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 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 identity permutation, 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 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 essentially just 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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