. We project the surface onto a line, and then project the line onto an interior point of the cage. Thereby the lion is projected onto that same point.
4. The Bolzano-Weierstrass method. Divide the desert by a line running from N-S. The lion is then either in the E portion or in the W portion; let us assume him to be in the W portion. Bisect this portion by a line running from E-W. The lion is either in the N portion or in the S portion; let us assume him to be in the N portion. We continue this process indefinitely, constructing a sufficiently strong fence about the chosen portion at each step. The diameter of the chosen portions approaches zero, so that the lion ultimately surrounded by a fence of arbitrarily small perimeter.
5. The “Mengentheoretisch” method. We observe that the desert is a separable space. It therefore contains an enumerable dense set of points, from which can be extracted a sequence having the lion as limit. We then approach the lion stealthily along this sequence, bearing with us suitable equipment.
6. The Peano method. Construct, by standard methods, a continuous curve passing through every point of the desert. It has been remarked [1]that it is possible to traverse such a curve in an arbitrarily short time. Armed with a spear, we traverse the curve in a time shorter than that in which a lion to move a distance equal to its own length.
7. A topological method. We observe that a lion has at least the connectivity of a torus. We transport the desert into four-space. Then it is possible [2] to carry out such a deformation that the lion can be returned to three-space in a knotted condition. He is then completely helpless.
8. The Cauchy, for function theoretical, method. We examine a lion-valued function f(z). Let ζ be the cage. Consider the integral
where C represents the boundary of the desert. Its value is f(ζ), i.e. there is a lion in the cage [3].
9. The Wiener-Tauberian method. We obtain a tame lion, L0, from the class L(-¥, ¥), whose Fourier transform vanishes nowhere, and release it in the desert. L0 then converges toward our cage. By Wiener’s General Tauberian Theorem [4], any other lion, L (say), will converge to the same cage. Alternatively we can approximate arbitrarily closely to L by translating L0 through the desert [5].)
10. The Eratosthenian method. Enumerate all the objects in the desert. Examine them one by one, and discard all those that are not lions. A refinement will capture only prime lions.
II. Methods from theoretical physics
11. The Dirac method. We observe that wild lions are, ipso facto, not be observable in the Sahara desert. Consequently, if there are any lions at all in the Sahara, they are tame. We leave catching a tame lion as an exercise to the reader.
12. The Schroedinger method. At any given moment there is a positive probability that there is a lion in the cage. Sit down and wait.
13. The nuclear physics method. Place a tame lion into the cage, and apply a Majorana exchange operator [6] on it and a wild lion.
As a variant, let us suppose, to fix ideas, that we require a male lion. We place a tame lioness into the cage, and apply the Heisenberg exchange operator [7] which exchanges spins.
14. A relativistic method. We distribute about the deser lion bait containing large portions of the Companion of Sirius. When enough bait has been taken, we project a beam of light across the desert. This will bend right around the lion, who will hen become so dizzy that he can be approahced with impunity.
III. Experimental physics methods
15. The thermodynamics method. We construct a semi-permeable membrane, permeable to everything except lions, and sweep it across the desert.
16. The atom-splitting method. We irradiate the desert with slow neutrons. The lion becomes radioactive, and a process of disintegration set in. When the decay has proceeded sufficiently far, he will become incapable of showing fight.
17. The magneto-optical method. We plant a large lenticular bed of catnip (Nepeta cataria), whose axis lies along the direction of the horizontal component of the earth’s magnetic field, and place a cage at one of its foci. We distribute over the desert large quantities of magnetized spinach (Spinacia oleracea), which, as is well known, has a high ferric content. The spinach is eaten by herbivorous denizens of the desert, which in turn are eaten by lions. The lions are then oriented parallel to the earth’s magnetic field, and the resulting beam of lions is focus by the catnip upon the cage.
IV. References
[1] After Hilbert, cf. E. W. Hobson, “The Theory of Functions of a Real Variable and the Theory of Fourier’s Series” (1927), vol. 1, pp 456-457
[2] H. Seifert and W. Threlfall, “Lehrbuch der Topologie” (1934), pp 2-3
[3] According to the Picard theorem (W. F. Osgood, Lehrbuch der Funktionentheorie, vol 1 (1928), p 178) it is possible to catch every lion except for at most one.
