puzzles on permutations

7/30/2019 puzzles on permutations

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Permutation Puzzles

Consider five women, Anne, Betsy, Corrine, Donna, and Emily, who live

(not necessarily respectively) in the cities Anaheim, Baltimore,

Cleveland, Denver, and Enumclaw. Each woman has exactly one son, and

their names are Andy, Bob, Chuck, Dave, and Edward (again, not

necessarily in that order). The ages of the women are 41, 42, 43,45, and 46 (again, not necessarily in that order). We're given the

following information:

1. Emily lives in Anaheim. She's one year older than Chuck's

mother who lives in Cleveland.

2. Edward's mother is Donna. She's one year older than Anne whose

son isn't Dave.

3. The woman in Denver (who isn't Betsy) is younger than the woman in

Enumclaw.

4. Andy's mother (not Corrine) is 43.

5. The woman living in Baltimore is 45.

How old is each woman, who is her son, and where does she live?Moreover, is there an efficient formal technique for solving problems

like this? Perhaps if we think about the thought process we would

use to solve it "by hand", and then think about how we might program

a computer to do the same thing, the process could be formalized.

The most useful information in the particular puzzle stated above

seems to be the ages and their relations to each other. There are

two distinct pairs of women whose ages differ by only one, and the

list of ages contains only three such possibilities. Furthermore,

we know Chuck's mother isn't 45 (from her location), so we must have

either [Emily=42,Chuck's=41] or else [Emily=43,Chuck's=42]. Both of

these involve the age 42, so we have Donna=46 and Anne=45, hence

Anne is in Baltimore. Also, we must put Donna in Enumclaw to make

its resident older than the woman in Denver, who must be Corrine.Then we see Andy must be Emily's son, so she is 43, and the rest

falls into place. The result is

Emily Anne Betsy Corrine Donna

43 45 42 41 46

Andy Bob Chuck Dave Edward

Anaheim Baltimore Cleveland Denver Enumclaw

There are probably many different and equivalent ways of expressing

and formalizing the constraints. One way is in terms of compositions

of permutations. We have four groups (names, ages, son's names, and

locations) of five elements, and we can arbitrarily assign the numbers

1 to 5 to the elements of each group as shown below

City Woman Son Age

1 Anaheim Anne Dave 41

2 Baltimore Donna Chuck 42

3 Cleveland Emily Bob 43

4 Denver Betsy Andy 45

5 Enumclaw Corrine Edward 46


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We seek permutations of these four sets that, when aligned, satisfy

certain conditions. The solution can be expressed by the following

set of four permutations

Cities 12345 I

Women 31452

Sons 43215

Ages 34215

These permutations match each city with the resident woman, her

son, and her age. Of course, there are really 120 distinct ways of

permuting the five columns, and they all represent the same logical

solution. I just arbitrarily chose to arrange the columns so that

the cities have the identity permutation 12345. Three other ways of

expressing the same solution are

Cities 25134 Cities 43215 Cities 43125

Women 12345 I Women 54132 Women 54312

Sons 35421 Sons 12345 I Sons 12435

Ages 45321 Ages 12435 Ages 12345 I

where "I" denotes the category that has the identity permutation.

Obviously if the permutation from Cities to Women (for example) is

31452, then the inverse is 25134. Notice that the composition of

any permutation and its inverse is the identity permutation. More

generally, the composition of any "loop" of permutations must yield

the identity, and this fact can be used to infer information about

some of the permutations given information about some of the others.

To illustrate, consider the three categories Cities, Ages, and Women,

and note that we have the permutations

Cities to Women: 31452

Women to Sons: 35421

Sons to Cities: 43215

The composition of these three permutations, in sequence, necessarily

gives the identity. To show how this establishes conditions on the

solution, recall that we were told explicitly that Emily is in Anaheim,

so the permutation from Women to Cities is of the form {**1**}. In

addition, as noted above, the numerical clues about the Ages exclude

all but two permutations from Cities to Ages, namely, {24135} and

{34215}. Taking the second of these, we have

Cities to Ages: 3****

Ages to Women: *****

Women to Cities: **1**

The composition of these three permutations must be the identitypermutation, which clearly requires that the permutation from Ages

to Women must be of the form {**3**}. In other words, we've deduced

that the 43 year old woman is Emily, which determines the rest of

the solution.

Of course, there's nothing profound going on here. The leading "3"

in the permutation from Cities to Ages signifies that the woman in

Anaheim is 43, and the middle "1" in the permutation from Women to

Cities signifies that Emily is in Anaheim. These two facts in


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combination obviously imply Emily is 43. This just illustrates how

information on one basis is related to information on other bases.

It's important to recognize that there are three distinct aspects of

puzzles of this kind. The first is just general computation. For

example, the particular puzzle above required making use of numerical

relations between a given set of numbers. More generally, conditions

could be imposed such as "The age of person A has exactly two prime

divisors in common with the age of person B". Before even beginning

to solve the puzzle, we must first resolve all the computational

"clues" of this kind, and express them as constrants on the allowable

permutations. These computational aspects could be arbitrarily

difficult - or even practically impossible - to resolve. Fro example,

it might be specified that person A lives in city D if and only if

the 100 trillionth decimal digit of pi is 7. This is a perfectly

deterministic and well-defined "clue", but it is practically useless.

Assuming we can solve the computational aspects of the puzzle, the

second task is to determine which permutations between two given sets

are allowed based on a set of clues. This has nothing to do with

transferring information from one basis to another. It is essentiallyjust the classical "satisfiability problem", i.e., given a set of

logical variables feeding into a fixed system of logic gates with a

single logical output variable, determine which (if any) input states

set the output TRUE. (It is this kind of reasoning that enables us

to say the permutation from Cities to Ages must be either {24135} or

{34215}, without even taking any other information into account.)

This is known to be NP-complete, so we can't expect there to exist

any simple recipe that would work in polynomial time in all cases.

The third task involved in solving such puzzles is to relate the

information that has been provided on different bases. It is only

this third task (the simplest of the three) that can be formalized

in terms of the compositions of permutations.

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puzzles on permutations