WHO KILLED THE COOK?

WHO KILLED THE COOK?Author(s): JAMES J. CORBET and J. SUSAN MILTONSource: The Mathematics Teacher, Vol. 71, No. 4 (APRIL 1978), pp. 263-266Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961243 .

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WHO KILLED THE COOK?

An example of the use of probability that should capture the attention of students.

By JAMES J. CORBET andJ. SUSAN MILTON

Rad ford College Radford, VA 24142

In this article the idea of "complete per mutations" of distinct objects is explored in order to investigate an interesting appli cation of the concept that arises in Rex Stout's mystery novel, Too Many Cooks. A

complete permutation is simply a per mutation that leaves no object fixed. A re

cursive formula for finding the number of such permutations for fixed is developed using the discovery method.

Consider the following problem: If dis tinct objects are available, in how many ways can they be permuted so that no ob

ject remains in its original position? Let us

denote this number by Wn, > 1 and con

sider the following possibilities:

It is obvious that Wx = 0.

2. = 2

It is also evident that W2 =

1, since only a simple interchange of objects is allow able.

3. = 3

For this case, visualize three colored cubes: one red, one yellow, and one

blue; they are placed originally in the order mentioned in figure 1. In order to

Fig. 1

see how many complete permutations exist, consider the tree diagram, which lists all permutations (fig. 2). It can eas

ily be seen that only the starred branches

yield complete permutations, and hence

W3 = 2.

Original Color

Yellow Blue

Yellow-Blue

Blue

Red

Blue

Red

-Yellow

-Blue

-Red *

-Yellow *

Yellow -Red

Fig. 2

4. = 4

The previous procedure is a bit awk ward. Hence the following argument is

proposed. Let the original arrangement of cubes be yellow, red, blue, white. Two

mutually exclusive methods may be used to generate complete permutations. The

yellow cube is to be interchanged with another cube, or else it is not to be inter

changed with another. Consider each case separately (fig. 3).

W

Fig. 3

If the yellow cube is interchanged with any of the other three cubes, then the remaining cubes may be

completely permuted in the W2 = 1

way. Thus a simple interchange gene rates 3 ?

W2 = 3 ? 1 = 3 complete

permutations (fig. 4).

W

W

W

Fig. 4

April 1978 263



b. If the yellow cube is not interchanged with another cube, three choices are left for the first position. After the first position has been filled by either the red, blue, or white cube, then the

three remaining cubes may be com

pletely permuted in Wz = 2 ways. Thus 3 ?

Wz = 6 additional complete permutations are generated. Hence

W4 = 3 ? W2 + 3 ?

Wz = (4

- 1)^4-2

+ (4 -

1) W<_x = 9 (fig. 5).

W

w

w

w

w

Fig. 5

5. > 5 In a similar manner it can be argued that for > 5 there are {

- 1) ways to

interchange the first cube with another and then Wn_2 ways to completely per mute the remaining

? 2 objects; if no

interchange is used, the first object may be replaced by any one of - 1 objects, with the remaining

? 1 objects com

pletely permuted in Wn_x ways. Hence the following recursive formula for >

5 is obtained:

Wn = (n- X)Wn-2 + (n -

DWn^.

In summary, the following set of equa tions has been shown to hold:

Wx = 0

W2 = 1

Wz =2

Wn =(n- \)Wn_2 + (n- \)Wn^ for > 4.

The classic example used to illustrate

complete permutations is the familiar "hatcheck" problem found in Kemeny, Snell, and Thompson's Introduction to Fi

nite Mathematics (1956). This problem is

generally framed as follows: A hatcheck

girl has checked hats. She returns them to her customers at random. What is the

probability that no person receives the

correct hat? The solution to this problem is

obviously

number of ways to completely permute hats

total number of ways to permute hats

Wn

n\

The following example presents the clas sic "hatcheck" problem in a new and un

usuaj context. It has been presented suc

cessfully as a talk to mathematics clubs as a means of developing interest in probability at both the secondary and college levels.

