### 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.

- 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

- 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

- Methods from Experimental Physics
- 3.1 The thermodynamical method
- 3.2 The atom-splitting method
- 3.3 The magneto-optical method

- 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

- A New Method of Catching a Lion
- On a Theorem of H. Pétard
- 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

- 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

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.

### 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 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 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

possible^{2} 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 theorem^{4},

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 desert^{5}.

### 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

operator^{6} 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 operator^{7}, 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.

### 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.

### 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.)

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.

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 showed^{3}

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

*be the category of caged lions. The subcategory*

**l***is clearly complete, is*

**l**nonempty (by inspection), and has both generator and cogenerator

^{8}. Let

*F*:

*->*

**l***be the forgetful functor, which forgets the cage. By the Adjoint*

**L**Functor Theorem

^{9}, 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.

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

Hunting^{10}

## 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

Desert^{11}. 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

known^{12} 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 theorem^{13} 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 kittygory^{9} 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 handlebody^{14}.

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-section^{15} of the sheaf of germs of lions^{16}

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 fight^{18}.

**7.7 Postlikov method**

A male lion is quite hairy^{19} 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 decomposition^{20}

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 operations^{21}. 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 conspiracy^{22}

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^{20}. 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 lion^{23}. 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 strategy^{24}.

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

Feit^{25}.

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 lion^{26}. The lion,

now thinking that he is a planarian, will proceed to subdivide. This process, while natural

for the planarian, is disastrous to the lion^{27}.

*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.

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.

**Footprints:**

- 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.
- H. Seifert and W. Threlfall, ‘Lehrbuch der Topologie’ (1934) pp. 2-3.
*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.- N. Wiener, ‘The Fourier Integral and Certain of its Applications’ (1933) pp. 73-74.
*Ibid.*, p. 89.- See, for example, H. A. Bethe and R. F. Bacher, ‘Reviews of Modern Physics’
**8**(1936) pp. 82-229; especially pp. 106-107. *Ibid.*- Moses, The Book of Genesis, vii, 15-16.
- P. Freyd, ‘Abelian Categories’, Harper and Row, New York, 1964.
- This report was supported by grant #007 from Project Leo of the War on Puberty.
- This restriction of the habitat does not affect the generality of the results because of Brouwer’s theorem on the invariance of domain.
- Kervaire and Milnor, ‘Groups of homotopy spheres’, I, Ann. of Math. (1963).
- P. J. Cohen, ‘The independence of the continuum hypothesis’, Proc. N. A. S. (63-64).
- S. Smale, ‘A survey of some recent developments in differential topology’, Bull. A. M. S. (1963).
- It has been experimentally verified that lions are cross.
- G. Brédon, ‘Sheaf Theory’, McGraw-Hill, New York, 1967.
- Montgomery and Zippin, ‘Topological Transformation Groups’, Interscience, 1955.
- E. Borden, ‘Characteristic classes of bovine spaces’, Peripherblatt für Math. (1966BC).
- Eddy Courant, ‘Sinking of the Mane’, Pantz Press, 1898.
- E. Spanier, ‘Algebraic Topology’, McGraw-Hill, New York, 1966.
- Steenrod and Epstein, ‘Cohomology Operations’, Princeton, 1962.
- Logistics of the Attorney-General’s list, Band Corp. (1776).
- Admiral, T. V., (USN Ret.), ‘How to deck a swab’, ONR tech. rep. (classified).
- von Neumann and Morgenstern, ‘Theory of Games…’, Princeton, 1947.
- Feit and Thompson, ‘Solvability of groups of odd order’, Pac. J. M. (1963).
- J. V. McConnell, ed., ‘The Worm Re-turns’, Prentice-Hall, Englewood Cliffs, N. J., 1965.
- 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.