If Julio leads the Thursday afternoon session, then for how many of the other lab assistants can one determine which ...

Ava on March 22 at 07:39PM

Approach

Can someone explain how to solve this question?

1 Reply

Skylar on March 22 at 08:48PM

@shafieiava, Of course, happy to help!

Let's start with the game setup:

JKLNOR

AM __ __ __
PM __ __ __
W T F

Rule #1: We must either have K or R
R K
Rule #2: We cannot have L or O
O L
Rule #3: N = PM
Rule #4: J = at least one day before O. I did not write this as J>O because I wanted to be sure to remind myself that we are talking about days, not just time slots.

Before proceeding to the questions, we should look for deductions we can make. Rule #1 tells us that K and R must go on the same day, which is a significant restriction. We should try to start there. Let's set up a few hypotheticals:

HYPO 1:
Let's see what happens if we place K and R on Wednesday. Remember, we don't know which variable is assigned to which slot on Wednesday. This gives us:
AM (K/R) __ __
PM (R/K) __ __
W T F
Rule #4 tells us that J is on an earlier day than O, so we know that J will go somewhere on Thursday, while O will go somewhere on Friday. This gives us:
AM (K/R) (J/?) (O/?)
PM (R/K) (?/J) (?/O)
W T F
Rule #2 tells us that L and O cannot go on the same day, so L must be assigned to Thursday. This leaves N to fill the last spot on Friday. Rule #3 tells us that N must go in the afternoon spot, which means that O must go in the morning spot. This gives us:
AM (K/R) (J/L) O
PM (R/K) (L/J) N
W T F

HYPO 2:
We can place K and R on Thursday and go through the same steps above to get:
AM (J/L) (K/R) O
PM (L/J) (R/K) N
W T F

HYPO 3:
We can place K and R on Friday and go through the same steps above to get:
AM (J/L) O (K/R)
PM (L/J) N (R/K)
W T F

These three hypotheticals set up all of our game possibilities and will save us significant time and effort when answering the questions. We were tipped off to setting up the hypotheticals by the significant restrictions imposed by the rules. However, if you did not get the hypotheticals when you completed the game, don't worry. It is still very possible to complete the game on time without these scenarios.

Now to the question: "If Julio leads the Thursday afternoon session, then for how many of the other lab assistants can one determine which sessions they lead?"

If we were able to deduce the hypotheticals above, we could solve this by referencing HYPO 1. In this, we have O on Friday morning and N on Friday afternoon. Since we are told that J is on Thursday afternoon, we can also deduce that L is on Thursday morning. Therefore, we can determine the placement of 3 other assistants, and (C) is our correct answer. As our hypothetical shows, we could not exactly place K or R, as we don't know which is on Wednesday morning and which is on Wednesday afternoon.

If we did not make the hypotheticals, we can work through this question as follows:
AM __ __ __
PM __ J __
W T F
Rule #4 tells us that J meets on an earlier day than O does, so we know that O must be sometime on Friday. This gives us:
AM __ __ (O/?)
PM __ J (?/O)
W T F
Rule #1 tells us that K and R must be on the same day. Thursday and Friday only have one open slot, so they could not accommodate both K and R. Therefore, K and R must be assigned to Wednesday, which has both slots open. This gives us:
AM (K/R) __ (O/?)
PM (R/K) J (?/O)
W T F
Rule #2 says that L and O cannot be placed on the same day, so L cannot go on Friday. This only leaves Thursday as an option, which gives us:
AM (K/R) L (O/?)
PM (R/K) J (?/O)
W T F
N is our last variable to place. The only open spot is sometime on Friday, so we know that N will go on Friday. Rule #3 tells us that N has to go in the afternoon, so we know that it will go specifically on Friday afternoon. This means that O has to go on Friday morning. This gives us:
AM (P/R) L O
PM (R/P) J N
W T F
Besides J (which is assigned in the question), we assigned L, O, and N. We did not definitively place P and R. Therefore, we could determine placements for exactly three of the other lab assistants, and (C) "three" is the correct answer.

Does that make sense? Please let us know if you have any other questions!