g-rh@cca.UUCP (Richard Harter) (03/03/86)
[------- The statement in this line is provably false ------] Here is the solution to the student placement puzzle. I was a little disappointed that no one posted a solution. This is, in my opinion, the most elegant logic puzzle ever created. Here is the puzzle restated: ] My old friend, Professor Flootersnoot at Arkham University, has three ] prize students in his Creative Ontology class, Jones, Smith, and Wilson. ] Recently he sent me a small puzzle about how they placed in a recent ] exam. The puzzle consisted of three statements: ] ] (1) No other employee of the Adelphi bookstore placed ahead of ] Wilson. ] ] (2) If I loathe pepperoni pizza, then the person who placed first ] has red hair. ] ] (3) If either Jones or Smith is second, then Smith finished ahead ] of the youngest of the three. ] ] Being the sly old dog that he is, Flootersnoot ommitted the relevant ] data. He was, however, kind enough to reassure me that the problem ] was well posed. That is, given the relevant data, all three of the ] above statements are necessary and sufficient for determining the ] order in which the three students placed. Knowing this, I was able ] to reconstruct the missing data and the order in which they placed. ] ] In what order did the three students place? ] Who works at the Adelphi bookstore? ] Who has red hair? ] Who is the youngest of the three? ] Do I loathe pepperoni pizza? (a) First consider statement two. Since Professor Flootersnoot posed the puzzle, the "I" of statement two is Flootersnoot. If he does not loathe pepperoni pizza then statement two contributes no information and therefore is not needed. Therefore: (1) Flootersnoot loathes pepperoni pizza. (2) The person who placed first has red hair. Since the "I" in the question "Do I loathe peppperoni pizza?" refers to me (Richard Harter) the answer to the question is "insufficient information". As a matter of record, I adore pepperoni pizza and loathe anchovy pizza -- Flootersnoot is perverted. (b) Now consider statement one. The wording of the statement implies that Wilson and at least one other student works at the Adelphi book store. If both worked there then statements one and three together would be sufficient to determine the order of placement and statement two would not be needed. Hence: (3) Either Jones or Smith, but not both, works at the Adelphi bookstore. This implies that Wilson did not finish last. (c) Now consider statement three. If Smith is the youngest of the three then statement three can be true only is Wilson is second. Since the other worker at the Adelphi finishes after Wilson statements one and three (together with the data) would uniquely specify the order of placement. Hence: (4) Smith is not the youngest of the three. (d) Now consider statement two. If all three have red hair then the statement contributes no information. Hence at least one of the three does not have red hair. If only Wilson has red hair then statement one is not needed. If Wilson does not have red hair then Wilson is not first and, hence, from statement one and (3) must be second. It follows that statements one and two alone would be sufficient to determine the order. Hence: (5) Wilson and one other person has red hair. (e) Now consider statements one and two together with what we know so far. Wilson and one other person, call him X, have red hair. There is a person, call him Y, who does not work at the Adelphi bookstore. From one, either Wilson or Y is first. From two, either Wilson or X is first. If X and Y are the same person then statement two is not needed. Hence X and Y are different people. It follows that neither X nor Y can be first, i.e. Wilson finished first. From statement three we conclude that: (6) Wilson finished first, Smith finished second, and Jones third. (7) Jones is the youngest of the three. (f) Finally, who has red hair? A careful check of the possible cases shows that the given information in the puzzle is insufficient to determine uniquely who has red hair. However we can state that: (8) Either Jones but not Smith has red hair and works at the Adelphi bookstore or Smith but not Jones has red hair and works at the Adelphi bookstore. [Note: This problem was given in "Problems Omnibus" by Hubert Philips (Caliban), an English puzzle writer, ARCO publications, London, 1960. The current presentation is a rewording of "Caliban's Will" by H. M. Newman. The solution given there has an error which is equivalent to asserting that Jones has red hair.] Now that's what I call a real puzzle. Richard Harter, SMDS Inc.
