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Problem: To Catch a Lion in the Sahara Desert.

Note: This problem was originally posed and solved mathematically in an article called ‘A Contribution to the Mathematical Theory of Big Game Hunting’. The author was said to be one H. Pétard, although apparently this was a pseudonym for Ralph Philip Boas, Jr. and some colleagues at the Princeton Institute for Advanced Study. The original article has been added to many times, and our document attempts to include as many different lion-hunting methods as we can find.
Methods 1.1-1.9,2.1-2.4 and 3.1-3.3 were included in the original article; all articles reproduced without permission, but with due credit.

1. Mathematical Methods
   • 1.1 The Hilbert, or axiomatic, method
   • 1.2 The method of inversive geometry
   • 1.3 The method of projective geometry
   • 1.4 The Bolzano-Weierstrass method
   • 1.5 The ‘Mengentheoretisch’ method
   • 1.6 The Peano method
   • 1.7 A topological method
   • 1.8 The Cauchy, or function-theoretical, method
   • 1.9 The Wiener Tauberian method
2. Methods from Theoretical Physics
   • 2.1 The Dirac method
   • 2.2 The Schrödinger method
   • 2.3 The method of nuclear physics
   • 2.4 A relativistic method
   • 2.5 The Newton method
   • 2.6 The Heisenberg method
   • 2.7 The Einstein method
   • 2.8 The quantum measurement method
3. Methods from Experimental Physics
   • 3.1 The thermodynamical method
   • 3.2 The atom-splitting method
   • 3.3 The magneto-optical method
4. Contributions from Computer Science
   • 4.1 The search method
   • 4.2 The parallel search method
   • 4.3 The Monte Carlo method
   • 4.4 The practical approach
   • 4.5 The common language approach
   • 4.6 The standard approach
   • 4.7 Linear search
   • 4.8 The Dijkstra method
5. A New Method of Catching a Lion
6. On a Theorem of H. Pétard
7. Some Modern Mathematical Methods in the Theory of Lion Hunting
   • 7.1 Surgical method
   • 7.2 Logical method
   • 7.3 Functorial method
   • 7.4 Method of differential topology
   • 7.5 Sheaf theoretic method
   • 7.6 Method of transformation groups
   • 7.7 Postlikov method
   • 7.8 Steenrod algebra method
   • 7.9 Homotopy method
   • 7.10 Covering space method
   • 7.11 Game theoretic method
   • 7.12 Group theoretic method
   • 7.13 Biological method
8. Futher Techniques in the Theory of Big Game Hunting
   • 8.1 Moore-Smith method
   • 8.2 Method of analytical mechanics
   • 8.3 Mittag-Leffler method
   • 8.4 Method of natural functions
   • 8.5 Boundary value method
   • 8.6 Method of moral philosophy

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The following paper was published in the American Mathematical
Monthly 45 (1938) pp. 446-447.

A contribution to the mathematical theory of big game hunting

H. Pétard Princeton, New Jersey

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 in 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
happy unifying effect of 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 indebtness to the Trivial Club,
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.

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1. Mathematical Methods

1.1 The Hilbert, or axiomatic, method

We place a locked cage at a given point of the desert.
We then introduce the following logical system.

Axiom 1: The class of lions in the Sahara Desert is non-void.
Axiom 2: If there is a lion in the Sahara Desert, there is a
lion in the cage.
Rule of procedure: If p is a theorem, and ‘p implies q’ is a theorem,
then q is a theorem.
Theorem 1: There exists a lion in the cage.

1.2 The method of inversive geometry

We place a spherical cage in the desert, enter it and lock it.
We perform an inversion with respect to the cage. The lion is then in the
interior of the cage, and we are outside.

1.3 The method of projective geometry

Without loss of generality, we may regard the Sahara Desert as a plane.
Project the plane into a line, and then project the line
into an interior point of the cage. The lion is projected into
the same point.

1.4 The Bolzano-Weierstrass method

Bisect the desert by a line running N-S. The lion is either in the E portion
or in the W portion; let us suppose him to be in the W portion. Bisect this
portion by a line running E-W. The lion is either in the N portion or in the
S portion; let us suppose 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
is ultimately surrounded by a fence of arbitrarily small diameter.

1.5 The ‘Mengentheoretisch’ [set-theoretical] 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.

1.6 The Peano method

Construct, by standard methods, a continuous curve passing through every point of the
desert. It has been remarked¹ 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 can move his
own length.

1.7 A topological method

We observe that the lion has at least the connectivity of the
torus. We transport the desert into four-space. It is then
possible² to carry out such a deformation that
the lion can be returned to three-space in a knotted condition.
He is then helpless.