[4] N. Wiener, “The Fourier Integral and Certain of its Applications” (1933), pp 73-74
[5] N. Wiener, ibid, p 89
[6] cf e.g. H. A. Bethe and R. F. Bacher, “Reviews of Modern Physics”, 8 (1936), pp 82-229, esp. pp 106-107
[7] ibid
* It is perhaps generally not known that Petard’s** full initials are H. W. O., standing for “Hoist With Own.”
** Actually, H. Petard is the pen-name for the mathematician E. S. Pondiczery***, who preferred to publish the paper pseudonymously.
*** In fact, Pondiczery himself was a fictitious mathematician invented by Ralph P. Boas and Frank Smithies.****
**** This was published in The American Mathematical Monthly, 1938, with one editorial alternation: a “footnote to a footnote was ruthlessly removed.” Consider this last footnote to a footnote to a footnote of a footnote as our nod to the masters.
Following H. Petard’s seminal 1938 paper, work in theoretical big-game hunting stalled. However, three decades later, a new generation of mathematicians were prompted to add to this literature. We have included a collection of these articles, including the famous paper by Morphy, viewed by many as the most pioneering work in the field since Petard.
A new method of catching a lion [1965]
– I. J. Good
Amer. Math. Monthly 72 (1965), p. 436.
In this note a definitive procedure will be provided for catching a lion in a desert (see [1]).
Let Q be the operator that encloses a word in (single) 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 it to 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.
References
[1] H. Petard. A contribution to the mathematical theory of big game hunting, Amer. Math. Monthly 45 (1938) p446-447.
On a theorem of H. Petard [1967]
– Christian Roselius
Tulane University
Amer. Math. Monthly 74 (1967), p. 838-839.
In a classical paper [4], H. Petard proved that it is possible to capture a lion in the Sahara desert. He further showed [4, no. 8, footnote] that it is in fact possible to capture every lion with at most one exception. Using completely new techniques, unavailable to Petard at the time, we are able to sharpen this result, and to show that every lion may be captured.
Let L denote the category whose objects are lions, with “ancestor” as the only nontrivial morphism. Let C be the category of caged lions. The subcategory C is clearly complete, is nonempty (by inspection), and has both a generator and cogenerator [3, vii, 15-16]. Let F : C → L be the forgetful functor, which forgets the cage. By the Adjoint Functor Theorem [1, 80-91] the functor F has a coadjoint G : L → C, which reflects each lion into a cage.
We remark that this method is obviously superior to the Good method [2], which only guarantees the capture of one lion, and which requires an application of the Weierkafig Preparation Theorem.
References
[1] P. Freyd, Abelian categories, New Yor, 1964.
[2] I. J. Good, A new method of catching a lion, Amer. Math. Monthly, 72 (1965) 436.
[3] Moses, The Book of Genesis.
[4] H. Petard. A contribution to the mathematical theory of big game hunting, Amer. Math. Monthly 45 (1938) p446-447.
Some modern mathematical methods in the theory of big-game hunting [1968]
–Otto Morphy, D.Hp.
(Doctor of Hypocrisy)
Amer. Math. Monthly 75 (1968), p. 185-187.
It is now 30 years since the appearance of H. Petard’s classic treatise [2] on the mathematical theory of big game hunting. These years have seen a remarkable development of practical mathematical techniques. It is, of course, generally known that it was Petard’s famous letter to the president in 1941 that led to the establishment of the Martini Project, the legendary crash program to develop new and more efficient methods for search and destroy operations against the axis lions. The Infernal Bureaucratic Federation (IBF) has recently declassified certain portions of the formerly top secret Martini Project work. Thus we are now able to reveal to the world, for the first time, these important new applications of modern mathematics to the theory and practice of lion hunting. As has become standard practice in the discipline [2] we shall restrict our attention to the case of lions residing in the Sahara Desert [3]. As noted by Petard, most methods apply, more generally, to other big game. However, method (3) below appears to be restricted to the genus Felis. Clearly, more research on this important matter is called for.
1. Surgical method. A lion may be regarded as an orientable 3-manifold with a nonempty boundary. It is known [4] that by means of a sequence of surgical operations (known as “spherical modification” in medical parlance) the lion can be rendered contractible. He may the be signed to a contract with Barnum and Bailey.