js2j@mhuxt.UUCP (sonntag) (03/06/86)
> ] My old friend, Professor Flootersnoot at Arkham University, has three > ] prize students in his Creative Ontology class, Jones, Smith, and Wilson. > ] Recently he sent me a small puzzle about how they placed in a recent > ] exam. The puzzle consisted of three statements: > ] > ] (1) No other employee of the Adelphi bookstore placed ahead of > ] Wilson. Note that this statement does *not* imply that more than one of Flootersnoot's prize students work for the Adelphi bookstore, as asserted by whoever posted the 'solution' to this problem. -- Jeff Sonntag ihnp4!mhuxt!js2j
g-rh@cca.UUCP (Richard Harter) (03/08/86)
In article <> js2j@mhuxt.UUCP (sonntag) writes: >> ] My old friend, Professor Flootersnoot at Arkham University, has three >> ] prize students in his Creative Ontology class, Jones, Smith, and Wilson. >> ] Recently he sent me a small puzzle about how they placed in a recent >> ] exam. The puzzle consisted of three statements: >> ] >> ] (1) No other employee of the Adelphi bookstore placed ahead of >> ] Wilson. > > Note that this statement does *not* imply that more than one of >Flootersnoot's prize students work for the Adelphi bookstore, as asserted >by whoever posted the 'solution' to this problem. Nice try, but you miss the point. The full statement of the puzzle states that all three conditions are necessary for the solution of the puzzle. If Wilson is the only one working at the Adelphi bookstore, then this statement does not contribute any information to the solution, i.e. the statement is irrelevant. The elegance of the puzzle resides precisely in the fact that the requirement that the unknown data be such that all of the statements are relevant and necessary makes it possible to deduce enough restrictions on the data to make it possible to solve the puzzle. [Now there is a convoluted sentence.] Richard Harter, SMDS Inc.
holmes@dalcs.UUCP (Ray Holmes) (03/08/86)
> > ] My old friend, Professor Flootersnoot at Arkham University, has three > > ] prize students in his Creative Ontology class, Jones, Smith, and Wilson. > > ] Recently he sent me a small puzzle about how they placed in a recent > > ] exam. The puzzle consisted of three statements: > > ] > > ] (1) No other employee of the Adelphi bookstore placed ahead of > > ] Wilson. > > Note that this statement does *not* imply that more than one of > Flootersnoot's prize students work for the Adelphi bookstore, as asserted > by whoever posted the 'solution' to this problem. > -- > Jeff Sonntag > ihnp4!mhuxt!js2j Not true see below. However the problem is flawed in another way: Flootersnoot lied when he said that all three statements were necessairy to determine the ordering of the three students. Statement 2 is NOT necessairy. Consider that we have only statements 1 and 3 and that they are necessairy. Consider statement 1: If Wilson is the only one of the three that works at Adelphi then this statement contains no information at all about the relative ordering of the three students and is not necessairy (also there is not a unique solution). Thus Wilson plus one or both of the others works at Adelphi. CASE 1: All three work at Adelphi. From #1, Wilson must be first, thus Jones and Smith must be second and third. #3 then inplies that Smith finished ahead of the youngest of the three. Thus the only possible ordering of the three is: Wilson 1st, Smith 2nd, and Jones 3rd. CASE 2: Wilson and one other work at Adelphi's. Wilson must then be either first or second. If Wilson is second then #3 gives no additional information and is not necessairy (if fact both orderings are possible). Thus Wilson must be first. Now the reasoning of CASE 1 gives us the same ordering. Hence, in any case, we get the correct ordering without ever considering statement #2. Ray
g-rh@cca.UUCP (Richard Harter) (03/09/86)