1.8 The Cauchy, or function-theoretical, method

We consider an analytic lion-valued function f(z).
Let zeta be the cage. Consider the integral

where C is the boundary of the desert; its value is f(zeta),
i.e., a lion in the cage.³

1.9 The Wiener Tauberian method

We procure a tame lion, L0 of class
L(-00, 00),
whose Fourier transform nowhere vanishes, and release it
in the desert. L0 then converges to our cage.
By Wiener’s General Tauberian theorem⁴,
any other lion, L (say), will then converge
to the same cage. Alternatively, we can approximate arbitrarily
closely to L by translating L0 about the desert⁵.

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2. Methods from Theoretical Physics

2.1 The Dirac method

We observe that wild lions are, ipso facto, not observable in the
Sahara Desert. Consequently, if there are any lions in the
Sahara, they are tame. The capture of a tame lion may be left as an exercise for
the reader.

2.2 The Schrödinger method

At any given moment there is a positive probability that there is a lion
in the cage. Sit down and wait.

2.3 The method of nuclear physics

Place a tame lion in the cage, and apply a Majorana exchange
operator⁶ between 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 in the
cage, and apply a Heisenberg exchange operator⁷, which exchanges the spins.

2.4 A relativistic method

We distribute about the desert 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 round the lion, who will then become so dizzy that he can be
approached with impunity.

2.5 The Newton method

Neglect friction and the lion and the cage will attract each other.

2.6 The Heisenberg method

You will disturb the lion if you observe it before its capture, so keep
your eyes closed.

2.7 The Einstein method

Run in the direction opposite to that of the lion. The relative velocity
makes the lion run faster, and hence it feels heavier.

2.8 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.

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3. Methods from Experimental Physics

3.1 The thermodynamical method

We construct a semi-permeable membrane, permeable to everything except lions,
and sweep it across the desert.

3.2 The atom-splitting method

We irradiate the desert with slow neutrons. The lion becomes
radioactive, and a process of disintegration sets in. When the decay has
proceeded sufficiently far, he will become incapable of showing fight.

3.3 The magneto-optical method

We plant a large lens-shaped bed of catnip (Nepata cataria)
whose axis lies along the direction of the horizontal
component of the earth’s magnetic field, and place a cage at one of
the field’s 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 the herbivorous
denizens of the desert, which are in turn eaten by lions. The lions are then
oriented parallel to the earth’s magnetic field, and the resulting beam of
lions is focused by the catnip upon the cage.

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4. Contributions from Computer Science

4.1 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.

4.2 The parallel search method

By using parallelism, we will be able to search in the direction to the north
much faster than earlier.

4.3 The Monte Carlo method

We pick a random number indexing the space we search. By excluding
neighbouring 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.

4.4 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.

4.5 The common language approach

If only everyone used ADA/Common Lisp/Prolog, this problem would be
trivial to solve.

4.6 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
harmonised standard to appear. Funding worth 10,000,000 pounds has been
provided for initial investigations into this standard’s development.

4.7 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 in 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.

4.8 The Dijkstra approach

The way the problem reached us 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.)
(End of approach.)

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The following paper was published in the
American Mathematical Monthly 72 (1965) p. 436.

5. A New Method of Catching a Lion

I. J. Good

In this note a definitive procedure will be provided for catching a lion
in a desert.

Let Q be the operator that encloses a word in quotation marks. Its square
Q^2 encloses a word in double quotes. The operator clearly satisfies the
law of indices, Q^m*Q^n=Q^(m+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.

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The following paper was published in the
American Mathematical Monthly 74 (1967) pp. 838-839.

6. On a Theorem of H. Pétard

Christian Roselius Tulane University

In a classical paper, H. Pétard proved that it is possible to capture a lion in the
Sahara desert. He further showed³
that it is in fact possible to catch every
lion with at most one exception. Using completely new techniques, not available to Pétard 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 l be the category of caged lions. The subcategory l is clearly complete, is
nonempty (by inspection), and has both generator and cogenerator⁸. Let
F:l->L be the forgetful functor, which forgets the cage. By the Adjoint
Functor Theorem⁹, 80-91,
the functor F has a coadjoint C:L->l,
which reflects each lion into a cage.

We remark that this method is obviously superior to the Good method, which only guarantees
the capture of one lion, and which requires an application of the Weierkäfig Preparation Theorem.

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The following paper was published in the American Mathematical
Monthly 75 (1968) pp. 185-187.

7. Some Modern Mathematical Methods in the Theory of Lion
Hunting¹⁰

Otto Morphy, D.Hp. (Dr. of Hypocrisy)

It is now 30 years since the appearance of H. Pétard’s classic
treatise 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 Pétard’s famous letter to the president in 1941
that led to the establishment of the Martini Project, the legendary crash programme 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 we shall
restrict our attention to the case of lions residing in the Sahara
Desert¹¹. As noted by
Pétard, 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.

7.1 Surgical method

A lion may be regarded as an orientable three-manifold
with a nonempty boundary. It is
known¹² that by means of a
sequence of surgical operations (known as ‘spherical modifications’
in medical parlance) the lion can be rendered contractible. He may then be signed to a contract
with Barnum and Bailey.