2. Logical method. A lion is a continuum. According to Cohen’s theorem [5] 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.
3. 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 [6] 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 clearly not of any concern to us.
4. Method of differential topology. The lion is a 3-manifold embedded in euclidean 3-space. This implies that he is a handlebody [7]. However, a lion which can be handled is tame and will enter the cage upon request.
5. Sheaf theoretic method. The lion is a cross-section [8] of the sheaf of germs of lions [9] 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.
6. Method of tranformation 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 isoptropy group of the nose (considered as a point) [10]. This represents the lion as a homogeneous space. That is, this representation homogenizes the lion. A homogenized lion is in no shape to put up a fight [11].
7. Postnikov method. A male lion is quite hairy [12] and may be regarded as being made up of fibers. Thus we may regard the lion as a fiber space. We may then construct a Postnikov decomposition [13] of the lion. This being done, the lion, being decomposed, is dead and in bad need of burial.
8. Steenrod algebra method. Consider the mod p cohomology ring of the lion. We may regard this as a module over the mod p Steenrod algebra. Doing this requires the use of the table of Steenrod cohomology operations [14]. Every element must be killed by some of these operations. Thus the lion will die on the operating table.
9. Homotopy method. The lion has the homotopy type of a one-dimensional complex and hence he is a K(pi,1) space. If pi is noncommutative, then the lion is not a member of the international communist conspiracy [15] and hence he must be friendly. If pi is commutative, then the lion has the homotopy type of the space of loops on a K(pi,2) space [13]. We hire a stunt pilot to loop the loops, thereby hopelessly entagling the lion and rendering him helpless.
10. Covering space method. Cover the lion by his simply connected covering space. In effect this decks the lion [16]. Grab him while he is down.
11. Game theoretic method. A lion is big game. Thus, a fortiori, he is a game. Therefore there exists an optimal strategy [17]. Follow it.
12. Group 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 work capable of a solution according to the work of Thompson and Feit [18].
We conclude with one significant nonmathematical method.
13. Biological method. Obtain a number of planarians (flat worms) 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 [19]. The lion, now thinking that he is a planarian, will proceed to subdivide. This process, while natural for the planarian, is disastrous for the lion.
Ed. Note: Prof. Morphy is the namesake of his renowned aunt, the author of the famous series of epigrams now popularly known ans Aunti Otto Morphisms, or euphemistically as epimorphisms.
Footprints
[1] This report was supported by grant #007 from Project Leo of the War on Puberty.
[2] H. Petard. A contribution to the mathematical theory of big game hunting, Amer. Math. Monthly 45 (1938) p446-447.
[3] This restriction of the habitat does not affect the generality of the results because of Brower’s theorem on the invariance of domain.
[4] Kervaire and Milnor, Groups of homotopy spheres, I, Ann. of Math., (1963).
[5] P. J. Cohen, The independence of the continuum hypothesis, Proc. N.A.S. (63-64).
[6] P. Freyd, Abelian Categories, Harper and Row, New York, 1964.
[7] S. Smale, A survey of some recent developments in differential topology, Bull. A.M.S. (1963).
[8] It has been experimentally verified that lions are cross.
[9] G. Bredon, Sheaf Theory, McGraw-Hill, new York, 1967.
[10] Montgomery and Zippin, Topological Information Groups, Interscience, 1955.
[11] E. Bordern, Characteristic classes of bovine spaces, Peripherblatt fur Math., (1966BC).
[12] Eddy Courant, Sinking of the Mane, Pantz Press, 1898.
[13] E. spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
[14] Steenrod and Epstein, Cohomology Operations, Princeton, 1962.
[15] Logistics of the Attorney General’s list, Band Corp. (1776).
[16] Admiral, T.V., (USN Ret.), How to deck a swab, ONR tech. rep. (classified).
[17] von Neumann and Morgenstern, Theory of Games…, Princeton, 1947.
[18] Feit and Thompson, Solvability of groups of odd order, Pac. J. M. (1963).
[19] J. V. McConnell, ed., The Worm Re-Turns, Prentice-Hall, Englewood Cliffs, N.J., 1963.