In article <> holmes@dalcs.UUCP (Ray Holmes) writes: > > ....... [Sonntag's comments deleted] ..... > > Not true see below. However the problem is flawed in another way: >Flootersnoot lied when he said that all three statements were necessary to >determine the ordering of the three students. Statement 2 is NOT necessary. >Consider that we have only statements 1 and 3 and that they are necessary. > >Consider statement 1: > If Wilson is the only one of the three that works at Adelphi then >this statement contains no information at all about the relative ordering >of the three students and is not necessary (also there is not a unique >solution). Thus Wilson plus one or both of the others works at Adelphi. > >CASE 1: All three work at Adelphi. > From #1, Wilson must be first, thus Jones and Smith must be second > and third. #3 then inplies that Smith finished ahead of the youngest > of the three. Thus the only possible ordering of the three is: > > Wilson 1st, Smith 2nd, and Jones 3rd. > >CASE 2: Wilson and one other work at Adelphi's. > Wilson must then be either first or second. If Wilson is second then > #3 gives no additional information and is not necessary (if fact > both orderings are possible). Thus Wilson must be first. Now the > reasoning of CASE 1 gives us the same ordering. > >Hence, in any case, we get the correct ordering without ever considering >statement #2. > Sorry, this is another logical trap, albeit a good deal more subtle. You are correct that case 1 is ruled out. Now the possible data cases are: A: Wilson and Smith work at the Adelphi book store. Wilson and Smith have red hair. Jones is the youngest of the three. B: Wilson and Jones work at the Adelphi book store. Wilson and Jones have red hair. Jones is the youngest of the three. Cases A and B are isomorphic, so we will consider case A only. The following table gives the only orderings consistent with each statement, considered singly: (1) Wilson, Smith, Jones Wilson, Jones, Smith Jones, Wilson, Smith (2) Wilson, Smith, Jones Wilson, Jones, Smith Smith, Wilson, Jones Smith, Jones, Wilson (3) Wilson, Smith, Jones Smith, Wilson, Jones Jones, Wilson, Smith Now consider, any two statements, taken as pairs. The following table gives the orderings consistent with each pair of statements: (1,2) Wilson, Smith, Jones Wilson, Jones, Smith (1,3) Wilson, Smith, Jones Jones, Wilson, Smith (2,3) Wilson, Smith, Jones Smith, Wilson, Jones That is, no pair of statements is sufficient to determine the order; all three statements are necessary. Where, then, is the fallacy in your argument? You say: "Wilson must then be either first or second. If Wilson is second then #3 gives no additional information and is not necessary (if fact both orderings are possible). Thus Wilson must be first. Now the reasoning of CASE 1 gives us the same ordering." Now it is true that if Wilson were known to be second then #3 would be unnecessary. However it is not known (from #1) whether Wilson is first or second. Given statement 1, #3 contributes the information that the order (Wilson, Jones, Smith) is not admissable. Your error lies in insisting that statement 3 must give information about each possible placement of Wilson. However all that is required of 3 is that it eliminate some of the possible orders implied by #1. What you are saying is equivalent to this: Premise 1: If I had the datum, 'Wilson is second', then statement three is not needed. Premise 2: Statement three is needed. Conclusion: Wilson is not second. However the correct conclusion is that you do not have the datum, 'Wilson is second', which is, in fact, the case. I hope this clarifies the matter. Tricky little bugger, ain't it. Richard Harter, SMDS Inc.
lewish@acf2.UUCP (Henry M. Lewis) (03/11/86)
> Flootersnoot lied when he said that all three statements were necessairy to > determine the ordering of the three students. Statement 2 is NOT necessairy. Statement 2 is necessary; it disqualifies the case wherein Wilson and Smith tied for first, and Jones came in second. --Hank Lewis ihnp4!cmcl2!acf2!lewish
holmes@dalcs.UUCP (Ray Holmes) (03/14/86)
In article <1350001@acf2.UUCP> lewish@acf2.UUCP (Henry M. Lewis) writes: >> Flootersnoot lied when he said that all three statements were necessairy to >> determine the ordering of the three students. Statement 2 is NOT necessairy. > >Statement 2 is necessary; it disqualifies the case wherein Wilson and Smith >tied for first, and Jones came in second. If Wilson and Smith tied for first then Jones could do no better than third! Ray