7.2 Logical method

A lion is a continuum. According to Cohen’s theorem¹³ 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.

7.3 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 kittygory⁹ 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.

7.4 Method of differential topology

The lion is a
three-manifold embedded in Euclidean 3-space. This implies
that he is a handlebody¹⁴.
However, a lion which can
be handled is tame and will enter the cage upon request.

7.5 Sheaf theoretic method

The lion is a cross-section¹⁵ of the sheaf of germs of lions¹⁶
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.

7.6 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)¹⁷.
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 fight¹⁸.

7.7 Postlikov method

A male lion is quite hairy¹⁹ 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 Postlikov decomposition²⁰
of the lion. This being done, the lion, being decomposed, is dead and in bad need of burial.

7.8 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 operations²¹. Every element must be killed
by some of these operations. Thus the lion will die on the operating table.

7.9 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 commutist conspiracy²²
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)
space²⁰. We hire a stunt pilot to
loop the loops, thereby hopelessly entangling the lion and rendering him helpless.

7.10 Covering space method

Cover the lion by his simply
connected covering space. In effect
this decks the lion²³. Grab him while
he is down.

7.11 Game theoretic method

A lion is big game. Thus,
a fortiori, he is a game. Therefore
there exists an optimal strategy²⁴.
Follow it.

7.12 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 situation capable of solution according to the work of Thompson and
Feit²⁵.

We conclude with one significant nonmathematical method:

7.13 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 lion²⁶. The lion,
now thinking that he is a planarian, will proceed to subdivide. This process, while natural
for the planarian, is disastrous to the lion²⁷.

Ed. note: Prof. Morphy is the namesake of his renowned aunt, the author of the famous
series of epigrams now popularly known as Auntie Otto Morphisms or euphemistically as
epimorphisms.

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The following paper was published in the
American Mathematical Monthly 75 (1968) pp. 896-897.

8. Further Techniques in the Theory of Big Game Hunting

Patricia L. Dudley, G. T. Evans, K. D. Hansen and I. D. Richardson
Carleton University, Ottawa

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.

8.1 Moore-Smith method

Letting A = Sahara Desert, one can construct a
net in A converging to any point in the closure of A. Now lions are
unable to resist tuna fish, on account of the charge 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.

8.2 Method of analytical mechanics

Since the lion has nonzero
mass it has moments of inertia. Grab it during one of them.

8.3 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.

8.4 Method of natural 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.

8.5 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.

8.6 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.

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Footprints:

1. By Hilbert. See 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. N.B. By Picard’s Theorem (W. F. Osgood, ‘Lehrbuch der Funktionentheorie’ (1928) vol. 1, p. 178), we can catch every lion with at most one exception.
4. N. Wiener, ‘The Fourier Integral and Certain of its Applications’ (1933) pp. 73-74.
5. Ibid., p. 89.
6. See, for example, H. A. Bethe and R. F. Bacher, ‘Reviews of Modern Physics’ 8 (1936) pp. 82-229; especially pp. 106-107.
7. Ibid.
8. Moses, The Book of Genesis, vii, 15-16.
9. P. Freyd, ‘Abelian Categories’, Harper and Row, New York, 1964.
10. This report was supported by grant #007 from Project Leo of the War on Puberty.
11. This restriction of the habitat does not affect the generality of the results because of Brouwer’s theorem on the invariance of domain.
12. Kervaire and Milnor, ‘Groups of homotopy spheres’, I, Ann. of Math. (1963).
13. P. J. Cohen, ‘The independence of the continuum hypothesis’, Proc. N. A. S. (63-64).
14. S. Smale, ‘A survey of some recent developments in differential topology’, Bull. A. M. S. (1963).
15. It has been experimentally verified that lions are cross.
16. G. Brédon, ‘Sheaf Theory’, McGraw-Hill, New York, 1967.
17. Montgomery and Zippin, ‘Topological Transformation Groups’, Interscience, 1955.
18. E. Borden, ‘Characteristic classes of bovine spaces’, Peripherblatt für Math. (1966BC).
19. Eddy Courant, ‘Sinking of the Mane’, Pantz Press, 1898.
20. E. Spanier, ‘Algebraic Topology’, McGraw-Hill, New York, 1966.
21. Steenrod and Epstein, ‘Cohomology Operations’, Princeton, 1962.
22. Logistics of the Attorney-General’s list, Band Corp. (1776).
23. Admiral, T. V., (USN Ret.), ‘How to deck a swab’, ONR tech. rep. (classified).
24. von Neumann and Morgenstern, ‘Theory of Games…’, Princeton, 1947.
25. Feit and Thompson, ‘Solvability of groups of odd order’, Pac. J. M. (1963).
26. J. V. McConnell, ed., ‘The Worm Re-turns’, Prentice-Hall, Englewood Cliffs, N. J., 1965.
27. This method must be carried out with extreme caution for, if the lion is large enough to approach critical mass, this fissioning of the lion may produce a violent reaction.