[20] This method must be carried out with extreme caution, for if the lion is large enough to approach critical mass, the fissioning of the lion may produce a violent reaction.
Further techniques in the theory of big-game hunting [1968]
–Patricia L. Dudley
G. T. Evans
K. D. Hansen
I. D. Richardson
Carleton University, Ottawa
Amer. Math. Monthly 75 (1968), p. 896-897.
Interest in the problem of big game hunting has recently been reawakened by Morphy’s paper in this Monthly, Feb. 1968, p. 185. We outline below several new techniques, including one from the humanities. We are also in possession of a solution by means of Bachmann geometry which we shall be glad to communicate to anyone who is interested.
1. 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 the lion. As noted above, one can construct a net converging to any point in a bar; in this net enmesh the lion.
2. Method of analytical mechanics. Since the lion has nonzero mass it has moments of inertia. Grab it during one of them.
3. 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.
4. Method of trigonometric functions. The lion, having spent his life under the Sahara sun, will surely have a tan. Induce him to lie on his back; he can then, by virtue of his reciprocal tan, be cot.
5. Boundary value method. As Dr. Morphy pointed out, Brower’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.
6. 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.
Published a decade after the late-60s explosion in theoretical bug-game hunting, the following survey article provides an overview of key classic and new results.
Fifteen new ways to catch a lion [1981]
— John Barrington*
This, O Best Beloved, is another tale of the High and the Far-Off Times. In the blistering midst of the Sand-Swept Sahara lived a Pride of Lions. There was a Real Lion, and a Projective Lion, and a pair of Parallel Lions; and all manner of Lion Segments. And on the edge of the Sand-Swept Sahara there lived a ‘nexorable Lion Hunter…
I make no apologies for raising once again the problems of the mathematical theory of big game hunting. As with any branch of mathematics, much progress has been made in the last decade.
The subject started in 1938 with the epic paper of Petard [1]. The main problem is usually formulated as follows: In the Sahara desert there exist lions. Devise methods for capturing them. Petard found ten mathematical solutions, which we can paraphrase as follows.
1. The Hilbert method. Place a locked cage in the desert. Set up the following axiomatic system.
- The set of lions is nonempty.
- If there is a lion in the desert, then there is a lion in the cage.
THEOREM 1. There is a lion in the cage.
2. The method of inversive geometry. Place a locked, spherical cage in the desert, empty of lions, and enter it. Invert with respect to the cage. This maps the lion to the interior of the cage, and you outside it.
3. The projective geometry method. The desert is a plane. Project this to a line, then project the line to a point inside the cage. The lion goes to the same point.
4. The Bolzano-Weierstrass method. Bisect the desert by a line running N-S. The lion is in one half. Bisect this half by a line running E-W. The lion is in one half. Continue this process indefinitely, at each stage building a fence. The lion is enclosed in a fence of arbitraily small length.
5. The general topology method. Observe that the desert is a separable metric space, so has a countable dense subset. Some subsequence converges to the lion. Approach stealthily along it, bearing suitable equipment.
6. The Peano method. There exists a space-filling curve passing through every point of the desert. It has been remarked [2] that such a curve may be traversed in as short a time as we please. Armed with a spear, traverse the curve faster than the lion can move his own length.
7. A topological method. The lion has at least the connectivity of a torus. Transport the desert to 4-space. It can now be deformed in such a way as to knot the lion [3]. He is now helpless.
8. The Cauchy method. Let f(z). Let ζ be the cage. Consider the integral
where C is the boundary of the desert. Its value is f(ζ),
that is, there is a lion in the cage.9. The Wiener Tauberian method. Procure a tame lion L0 of class L(-∞, ∞) whose Fourier transform (Furrier transform?) nowhere vanishes, and set it loose in the desert. Being tame, it will converge to the cage. By Wiener [4] every other lion will converge to the same cage.
10. The Eratosthenian method. Enumerate all objects in the desert; examine them one by one; discard all those that are not lions. A refinement will capture only prime lions.