The main character in Rex Stout's mys tery series is the 312-pound detective, Nerq

Wolfe. Mr. Wolfe, who charges exorbitant

fees, is a gourmet who abhors physical ex ercise. He seldom leaves his home to solve a

crime, leaving most of the legwork to his

assistant, Archie Goodwin. In the book Too Many Cooks (Stout 1963), Mr. Wolfe is persuaded to leave home in the company of Mr. Goodwin to be the guest speaker at a meeting of Les Quinze Ma?tres, a group of internationally known chefs and gour mets. Ten members of the group are pres ent at the meeting, among them Philip Las

zio, who has in the past aroused a great deal of animosity among his colleagues. The professional rivalry among members of the group is intense, and the atmosphere is

charged with tension. During the course of the evening Laszio proposes the following experiment:

On the dining table would be placed nine dishes of Sauce Printemps on warmers, each lacking one of the nine seasonings known to be included in the recipe. Also there would be placed a server of squabs and plates and other utensils. Each of the members of Les Quinze Ma?tres (except Laszio) and Nero Wolfe would slice pieces of squab and taste once from each dish. In front of each dish would be a numbered

card, from 1 to 9. Each taster would be

264 Mathematics Teacher



provided with a slip of paper on which the nine seasonings were listed, and after each

seasoning he would write the number of the dish in which that seasoning was missing. Laszio, who had prepared the sauce, would be in the room to preside. He would record the correct answers prior to the experiment. Those who had tasted were not to converse with those who had not tasted until all were finished.

One member of Les Quinze Ma?tres harbors such hatred of Laszio that he re fuses to take part in the experiment saying, "Under no circumstances will I eat any thing prepared by him." Thus a total of nine men agree to the experiment.

During the course of the experiment Las zio disappears and is later discovered by

Wolfe in the tasting room, murdered with one of the knives being used in the experi ment. The question, obviously, is, "Who killed the cook?" After much questioning of the guests, the local police come up with two prime suspects, Berin and Vukcic, both

participants in the experiment. However,

they can prove nothing and are forced to turn to Nero Wolfe for help. His first sug gestion is to check the answer slips of all

participants against the correct list found in Laszio's pocket. For as Mr. W?lfe so ably puts it (Stout 1963, p. 67),

We were permitted but one taste from each dish?

only one! Have you any conception of the delicacy and sensitivity required? It took the highest degree of

concentration and receptivity of stimuli. So, compare those lists. If you find that Berin and Vukcic were

substantially correct?say seven or eight out of nine?

they are eliminated. Even six. No man about to kill

another, or just having done so, could possibly con

trol his nervous system sufficiently to perform such a

feat.

The examination reveals that five men,

including Vukcic, got all the answers cor

rect; three, including Wolfe, missed exactly two; and Berin got only two correct. Berin is immediately arrested for murder.

Several questions in elementary proba bility arise here:

A. What is the probability that one could

get all answers correct or miss exactly two if he is in fact simply guessing? (If this number is small, then the inference is that the first

men mentioned can distinguish one spice from another and are not merely lucky.)

B. What is the probability that one could do as poorly as Berin if in fact, he is simply guessing? (If this number is large, perhaps Berin is not really a gourmet and simply guessed poorly.)

Question A is a relatively simple exercise in classical probability. If guessing is in

volved, then any one of the 9! possible ar

rangements of the dishes is equally likely. There is only one way to get all answers

correct. There are ^^ways

to get exactly two wrong, since the only way that this can occur is by means of a simple interchange of two spices. Thus

(all correct or miss exactly two/guessing) =

(all correct/guessing) + (miss exactly two-guessing)

= .000 101 96.

Conclusion: The first eight gourmets are able to distinguish the spices.

Question is a bit more complex and is, in fact, the old "hatcheck" problem re

visited. Note that if we let

(get at most two correct/guessing) =

p,

then

= (miss all/guessing)

+ (get exactly 1 correct/guessing) + (get exactly 2 correct/guessing).

p-^Hl/^ +

V2/'9T where W7, Ws, W9 are as previously de fined. Simplifying,

= 133 496 9(14 833) 36(1854) 9! 9! 9!

That is, by guessing one will almost assur

edly get fewer than three answers correct.

Perhaps Berin was guessing? Because of his

reputation as a gourmet, this conclusion is

April 1978 265



untenable, Wolfe's explanation of the poor

performance is accepted, and Berin appears doomed by circumstantial evidence. It is

left to Wolfe to get to the bottom of the

mystery.

REFERENCES

Kem?ny, John G., Laurie J. Snell, and Gerald L.

Thompson. Introduction to Finite Mathematics.

Englewood Cliffs, N.J.: Prentice-Hall, 1956.

Stout, Rex. Too Many Cooks. New York: Pyramid, 1963.

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WHO KILLED THE COOK?