If Paton and Madison have a specialty in common, then which one of the following must be true?

Irina on October 9, 2019

@ali,

This is a fairly straightforward linear game that requires us to determine the order of hire for 7 professors - M N O P R S T hired in the years 1989 through 1995. The game leaves open the possibility that more than one professor could be hired in the same year and no professors could be hired in some of the years.

___ ___ ___ ___ ___ ___ ___
89 90 91 92 93 94 95

Each professor has one or more specialties and any two professors hired in the same years or in consecutive years do not have a specialty in common.

The following rules apply:
(1) M was hired in 93, R in 91

___ ___ R ___ M ___ ___
89 90 91 92 93 94 95

(2) There is at least one specialty that M, O, and T have in common.

This rule tells that neither O nor T could be hired in the same or consecutive years as M. Thus, we can conclude that neither O nor T could be hired in 92, 93, or 94.

___ ___ R ___ M ___ ___
89 90 91 92 93 94 95
~O ~ O ~O
~T ~T ~T

(3) N shares a specialty with R.

This rule tells us that N cannot be hired in years 90, 91 or 92.


___ ___ R ___ M ___ ___
89 90 91 92 93 94 95
~N ~N ~O ~ O ~O
~T ~T ~T
~N

(4) P and S were each hired at least one year before M and at least one year after N.

This rule tells us that neither P nor S could be hired in 93, 94, 95 since M is hired in 93, and not in 89 since they must be hired at least one year after N. It also tells us that N could not be hired in 93, 94, or 95 since P must be hired after N and P cannot be hired after M.


___ ___ R ___ M ___ ___
89 90 91 92 93 94 95
~P ~N ~N ~O ~ O ~O
~S ~T ~T ~T
~N ~P ~P ~P
~S ~S ~S
~N ~N ~N

We can see from this diagram that the only year N could be hired is 89.


(5) O, who shares specialty with S, was hired in 1990.

This rule tells us that S cannot be hired in 90, 89, or 91.
If we look back at rule (2), we can also infer that T cannot be hired in 89, 90 or 91 because he shares at least one specialty with O.


N O R ___ M ___ ___
89 90 91 92 93 94 95
~T ~T ~T ~T ~T ~T
~P ~P ~P ~P
~S ~S ~S ~S ~S ~S

Just by looking at this diagram, we can tell that the only year T could be hired is 1995, and the only year S could be hired in 1992.


N O R S M ___ T
89 90 91 92 93 94 95
~P ~P ~P ~P

P is the only professor left unassigned and he could be hired in either 90, 91 or 92.

The question gives us an additional fact - if P & M have a specialty in common, which of the following must be true?

Well, we can tell that if P & M have a specialty in common, P can no longer be hired in 92, and must be hired either in 90 or 91.

Let's look at the answer choices:

(A) N does not share a specialty with P.

This does not have to be true, because P could be hired in 91 and N is 89, thus they are not hired in consecutive years and could share a specialty.

(B) Exactly one professor was hired in 90.

This does not have to be true, P could be hired in 90 or 91.

(C) Exactly one professor was hired in 91.

This does not have to be true, P could be hired in 90 or 91.

(D) Exactly two professors were hired in each of two years.

This does not have to be true. P and either O or R would be the only pair of professors hired in the same year.

(E) P was hired at least one year before S.

Correct. P must be hired in 90 or 91 under this scenario, and S is hired in 1992, so it must be true that P is hired at least one year before S.

Let me know if this makes sense and if you have any further questions.