Petard also gives one physical method with strong mathematical content:
11. The Schrodinger method. At any instant there is a nonzero probability that a lion is in the cage. Wait.
The next work of any significance is that of Morhpy [5]. I confess that I do not find all of his methods convincing. The best are:
12. Surgery. The lion is an orientable 3-mainfold with boundary and so [6] may be rendered contractible by surgery. Contract him to Barnum and Bailey.
13. The cobordism method. For the same reasons the lion is a handlebody. A lion that can be handled is trivial to capture.
14. The sheaf-theoretic method. The lion is a cross-section [8] of the sheaf of the germ of lions in the desert. Re-topologize the desert to make it discrete: the stalks of the sheaf fall apart and release the germs, which kill the lion.
15. The Postnikov method. The lion, being hairy, may be regarded as a fiber space. Construct a Postnikov decomposition [9]. A decomposed lion must, of course, be long dead.
16. The universal covering. Cover the lion by his simply-connected covering space. Since this has no holes, he is trapped!
17. The game-theory method. The lion is big game, hence certainly a game. There exists an optimal strategy. Follow it.
18. The Feit-Thompson method. If necessary add a lion to make the total odd. This renders the problem soluble [10].
Recent, hitherto unpublished, work has revealed a range of new methods:
19. 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 L is nonzero. The element 1 may be located just to the right of 0 in the prime subfield. Pry it apart into L L-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 3 lions is a solid, and so on. It follows that every lion is transcendental. Modern mathematics permits algebraic lions.)
20. The kittygory method. Form the category whose objects are the lions in the desert, with trivial morphism. This is a small category (even if lions are big cats) and so can be embedded in a concrete category [11]. There is a forgetful functor from this to the category of sets: this sets the concrete and traps the embedded lions.
21. Backward induction. We prove by backward induction the statement L(n): “It is possible to capture n lions.” This is true for sufficiently large n since 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.
22. 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!
23. The Moore-Smith method. Like (5) above, but this applies to non-separable deserts: the lion is caught not by a sequence, but by a net.
24. For those who insist on sequences. The real lion is non-compact and so contains non-convergent subsequences. To overcome this let Ω be the first uncountable ordinal and insert a copy of the given lion between A and A+1 for all ordinals A less than Ω. You now have a long lion in which all sequences converge [12]. Proceed as in (5).
25. The group ring method. Let G be the free group on the set L of lions, and let ZG be its group ring. The lions now belong to a ring, so are circus lions, and hence tame.
26. The Bourbaki method. The capture of a lion in the 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.
27. 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 solve [13].
28. The PL method. The lion is a 3-manifold with nonempty boundary. Triangulate it to get a PL manifold. This can be collared [14], which is what we wish to achieve.
29. 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 (7) the problem becomes trivial.
30. The measure-theoretic method. Assume for a contradiction that no lion can be captured. Since capturable lions are imaginary, all lions are real. On any real lion there exists a nontrivial invariant measure m, namely Harr or Lebesgue measure. Then the product m x m is a Baire measure on L x L by [15]. Since a product of lions cannot be a bear, the Baire measure on L x L is zero. Hence, m=0, a contradiction. Thus, all lions may be captured.
31. 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 the tame lion and catch all the lions it meets: in this way, every lion in the desert will be captured.
32. The Thom-Zeeman method. A lion loose in the desert is an obvious catastrophe [16]. It has three dimensions of control (2 for position, 1 for time) and one dimension of behavior (being parametrized 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.
33. The Australian method. Lions are very varied creatures, so there is a variety of lions in the desert. This variety contains free lions [17], which satisfy no nontricial identities. Select a lion and register it as “Fred Lion” at the local register Office: now it has a nontrivial identity, and hence cannot be free. If it is not free it must be captive. (If “Fred Lion” is thought to be a trivial identity, call it “Albert Einstein.”)
Bibliography
[1] H. Petard. A contribution to the mathematical theory of big game hunting, Amer. Math. Monthly 45 (1938) 446-7.
[2] E. W. Hobson, The theory of functions of a real variable and the theory of Fourier’s series, 1927.
[3] H. Seifert and W. Threlfall, Lehrbuch der Topologie, 1934.
[4] N. Wiener, The Fourier integral and certain of its applications, 1933.
[5] O. Morphy, Some modern mathematical methods in the theory of big game hunting, Amer. Math. Monthly 75 (1968) 185-7.
