If Hakeem works at the gallery on Wednesday, then Joe must work at the gallery on which one of the following pairs of...

AndreaK on April 10, 2020

Hi @JakeT

This is a very tricky game! For this one, we want to start with a setup.

You may be wondering where to begin. I like to start with rules that put a lot of restrictions on our possible combinations.

Rule 6 works in combination with rules 2 and 3. You can tell that’s the case because they involve the same players (G and L).

Let’s start with rule 2, because there are only so many places those two Ls can go. Below, the bottom tier of my diagram reflects the first shift, and the top tier reflects the second.

L L _ _ _
_ _ _ _ _
MTWTF

_ L L _ _
_ _ _ _ _
MTWTF

_ _ L L _
_ _ _ _ _
MTWTF

_ _ _ L L
_ _ _ _ _
MTWTF

As we see here, there are only four worlds that accommodate that second rule. Now, when we combine that rule with rule 3 and rule 6, we can expect to further restrict our game and possibly (probably!) even eliminate one or more of those worlds. Next, let’s apply rule 3 keeping in mind rule 6, that G and L cannot be on the same day.

L L _ _ _
_ _ G _G
MTWTF

_ L L _ _
G _ _ _ _


_ _ L L _
_ _ _ _ G

_ _ _ L L
G _G _ _

An easy one to add from here is rule 4, K works on Tuesday and Friday. We won’t know where K goes every time, but because Tuesday and Friday are half filled in some of our worlds, we know K must take the remaining shift on those days.

L L _ _ K
_ KG _ G
MTWTF

_ L L _ _
G K _ _ _


_ _ L L K
_ _ _ _ G

_ _ _ L L
G _G _ K

Now let’s take a look at rule 5. This rule is particularly restrictive because it creates a vertical block. That means that on at least one day, we need to have both shifts left open—one for H and one for J. In some of these worlds, there is only one day left available with both shifts open. So in those worlds, we know H and J must go there. (Note: In my diagram, I am going to put J on the first shift and H on the second shift. However, either of those players could go in either shift within that day. I am simply putting one on top and one on the bottom so that the game board here stays clean. In real life on a piece of paper, I might use a double arrow or a slash (or maybe even draw a vertical box) to remind me that they are interchangeable within that day.

L L_ HK
_ KGJG
MTWTF

_L L_ _
GK _ _ _


_ _ L L K
_ _ _ _ G

_ H _ L L
G J G _ K

Notice something funny about the last world here? The only day left with both shifts open for the H/J block is Tuesday. However, rule 4 says K must go sometime on Tuesday. That means this world is impossible, and we can toss it out of the game. We’re left with the below:

Game world 1:
L L _ HK
_ K GJG
MTWTF

Game world 2:
_ L L_ _
GK _ _ _

Game world 3:
_ _ L LK
_ _ _ _ G

With these worlds, keep in mind that there must be at least one day with both shifts open in worlds 2 and 3 that accommodates the H/J vertical block (rule 5). We also know that K must go once on Tuesday and once on Friday (rule 4). Because rule 5 requires us to have a whole day just for H and J, we know that the H/J block then cannot be on Tuesday or Friday (because K has to take up one of those shifts on each of those days.) So now, game worlds 2 and 3 look like this:

Game world 2:
_ L L H _
G K _ J _

Game world 3:
H_ L L K
J _ _ _ G

Now let’s think about rule 3 and rule 6 again. G cannot go on the same day as L, and on two NONconsecutive days, G works the FIRST shift.

That further restricts worlds 2 and 3.

In world 2, we know then that G must go in the FIRST shift on Friday, because the only other first shift spot open is on Wednesday, but G cannot go with L on Wednesday. So, if G is now on the first shift of Friday, and K also has to go on Friday (rule 4), then we know that K is on the second shift Friday. Our last two players left in world 2 are H and J, who can alternate between Monday and Wednesday.

Game world 2:
_ L L H K
G K _ JG

In world 3, we see something very similar happen. We know that G must go in the FIRST shift on Tuesday, because G cannot go with L on Wednesday or Thursday. K also has to go on Tuesday (rule 4). Again, our last two players in world 3 are H and J, who can alternate between the first shifts on Wednesday and Thursday.

Game world 3:
H K L LK
J G _ _ G

So, that’s our setup! Phew! That was a long one.

Our final scenarios:

Game world 1:
L L_ HK
_ KGJG
MTWTF

Game world 2:
_ L LHK
GK _ JG

Game world 3:
H K L LK
J G _ _ G

Now, for the question...

The stem of this one asks us: If H works on Wednesday, then J must work at the gallery on which one of the following pairs of days?

Let’s tackle that by plugging H into Wednesday in our three worlds, and seeing what happens.

Game world 1:
L L H H K
J K G J G
MTWTF

In world one, H takes the last open Wednesday shift (shift 2). That leaves only the first shift on Monday open, which in tern must be filled by our last remaining player, J. Note that in this world, the other J must be on Thursday.

Game world 2:
J L L H K
G K H J G

In world 2, H takes the last open Wednesday shift (shift 1). That leaves only the second shift on Monday open, which in tern must be filled by our last remaining player, J. Note that in this world, the other J must be on Thursday.

Game world 3:
H K L L K
J G H J G

In world 3, H takes the last open Wednesday shift (shift 1). That leaves only the first shift on Thursday open, which in tern must be filled by our last remaining player, J. Note that in this world, the other J must be on Monday.

That’s how we get to answer choice B, Monday and Thursday. If H works on Wednesday, J must work on Monday and Thursday.

This is a really hard game @JakeT. Hope this helps you understand it better! Feel free to follow up if you have anymore questions. We’re here to help.