[6] M. Kervaire and J. Milnor, Groups of homotopy spheres I, Ann. of Math. 1963.
[7] This footnote has been censored by the authorities.
[8] It has been verified experimentally that lions are cross.
[9] E. Spanier, Algebraic Topology, McGraw-Hill 1966.
[10] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pac. J. Math. 1963.
[11] P. Freyd, Abelian Categories.
[12] J. L. Kelley, General Topology.
[13] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, 1973.
[14] C. P. Rourke and B. L. Sanderson, Introduction to Piecewise Linear Topology, 1973.
[15] S. K. Berberian, Topological Groups.
[16] R. Thom, Stabilite Structurelle et Morphogenese, 1972.
[17] Hanna Neumann, Varieties of Groups, 1972.
* From Seven Years of Manifold, 1968-1980, Ian Stewart and John Kaworski, eds., Cheshire, England, Shiva Publishing Limited, 1981, pp. 36-39.
John Barrington is, in fact, a pseudonym for Ian Stewart.
Contributions to the theory of big-game hunting from computer science [1993]
— John Ioannidis
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.
2. The parallel search method. By using parallelism we will be able to search in the direction to the north much faster than earlier.
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.
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. The common language approach. If only everyone used ADA/Common Lisp/Prolog, this problem would be trivial to solve.
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.
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.
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.
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 esnures that the loop terminates.
Exercise 0:
Refine the step “Approach lion under invariance of P1”.
Note 2. The program is robust in the sense that it will lead to abortion if the value of L is “lioness”.
Remark 0. This may be a new sense of the word “robust” for you.
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.
Application of the theory of big-game hunting to other disciplines [2000]
Much work has been made on the many advances mathematics can make in the theory of big game hunting. This survey paper briefly describes the further contributions to the theory made by other disciplines.
Mathematicians hunt lions by going to Africa, throwing out everything that is not an lion, and catching one of whatever is left.
Experienced mathematicians will attempt to prove the existence of at least one lion before proceeding to step 1 as a subordinate exercise.
Professors of mathematics will prove the existence of at least one lion and then leave the detection and capture of an actual lion as an exercise for their graduate students.
Computer scientists hunt lions by exercising Algorithm A:
- Go to Africa.
- Start at the Cape of Good Hope.
- Work northward in an orderly manner, traversing the continent alternately east and west.
- During each traverse pass,
- Catch each animal seen.
- Compare each animal caught to a known lion.
- Stop when a match is detected.
Experienced computer programmers modify Algorithm A by placing a known lion in Cairo to ensure that the algorithm will terminate.
Assembly language programmers prefer to execute Algorithm A on their hands and knees.
Engineers hunt lions by going to Africa, catching yellow animals at random, and stopping when any one of them weighs within plus or minus 15 percent of any previously observed lion.
Economists don’t hunt lions, but they believe that if lions are paid enough, they will hunt themselves.
Statisticians hunt the first animal they see N times and call it an lion.
Consultants don’t hunt lions, and many have never hunted anything at all, but they can be hired by the hour to advise those people who do.
Operations research consultants can also measure the correlation of hat size and bullet color to efficiency of lion-hunting strategies, if someone else will only identify the lions.
Politicians don’t hunt lions, but they will share the lions you catch with the people who voted for them.
Lawyers don’t hunt lions, but they do follow the prides around arguing about who owns the droppings.
Software lawyers will claim that they own an entire pride based on the look and feel of one dropping.
Vice presidents of engineering, research and development try hard to hunt lions, but their staffs are designed to prevent it. When the vice president does get to hunt lions, the staff will try to ensure that all possible lions are completely nonprehunted lions, afterwhich the staff will (1) compliment the vice president’s keen eyesight and (2) enlarge itself to prevent any recurrence.
Senior managers set broad lion-hunting policy based on the assumption that lions are just like field mice, but with deeper voices.
Quality assurance inspectors ignore the lions and look for mistakes the other hunters made when they were packing the jeep.
Sales people don’t hunt lions but spend their time selling elephants they haven’t caught, for delivery two days before the season opens.
Software sales people ship the first thing they catch and write up an invoice for an lion.
Hardware sales people catch rabbits, paint them yellow, and sell them as desktop lions.
